In this section, we consider that every object can be considered as a dot without volume.

The position

The motionis the history of the position of an object, the succession of the positions of this object over the time. To define the position of the object, we need a reference system in which we can give the position: coordinates. Several systems of reference exist and there are several ways to calculate the coordinates of objects. All of them are corrects but some are more convenient than the others. During a trip, you won’t give your position with regard to the sun, the same is true in physics.

Several coordinates are generally necessary to determine the exact position of one object. If you say that you are 10km from Paris, you give an information on your position but we are lacking at least one coordinate to determine your position. Usually you need one coordinate by dimension of the system.

One dimension

We choose one origin to the coordinate system, the zero point. It is convenient to choose the initial position of the object A as the origin but it is not mandatory. Next we choose a direction that will be the positive positions. In the opposite direction we have the negative positions. Still for convenience, the positive positions are placed in the direction we guess the object will move towards. Imagine that the object moves towards another object B placed at a distance d. The position vector indicates the distance between an object and the origin, and points towards the object with an arrow. The symbol for vectors is topped by an arrow pointing to the right. The position vector for B is

Where  is the unit vector in the direction x (the single direction in this problem).

Two dimensions

The second coordinate is usually orthogonal, perpendicular to the first coordinate to avoid a maximum of angle problems and to benefit the simplicity of the calculation for right triangles. Each coordinate has a direction.

The position vector is now defined by two components from which we can calculate its length if desired.

The addition sign between the two unit vectors is seen as “followed by” and not like the addition of two usual numbers. Another way to write coordinates is to put them in brackets. If we do this, then we don’t write the unit vectors.

This system of reference, the Cartesian system, is not the single one that can be used at two dimensions to determine the position of an object. We can also position the object from its distance r to the origin point and an angle θ from one axis. This reference system is called the polar system.

It is possible to determine the relation between the coordinates X;Y in the Cartesian system and r;θ in the polar system using the relation defining the cosine, the sinus and the tangent:

The unity vector  can thus be calculated.

or

We can also define the unit vector :

Both polar unity vectors depend thus upon θ that should thus be chosen conscientiously.

Three dimensions

A third coordinate is added in the reference systems. In the Cartesian system, we add a coordinate that is orthogonal to the two previous ones.

In the polar system, we need a second angle to determine the position of an object. We take the first from the axis x in the xy plane and the second angle is taken from the z axis in the zr plane of the object.

Again, the Cartesian system can be associated to the polar system.

The unit vectors are defined as

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On the other hand, we can measure a distance in meters or in miles, the weight in grams or in pounds. All these units have a well-defined value and can serve as reference units. While some countries as the UK or the USA use some different units, international conventions defined an international system of units (SI base units). Their biggest advantage is the simple relation between the units of different quantities. In SI, one millilitre of water occupies one cubic centimetre, weighs one gram, and requires one calorie of energy to heat up by one degree centigrade, which is one percent of the difference between its freezing point and its boiling point. In the American system, you will need to make huge calculations to calculate how much energy it takes to boil a room-temperature gallon of water because you can’t directly relate any of those quantities.

The International Bureau of Weights and Measures (French: Bureau international des poids et mesures) is an intergovernmental organization, established to maintain the International System of Units (SI) under the terms of the Metre Convention (Convention du Mètre, May 20^th 1875). The organisation is usually referred to by its French initialism, BIPM. Its role is to

• establish fundamental standards and scales for the measure of main physical quantities and to conserve the international prototypes;
• compare international standards with national standards;
• ensure the coordination of the corresponding techniques of measurement;
• measure and coordinate the measures of the fundamental, physical constants involved in the above activities.

SI Units

The definitions of the reference units are mainly made to give them a well-known and fixed value.

These definitions fix the speed of light c at 299792458 m/s et the permeability of the void μ₀ at 4π 10^-7 H/m exactly. They also sometimes require some precisions. For instance, we point out that the cesium atom is at rest, that the carbon atoms are not connected, are at rest and in their fundamental state.

Deriving units

By commodity, some units are the combination of the SI units to express frequently used units.

Finally, there are some quantities without specific unit names

We can point out a few specific units:

To put an end to this section, we will list the prefix of multiples of the SI units.

Dimensional analysis

The existence of the IS system means that all the other physical quantities have units that are homogeneous functions of these base units. A function is homogeneous if, making a scale change on all of its variables: x₁ → λ₁x₁, x₂ → λ₂x₂, x₃ → λ₃x₃, … the function itself changes of scale: f(λ₁x₁, λ₂x₂, λ₃x₃,…) = λ₁^α1 λ₂^α2 λ₃^α3…f(x₁, x₂, x₃,…). The units of any quantity, let’s call it Q_r is thus always expressed by a relation like

So if we know the units of one physical quantity, and admitting that there is a relation between this quantity and other variables, and knowing the units of those variables, we can guess the relation between the quantities.

For instance, we observe the swinging of one object attached to one string: a pendulum. The pendulum oscillates because it falls and it is restrained by the string. We want to determine the relation between the times the pendulum takes to make one oscillation, i.e. the period, to the parameters we guess as important: the mass M of the object, the length L of the string, and the gravity acceleration g that affects each object on the planet. The units of the variables are kg for the mass, m for the length, and m/s² for the acceleration. The period is a time and thus its unit is s. The relation between the variables and the period should be something like this:

As there is no kg at the left of the equality, and it is present at the right side of the equation with the exponent a, then we conclude that a=0. Looking at the seconds, their exponent is 1 at the left and -2b at the right, b is thus b=-1/2. Finally, there is no m at the left while it is present at the right side of the equation, thus 0=b+c. As we determined the value of b, we have that c=1/2 and that the global relation is

From our dimensional analysis, we determined that the mass of the object has no influence on the period of the oscillation of the pendulum. Note that we did not write the equal sign: the dimensional analysis doesn’t give the true law; it gives clues on the variables but there can still be numerical factors that can be determined experimentally. On the other hand, the dimensional analysis allows to identify wrong laws not respecting the units of the quantities.

Let’s analyse a second example: the period T of revolution of planets around the Sun. First we identify the important parameters involved in the problem: the mass M of the Sun, the distance R between the planet and the Sun, and a constant G giving the gravitational force. The units of the parameters are respectively kg, m and kg^-1m³s^-2. The period is given in seconds s. The law should be of the form T ~ M^aG^bR^c. For the units, we have the relation

The next step is to identify the exponent of each unit at the left and the right side of the equation:

The law is thus written

This relation is the expression of the Kepler’s law that describes the trajectory of planets of the solar system.

Scales and orders of magnitude

An experimental approach is to estimate the order of magnitude of variables that appear in physical processes. Either we measure the characteristics of one well known property of the matter, or we determine a transition zone between two models of description.

For instance, we can regroup the matter as solids, liquids and gases. One major difference between these three phases is their density, i.e. the mass of the matter for a given volume, given in kg/m³. We can thus regroup liquids as matter with a density with the order of magnitude around 10³kg/m³ (at T=293K, water: 1003kg/m³, olive oil: 910kg/m³, sulfuric acid: 1834kg/m³,…)while gases have a density of order 1 (at T=273K, air: 1.2kg/m³, CO₂: 1.98kg/m³, methane: 0.72kg/m³,…). Between solids and liquids, there is a factor 10 in density (at T=293K: iron: 7893kg/m³, copper: 8954kg/m³, gold: 19320kg/m³,…). We will want to determine the temperature at which a solid becomes liquid, i.e. its melting temperature.

As the interactions between particles of a solid differ from the interactions between particles in a gas, laws are not the same at the microscopic scale than in the astronomic scale, not because the interactions mysteriously disappear, but because we can neglect some interactions. For instance, imagine an interaction between particles that depends directly on the distance between two particles and one interaction that depends on the third power of the distance. If the distance is small, both interactions will have an effect, but as soon as the distance gets large, we can neglect the first interaction.

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“One of the noblest desire of the man is to know the laws ruling the Universe, and those who contributed to enlighten some of the mysteries were always admired by their peers; they appear as privileged, wearing on them the divine light, a through centuries the generations gaze upon their indelible work and rank them first amongst the glories of the humanity.”

From Achille Cazin (Hachette, 1881).

Physics can be defined as a science that studies the general properties of matter, space, and time, and establishes laws that describe natural phenomena. We will point out the word law first, and ask ourselves if we can clearly, and without ambiguity, define all the objects we are discussing, i.e. space, time and matter. Their definitions are (Oxford dictionaries)

• Space: A continuous area or expanse which is free, available, or unoccupied.
• Matter: Physical substance in general, as distinct from mind and spirit; (in physics) that which occupies space and possesses rest mass, especially as distinct from energy.
• Time: The indefinite continued progress of existence and events in the past, present, and future regarded as a whole.

It is not what we can call clearly defined terms and it doesn’t indicate any of the relations that may exist between them. If we go back in time, Isaac Newton gave its definitions (1687):

• The absolute space, which is without any relation with anything from the surroundings, is always unchanging and immobile. The relative space is any measure or mobile dimension of this space, which is defined with regards to its position with regards to objects that we consider as the immobile space…
• The quantity of matter is the measure that we obtain from its density and its volume…
• The absolute time, true and mathematic, is without relation to anything from the surroundings, and by its nature flows uniformly. The relative time, is any measure, accurate or not, of the duration of an event, that we use in place of the true time. I.e. the hour, day, month,…

It is difficult to avoid cross references and circular definitions. We will abandon the idea to define everything and accept the fact that some notions can make sense without being explicit. Physics is seeking for the laws that rule the reciprocal actions between one object and its surrounding. Enouncing those laws is not an easy job, and we have seen many changes in the history, as the understanding of the scientists evolved. For instance, L. Wouters gives in a school book of 1916 the law for the dilatation of bodies due to heat as

“The first effect that heat produces on bodies is to increase their volume, to dilate them”

This law is based upon its hypothesis on the nature of heat:

“Based on the modern hypothesis on the nature of heat, it results from the vibratory movement of the smallest molecules of the ponderable matter, and is transmitted via a fluid called aether. Aether is a subtle, perfectly elastic, substance that fills the intermolecular spaces as well as the so-called interplanetary voids. Heat is, at the end, a particular state of movement.”

Today, it is obvious that this law is false, mostly because of the notion of aether it is based on. The law itself is true with some exceptions. So, shall we give no definition nor law? Obviously we will, but laws should results from the simple relations obtained from experimentations: if I modify one physical quantity of my experiment, then another physical quantity changes, at this effect is repeated consistently if I repeat the experiment.

To end this introduction, I will extend the citation of Achille Cazin (Hachette, 1881) I opened the introduction with.

“No matter the study on which we work, there are some general rules that one must follow to avoid falling in annoying confusions. …

In physics, we observe all the circumstances around a natural phenomenon; we measure all the available quantities; we seek relations between theses quantities, and these relations are called the law of the phenomenon. When the phenomenon is too complex, and it looks like it is impossible to state one unique law, we modify the phenomenon, we make an experiment. Some circumstances seeming ancillary, are made negligible, and we observe the dominating quantities. From this experiment we obtain an approximate law, and by extension we seek the influence of the neglected circumstances, in which way they alter the law. Doing that, we find the limit law towards which tends the observed law when the circumstances become more and more negligible. Such a law is then used as a fundamental principle, seen as the temporary expression of a physical truth, temporary because one more accurate observation or a new phenomenon can modify the conclusions admitted as truth until now.”

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The mechanism of cell division varies in important details in different organism. This is particularly true of chromosomal separation mechanism, which differ substantially in protists and fungi from the process in plants and animals that we will describe here. Meiosis in a diploid organism consists of two rounds of division, mitosis of one. Although meiosis and mitosis have much in common, meiosis has three unique features: synapsis, homologous recombination, and reduction division.

Synapsis :

The first unique feature of meiosis happens early during the first nuclear division. Following chromosome replication, homologous chromosomes, or homologous pair all along their length. The process of forming these complexes of homologous chromosomes is called synapsis.

Homologous Recombination :

The second unique feature of meiosis is that genetic exchange occurs between the homologous chromosomes while they are thus physically joined. The exchange process that occurs between paired chromosomes is called crossing over. Chromosomes are then drawn together along the equatorial plane of the dividing cell; subsequently, homologues are pulled by microtubules toward opposite poles of the cell. When this process is complete, the cluster of chromosomes at each pole contains one of the two homologues of each chromosome. Each pole is haploid, containing half the number of chromosomes present in the original diploid cell. Sister chromatids do not separate from each other in the first nuclear division, so each homologue is still composed of two chromatids.

Reduction Division :

The third unique feature of meiosis is that the chromosomes do not replicate between the two nuclear divisions, so that at the end of meiosis, each cell contains only half the original complement of chromosomes. In most respects, the second meiotic division is identical to a normal mitotic division. However, because of the crossing over that occurred during the first division, the sister chromatids in meiosis 2 are not identical to each other.

Meiosis is a continuous process, but it is most easily studied when we divide it into arbitrary stages. The stages of meiosis are traditionally called meiosis 1 and meiosis 2. Like mitosis, catch stage is subdivided further into prophase, metaphase, anaphase and telophase. In meiosis, however, prophase 2 is more complex than in mitosis. In meiosis, homologous chromosomes become intimately associated and do not replicate between the two nuclear divisions.

The Red Queen Hypothesis. One evolutionary advantage of sex may be that it allow populations to “store” recessive alleles that are currently bad but have promise for reuse at some time in the future. Because populations are constrained by a changing physical and biological environment, selection is constantly acting against such alleles, but in sexual species can never get rid of those sheltered in heterozygotes. The evolution of most sexual species, most of the time, thus manages to keep pace with ever-changing physical and biological constraints. This “treadmill evolution” is sometimes called the “Red Queen hypothesis”, after the Queen of Hearts in Lewis Carroll’s Through the Looking Glass, who tells Alice, “Now, here, you see, it takes all the running you can do, to keep in the same place.”

Miller’s Ratchet. The geneticist Herman Miller pointed out in 1965 that asexual populations incorporate a kind of mutational ratchet mechanism – once harmful mutations arise, asexual populations have no way of eliminating them, and they accumulate over time, like turning a ratchet. Sexual populations, on the other hand, can employ recombination to generate individuals carrying fewer mutations, which selection can then favor. Sex may just be a way to keep the mutational load down.

The Evolutionary Consequences of Sex

While our knowledge of how sex evolved is sketchy, it is abundantly clear that sexual reproduction has an enormous impact on how species evolve today, because of its ability to rapidly generate new genetic combinations. Independent assortment, crossing over, and random fertilization each help generate genetic diversity.

Whatever the forces that led to sexual reproduction, its evolutionary consequences have been profound. No genetic process generates diversity more quickly; and, as you will see in later chapters, genetic diversity is the raw material of evolution, the fuel that drives it and determines its potential directions. In many cases, the pace of evolution appears to increase as the level of genetic diversity increases. Programs for selecting larger stature in domesticated animals such as cattle and sheep, for example, proceed rapidly at first, but then slow as the existing genetic combinations are exhausted; further progress must then await the generation of new gene combinations. Racehorse breeding provides a graphic example: thoroughbred racehorses are all descendants of a small initial number of individuals, and selection for speed has accomplished all it can with this limited amount of genetic variability – the winning times in major races ceased to improve decades ago.

Paradoxically, the evolutionary process is thus both revolutionary process is thus both revolutionary and conservative. It is revolutionary in that the pace of evolutionary change is quickened by genetic recombination, much of which results from sexual reproduction. It is conservative in that evolutionary change is not always favored by selection, which may instead preserve existing combinations of genes. These conservative pressures appear to be greatest in some asexually reproducing organism that do not move around freely and that live in especially demanding habitats. In vertebrates, on the other hand, the evolutionary premium appears to have been on versatility, and sexual reproduction is the predominant mode of reproduction by an overwhelming margin.

The close association between homologous chromosomes that occurs during meiosis may have evolved as mechanism to repair chromosomal damage, although several alternative mechanism have also been proposed.

The evolutionary origin of sex is a puzzle.

Why Sex?

Not all reproduction is sexual. In asexual reproduction, an individual inherits all of its chromosomes from a single parent and is, therefore, genetically identical to its parent. Bacterial cells reproduce asexually, undergoing binary fission to produce two daughter cells containing the same genetic information. Most protists reproduce asexually except under conditions of stress; then they switch to sexual reproduction. Among plants, asexual reproduction is common, and many other multicellular organism are also capable of reproducing asexually. In animals, asexual reproduction often involves the budding off of a localized mass of cell, which grows by mitosis to form a new individual.

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Prophase 1

In prophase 1 of meiosis, the DNA coils tighter, and individual chromosomes first become visible under the light microscope as a matrix of fine threads. Because the DNA has already replicated before the onset of meiosis, each of these threads actually consist of two sister chromatids joined at their centromeres. In prophase 1, homologous chromosomes become closely associated in synapsis, exchange segments by crossing over, and then separate.

Prophase 1 is traditionally divided into five sequential stages: leptotene, zygotene, pachytene, diplotene, and diakinesis.

Leptotene: during which each chromosome becomes visible as two fine threads  (chromatids)

Zygotene: A lattice of protein is laid down between the homologous chromosomes in the process of synapsis, forming a structure called a synaptonemal complex.

Pachytene: Pachytene begins when synapsis is complete (just after the synaptonemal complex forms and lasts for days. This complex, about 100 nm across, holds the two replicated chromosomes in precise register, keeping each gene directly across from its partner on the homologous chromosome. Within the synaptonemal complex, the DNA duplex unwind at certain sites, and single strands of DNA from base-pairs with complementary strands on the other homologue. The synaptonemal complex thus provides the structural framework that enables crossing over between the homologues chromosomes. As you will see, this has a key impact on how the homologues separate later in meiosis.

Diplotene: the fourth stage of the prophase I of meiosis, following pachytene, during which the paired chromosomes begin to seperate (the synaptonemal complex disassembles) into two pairs of chromatids.Diplotene is a period of intense cell growth. During this period the chromosomes decondense and become very active in transcription

Diakinesis: At the beginning of diakinesis, the transition into metaphase, transcription ceases and the chromosomes recondense.

Synapsis: During prophase, the ends of the chromatids attach to the nuclear envelope at specific sites. The sites the homologous attach to are adjacent, so that the members of each homologous pair of chromosomes are brought close together. They then line up side by side, apparently guided by heterochromatin sequences, in the process called synapsis.

Crossing Over

Within the synaptonemal complex, recombination is thought to be carried out during pachytene by very large protein assemblies called recombination nodules. A nodule’s diameter is about 90 nm, spanning the central element of the synaptonemal complex. The details of the crossing over process are not well understood, but involve a complex series of events in which DNA segments are exchanged between nonsister or sister chromatids. In humans, an average of two or three such crossover events occur per chromosome pair.

When crossing over is complete, the synaptonemal complex breaks down, and the homologous chromosomes are released from the nuclear envelope and begin to move away from each other. At this point, there are four chromatids for each other. At this point, there are four chromatids for each type of chromosome (two homologous chromosomes, each of which consists of two sister chromatids). The four chromatids do not separate completely, however, because they are held together in two ways: (1) the two sister chromatids of each homologue, recently created by DNA replication, are held near by their common centromeres; and (2) the paired homologues are held together at the points where crossing over occurred within the synaptonemal complex.

Chiasma Formation

Evidence of crossing over can often be seen under the light microscope as an X-shaped structure known as a chiasma. The presence of a chiasma indicates that two chromatids (one from cach homologue) have exchanged parts. Like small rings moving down two strands of rope, the chiasmata move to the end of the chromosome arm as the homologous chromosomes separate.

Synapsis is the close pairing of homologous chromosomes that takes place early in prophase 1 of meiosis. Crossing over occurs between the paired DNA strands, creating the chromosomal configurations known as chiasmata. The two homologues are locked together by these exchanges and they do not disengage readily.

Metaphase 1

By metaphase 1, the second stage of meiosis 1, the nuclear envelope has dispersed and the microtubules form a spindle, just as in mitosis. During diakinesis of prophase 1, the chiasmata move down the paired chromosomes from their original points of crossing over, eventually reaching the ends of the chromosomes. At this point, they are called terminal chiasmata. Terminal chiasmata hold the homologous chromosomes together in metaphase 1, so that only one side of each centromere faces outward from the complex; the other side is turned inward toward the other homologue. Consequently, spindle microtubules are able to attach to kinetochore proteins only on the outside of each centromere, and the centromeres of the two homologues attach to microtubules originating from opposite poles. This one-sided attachment is in marked contrast to the attachment in mitosis, when kinetochores on both sides of a centromere bind to microtubules.

Each joined pair of homologues then lines up on the metaphase plate. The orientation of each pair on the spindle axis is random: either the maternal or the paternal homologue may orient toward a given pole.

Chiasmata play an important role in aligning the chromosomes on the metaphase plate.

Completing Meiosis

After the long duration of prophase and metaphase, which together make up 90% or more of the time meiosis 1 takes, meiosis 1 rapidly concludes. Anaphase 1 and telophase 1 proceed quickly, followed – without an intervening period of DNA synthesis – by the second meiotic division.

Anaphase 1

In anaphase 1, the microtubules of the spindle fibers begin to shorten. As they shorten, they break the chiasmata and pull the centromeres toward the poles, dragging the chromosomes along with them. Because the microtubules are attached to kinetochores on only one side of each centromere, the individual centromeres are not pulled apart to form two daughter centromeres, as they are in mitosis. Instead, the entire centromere moves to one pole, taking both sister chromatids with it. When the spindle fibers have fully contracted, each pole has a complete haploid set of chromosomes consisting of one member of each homologous pair. Because of the random orientation of homologous chromosomes on the metaphase plate, a pole may receive either the maternal or the paternal homologue from each chromosome pair. As a result, the genes on different chromosomes assort independently; that is, meiosis 1 result in the independent assortment of maternal and paternal chromosomes into the gametes.

Telophase 1

By the beginning of telophase 1, the chromosomes have segregated into two clusters, one at each pole of the cell. Now the nuclear membrane re-forms around each daughter nucleus. Because each chromosome within a daughter nucleus replicated before meiosis 1 began, each now contains two sister chromatids attached by a common centromere. Importantly, the sister chromatids are no longer identical, because of the crossing over that occurred in prophase 1. Cytokinesis may or may not occur after telophase 1. The second meiotic division, meiosis 2, occurs after an interval of variable length.

The Second Meiotic Division

After a typically brief interphase, in which no DNA synthesis occurs, the second meiotic division begins.

Meiosis 2 resembles a normal mitotic division. Prophase 2, metaphase 2, anaphase 2, and telophase 2 follow in quick succession.

Prophase 2. At the two poles of the cell the clusters of chromosomes enter a brief prophase 2, each nuclear envelope breaking down as a new spindle forms.

Metaphase 2. In metaphase 2, spindle fibers bind to both sides of the centromeres.

Anaphase 2. The spindle fibers contract, splitting the centromeres and moving the sister chromatids to opposite poles.

Telophase 2. Finally, the nuclear envelope re-forms around the four sets of daughter chromosomes.

The final result of this division is four cells containing haploid sets of chromosomes. No two are alike, because of the crossing over in prophase 1. Nuclear envelopes then form around each haploid set of chromosomes. The cells that contain these haploid nuclei may develop directly into gametes, as they do in animals. Alternatively, they may themselves divide mitotically, as they do in plants, fungi, and many protists, eventually producing greater numbers of gametes or, as in the case of some plants and insects, adult individuals of varying ploidy.

During meiosis 1, homologous chromosomes move toward opposite poles in anaphase 1, and individual chromosomes cluster at the two poles in telophase 1. At the end of meiosis 2, each of the four haploid calls contains one copy of every chromosome in the set, rather than two. Because of crossing over, no two cells are the same. These haploid cells may develop directly into gametes, as in animals, or they may divide by mitosis, as in plants, fungi, and many protists.

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The states are separated by full lines, the curves of transition of phase.

The curve between the liquid and the solid is called the curve of fusion.

The curve between the liquid and the gas is called the curve of vaporisation/coexistence.

The curve between the solid and the gas is called the curve of sublimation.

Curve of vaporisation

In a closed system containing a liquid and a gas, there is an equilibrium between the two phases. The gas becomes liquid at a given speed v_GL and the liquid becomes gas at a given speed v_LG. The equilibrium is reached when those speeds are equal.

The liquid becomes a gas if the molecules have a kinetic energy large enough. If this energy is not large enough, the molecules remain liquid because the interactions between the molecules make it more favourable. The kinetic energy can be provided by heating or excitation. Typically, we have to heat water to make it boil.

The other reaction, from a gas to a liquid is favourable but is not complete because of the entropy. The entropy is an expression of the disorder in the system. The second law of thermodynamics (the thermodynamics will be explained in its own chapter) imposes that the universe evolves towards a greater disorder. It means that the sum of the variations of entropy inside and outside the system has to be positive.

There is thus a competition between the increase of entropy and the decrease of the energy in the system.

The free enthalpy of Gibbs represents/reflects this competition:

The Gibbs free enthalpy is thus a thermodynamic potential taking into account the implications of the system on its environment.

The vapour pressure in the system at the equilibrium p° (also called the pressure of Clausius-Clapeyron) is

The entropic terms are present in the pre-exponential constant and the enthalpy is in the exponential term. As the temperature increases, the vapour pressure increases exponentially. Experimentally, we can find the enthalpy of vaporisation by plotting the curve ln p° vs 1/T.

The slope of the curve is –ΔH/R. The value of ΔH^vap depends on the interactions between the molecules. Strong interactions mean that the molecules are not easily getting away from each other and that the vaporisation requires more energy, i.e. ΔH^vap is large and p° is small. It is logical: if the molecules don’t easily vaporise, there are not a lot of molecules in the gas and the pressure of the vapour is thus low.

In the cyclohexane, there are barely no interactions between the molecules and its ΔH^vap is small. The cyclohexane is a volatile liquid.

The boiling temperature of a liquid is the temperature at which the vapour pressure of a liquid is equal to the atmospheric pressure. For the water, this temperature is 373K (100°C) if the atmospheric pressure is 1 atm. In altitude, the atmospheric pressure decreases and so does the boiling temperature. For example, the boiling temperature is 363K (90°C) at an altitude of 3000m. In a pressure cooker, the pressure is greater than 1 atm and the water boils at a temperature larger than 100°C.

Curve of sublimation

The sublimation is the transition of phase between a solid and a gas

We can define a vapour pressure for solids too, but it is much lower than for a liquid because interactions in a solid are stronger than in a liquid. The equation is similar

But the constant is different and ΔH^sub>>ΔH^vap, for the exact same reasons.

Curve of fusion

The fusion is the transition of state from a solid to a liquid.

We can also define a pressure such as

The enthalpy of fusion is very small and the curve is almost a right vertical line. In general, the slope is positive, meaning that when a liquid becomes solid (by decreasing its temperature for example), its volume decreases. It is however not always true.

A particular case is the water: the volume of 1kg of water is smaller than the volume of 1kg of ice. It is a tip of the barmen to decrease the volume of drink that they give to their clients but it is also what allow us to ice-skate:

When we skate, all our weight is applied on a very small surface. The ice at this place instantly becomes liquid because of this pressure. If the ice did not change into liquid, the friction between our skates and the ice would be much larger, as it is in general the case between two solids. The small puddle of water that is formed lubricates the ice and allow us to slide with almost no friction. Because the volume of water was smaller than the volume of ice, the water remains in the small hole that our skates dug in the ice. After our passage, the water turns back into ice at the same place where we skated. If the volume was the water was larger than the one of the ice, the puddle would extend near the skates and deform the surface of the ice. If it was the case, in a skating rink, the surface of the ice would not remain flat for a very long time.

The carbon dioxide in an extinguisher is in a liquid state, near the triple point. When we use the extinguisher, the pressure drops suddenly. We should thus expect a gas outside of the tube. However, there is a detente of the gas during the process (moving from a narrow space to a large space) that involves a decrease of the temperature, making the gas become a solid.

Variance

It is the number of degrees of freedom that characterise a state

Where c is the number of constituents of the matter and φ is the number of phases. Let’s take a look at different points on the phase diagram

• in the middle of a phase: V=1-1+2=2: we can modify the pressure or the temperature without transition of state.
• on a curve: V=1-2+2=1: if we want to modify the pressure or the temperature of the matter without transition of state, the other parameter (T or p) has to be fixed.
• on the triple point: V=1-3+2=0: if we want to keep the matter in this state, we can’t change the conditions of temperature and of pressure.

At the triple point, the enthalpy of sublimation equals the sum of the enthalpies of vaporisation and of fusion. The triple point for the water is at t=273,16K and p=0.006atm.

Another remarkable point is the critical point:

After this point, the liquid and gaseous phases become indistinguishable. In water, the critical point occurs at around 647 K and 218 atm.

In the vicinity of the critical point, the physical properties of the liquid and the vapour change dramatically, with both phases becoming ever more similar. For instance, liquid water under normal conditions is nearly incompressible, has a low thermal expansion coefficient, has a high dielectric constant, and is an excellent solvent for electrolytes. Near the critical point, water becomes compressible, expandable, a poor dielectric, a bad solvent for electrolytes, and prefers to mix with nonpolar gases and organic molecules.

At the critical point, only one phase exists. The heat of vaporization is zero.

Above the critical point (T>T_c and p>p_c) there is a state of matter that is continuously connected, without transition of phase, with both the liquid and the gaseous state: supercritical fluid. Such a fluid has usually properties between those of a gas and a liquid.

Case of mixtures

In a solution, the elements of the mixture don’t necessarily have the same boiling point. The element of lower boiling point will evaporate faster than the other components of the mixture. If we consider a binary mixture, we can plot the composition of the melange in function of the temperature on a diagram shown underneath

On this diagram, the temperature is reported on the ordinate and the mole fraction χ of the second component of the melange is set as abscise. At χ=0, the solution is a pure solution of the solute 1, and the boiling temperature of the solution is T_boil,1. At χ=1, the solution is a pure solution of the solute 2, and the boiling temperature of the solution is T_boil,2.  In between, the solution is a mixture of the two components. In this case, we considered a solution in which the solute 1 has a lower boiling temperature than the solute 2. If we increase the proportion of the solute 2, the boiling temperature of the mixture increases. This increase is not linear and one can see that there is an area in which both vapour and liquid coexist. If you look at a given temperature T’, the composition of the liquid and of the vapour is different: the composition of the vapour is composed by a larger fraction of the most volatile solute than the liquid.

The molar fractions of the melange can be determined with the law of Raoul. The total pressure of an ideal mix is the sum of the partial pressures of the gases

The partials pressures are proportional to the mole fraction of the compound in the solution χ^L_A multiplied by the vapour pressure of the pure compound p°_A:

The total pressure is thus

Given that the sum of the mole fractions in solution is equal to 1, i.e. χ^L_A+ χ^L_A=1, we find the mole fraction of one species from the pressures

Moreover, this allows us to connect the vapour and liquid mole fractions together by

Distillation

The distillation is a separation process based on the difference of boiling temperature of different solutes in a solution. Basically, if the initial solution has a mole fraction χ_L, we can boil it at the temperature T’. The goal is to collect the vapour and to liquefy it. The collected solution has a composition χ_V. We can keep doing this multiple times to obtain a virtually pure solution of the solute 1.

In practical, multiple distillations are done at once. The setup is composed by a balloon containing the initial solution, a heating system (as an infra-red lamp), a column of distillation, a cooling tube and a balloon to collect the distillate. Thermometers can be placed in the first balloon and at the top of the distillation column to control the temperature of the solution and of the vapour.

The cooling device is composed of two tubes, one thinner placed inside the other one. The inner tube has openings at both extremities and is connected to the column and the collecting balloon. The other tube has lateral openings connected to a counter flow of cool water. The flow of cool water goes this way to avoid an unfilled area and also because the water near the column, that has been heated by the vapour, is quickly evacuated. If the water was going in the opposite direction, the water gets heated by the vapour and the cooling power of the tube would be decreased.

The distillation column if filled with small pieces/tubes of glass to increase to surface of contact with the vapour. In some columns, needles are part of the column and no additional pieces have to be added.

In the distillation column, hot vapour goes up and meets the pieces of glass of lower temperature. At their contact, there is an exchange of heat and the vapour may turn back into liquid, not with the composition of the liquid in the balloon χ₀ (to keep the notation of the figure above), but the composition of the vapour χ₁. Droplets will eventually fall back into the balloon or evaporate on their way because of the hot vapour. Their composition will now be χ₂, as if two distillations had been performed. The vapour continues its way up to the top of the column, getting more pure, and enters the cooling tube where it turns into liquid, finishing its path inside the collecting balloon.

It is good to put some pumice stones in the balloon to avoid overheating and to obtain bubbles of a controled size.

Azeotropes

The diagram of boiling is not always that simple. Some mixtures are said to be azeotrope. An azeotrope mixture acts like a one-component solution, i.e. the mole fraction does not change during the boiling. On the diagram, the curves look like

If the initial solution has the azeotrope composition, the liquid and vapour compositions are identical and do not change over time. If the initial solution has a smaller mole fraction, we will eventually obtain a pure solution of the solute 1 but it is impossible to obtain a pure solution of the solute 2: the composition tends towards the azeotrope. The same is true if the initial composition of the mixture has a larger mole fraction than χ_az: it will be impossible to obtain a pure solution of the solute 1 in this case.

Two types of azeotropes can be sorted: the positive ones and the negative ones. The differences is simply their boiling temperature. On the diagram above, a positive azeotrope is represented: its boiling temperature is larger than the ones of the pure components of the mixture. A negative azeotrope has a lower boiling temperature.

Demixing

In mixtures, different interactions exist: the interactions between two identical molecules and between the different molecules of the mixture. If

then the mixture tends to separate. The expression above means that the interactions between identical molecules are more favourable than the interactions between different molecules. Imagine that in that case the molecules A and B are perfectly mixed together. The interactions between A and B are not necessarily unfavourable but with the Brownian motion, the molecules will tend to regroup between species and they will form clusters/bubbles to have the maximum amount of favourable interactions. That is how emulsions form. The clusters will merge together over time to improve the ratio volume/surface of the clusters and the solution will eventually end as two separated phases. It is the phenomenon of demixing.

For example, oil and water do not mix well. They form two separate phases. Yet, we can dilute a bit of water in oil or vice versa but if we continue to add water, the phases will separate. It is shown on the following figure.

If the oil is the component 1 and water the component 2, this figure shows that we can add water to oil up to the mole fraction χ₁ before a separation of phases. We can add oil into water without demixing for mole fractions larger than χ₂. As we can see, the demixing process depends on the temperature. The quantity of solute that we can dissolve before the demixing depends on the local temperature. Above a critical temperature (not the same as in the phase diagram), the mixture will remain as one phase no matter the composition.

Colligative properties

The colligative properties are properties that depend on the number of the particles in the solvent, of the solvent but not of the solute species. The fact that the boiling temperature of water increases when we add a bit of salt inside is a colligative property. It does not matter that it is salt or something else. The temperature of freezing decreases as well: we use salt on snow to make it melt. In the phase diagram, the coexistence curve has been translated down because of the addition of the salt. The ebullioscopic law says that

With k_b the ebullioscopic constant and m_s the molality of the solute. When we add salt into water, we add two species: the ions Cl^– and Na⁺. It is thus better to write this relation as

ν is called the van’t Hoff factor and is the number of ions found by unit formula. In fact, we observe that this factor is often smaller than its expected value: some of the molecules don’t dissolve in the solution.

That means that for 100 molecules of NaCl in solution, 90 will dissolve to produce the ions Cl^– and Na⁺ and 10 molecules don’t dissolve (making a total of 190 molecules instead of 200). ν can be determined from the constant of dissociation of the molecule. For example, all the molecules of a weak acid don’t dissociate in water.

The K_a of this reaction is

Let’s take a look at the concentration of the species before and after the reaction. In fact, instead of the concentrations, we will consider the concentrations/initial concentration of CH₃COOH.

The total number of molecules is

So, we know that there is a decrease of temperature due to the colligative properties. The variation of temperature is coupled to a variation of the vapour pressure of the mixture. Intuitively, the vapour pressure of the dissolved salt is in general very small with regards to the vapour pressure of the water. Some salts are not volatile at all. The pressure is equal to the sum of the partial pressures

The addition of a salt B into the solution A has thus lowered the pressure.

The mathematical demonstration follows:

Considering the dissolution of NaCl in water, the variation of pressure is

F_salt is a coefficient of activity of the salt, present for reasons similar to that of the van’t Hoff factor ν.

We can still simplify the expression:

On the phase diagram, the coexistence line is translated downwards by Δp. The ΔT can also be calculated:

The derivative gives

Math tells us that it also gives

So

dp°/dT is in fact equivalent to Δp/ΔT from the phase diagram. We obtain that

The difference of temperature is thus proportional to the square of the temperature and directly proportional to the amount of species in the solvent.

Similarly, it is possible to detect if solids are pure. If a solid melts at a temperature lower than its normal melting temperature, it shows that there are impurities in the solid.

The osmotic pressure is also a colligative property: in cells, the concentration of ions must be identical at each side of the cell membrane. Otherwise the cell expends or shrinks. The osmotic pressure of a solution is the difference in pressure between a solution containing solute particles and the pure liquid solvent when the two are in equilibrium across a semipermeable membrane, i.e. a membrane which allows the passage of solvent molecules but not of solute particles. If the two phases are at the same initial pressure, there is a net transfer of solvent across the membrane into the solution known as osmosis. The process stops and equilibrium is attained when the pressure difference equals the osmotic pressure.

Solubility

The solubility is the ability of a substrate to dissolve in a liquid. We are talking of solubility for solids, salts for instance, and for gases.

Solubility of gases

Some unnaturally sparkling mineral water sold in the market are enriched in carbonic gas. A given quantity of gas can be dissolved in a liquid by the direct injection of the gas into the liquid with a given flow. This quantity is not infinite and depends on the conditions of pressure and of temperature.  If a large pressure is applied, more gas can dissolve in the liquid. The quantity is proportional to the vapour pressure p° of the gas.

f_A is a function that depends on the molecule. The molar fraction of A in the solution is thus proportional to the pressure of the gas multiplied by a constant k_A=p°_A f_A. We can rewrite this in function of the concentration in the solution instead of the molar fraction as

A way to dissolve a gas inside a liquid is thus to increase the pressure of the gas.

Acidification of the oceans

We often hear about the greenhouse effect and about the massive impact that the modification of the concentrations of the greenhouse gases may have on our lives.  An important actor of the greenhouse effect is the CO₂. Animals exhale CO₂ and ours industries and locomotion middles (cars, trains, planes, etc) also produce a lot of this gas. Plants are consuming the CO₂ to produce O₂ that we breathe. It is a fragile equilibrium that we are menacing with the deforestation, but it is not our point here. The point is that oceans represent a colossal volume of water in which CO₂ may be absorbed (each day, it absorbs 4kg of CO₂/man, 25 billion kg CO₂/year). In the oceans, algae’s are playing the same role than the plants on the ground and their effect is probably as important with regard to the quantities of CO₂ absorbed. A global increase of the concentration of CO₂ would increase the quantity of CO₂ in the oceans, forming carbonic acid (H₂CO₃) that acidifies the water, slowly menacing the lives of oceanic species, including algae’s, planktons, corals, etc. Before the industrial revolution, the pH of oceans was 8.2. The actual pH is 8.05 and is expected to decrease to 7.8 at the end of this century.

Another way is to change the temperature.  At the beginning of this chapter, we discussed about the free enthalpy of Gibbs G=H-TS. The temperature is thus coupled with the entropy. The ΔG associated with the transition of phase is thus directly dependant on the temperature multiplied by the variation of entropy due to the transition of phase. The dissolution of a gas in a liquid decreases the entropy of the system: the disorder is greater in a gas where there is a lot of space to move and barely no interactions between the molecules than in a liquid where the space is limited and interactions may be strong. On the other hand, the dissolution is exothermic: new liaisons are generated. As a result, there is a competition between the entropy and the enthalpy of the dissolution. The enthalpy can be expressed by ΔH=-RTlnχ^L₂, from the van’t Hoff equation ln C_sat = -DH/(RT) + Const.

We can visualise the dissolution of a gas in a liquid with this diagram:

The energy is put as ordinate and the mole fraction of the gas in the liquid is put as abscise. A horizontal dashed line is traced at E=ΔH° and two dotted lines at E=TΔS, T₁<T₂. We will call those lines the enthalpic and the entropic lines. Those lines are in the negative values of the energy as explained before. Next, we trace the curve ΔG(χ^L₂)=TΔS- RTln χ^L₂. This curve has been plotted for two temperatures. The curves terminate for χ^L₂=1 at E=TΔS, i.e. it crosses the entropic line. On its path, the curve crosses the enthalpic line. At the crossing point, we find the maximum amount of gas χ₁ that the liquid can dissolve. For larger temperatures (red lines), the entropic line moves deeper in the negative values of E. The curve of ΔG translates the same way, still ending at E=TΔS. ΔH doesn’t change. The crossing point between ΔG and ΔH indicates now a smaller mole fraction of gas χ₂ into the liquid. An increase of temperature involves thus a decrease of the solubility of the gas. As a reminder from the phase diagram and the curve of coexistence, the heating of a liquid increases the kinetic energy of the molecules and molecules with a kinetic energy large enough become gaseous.

Solubility of solids

Salts dissolve into ions in polar liquids such as water.

In opposition to the solubility of gases, when solids dissolve in water, the entropy increases slightly: first, solutions are ordered (short range order) but solids are more ordered than them. This variation of entropy is small with regards to the variation of entropy of the dissolution of gases. Second, there are more molecules as one molecule of salt gives several ions. Also, the dissolution is endothermic: liaisons are broken to form the ions. Both ΔH and TΔS are thus positive. We determine the quantity of salt that can be dissolved the same way as for the gases. Now that the enthalpic and the entropic lines are in the positive energies, if the temperature increases, the solubility increases.

Some salts have a negative enthalpy. In this case, the enthalpic line is under the entropic lines whatever the temperature.

There is thus no crossing point between the curve of ΔG and ΔH. The salt is said to be infinitely soluble in this case.

Constant of solubility Ks

The dissolution of a solid inside a solution respects the rules of a chemical reaction: the stoichiometry is respected and the reaction has a constant of reaction defining the speed at which the process takes place.  Let’s define s as the maximum concentration of the salt that can be dissolved. If we put more solid in the solution, crystals will remain (in general) in the bottom of the solution.

In this situation, the concentration of the ions in solution can directly be related to s. For instance, the dissolution of AgCl into water leads to the following equation

With the constant of solubility K_s

As the solid is pure, its concentration is considered as equal to 1 in the equation. It is the case for any solid. When no more salt can be dissociated, the concentration of both ions Ag⁺ and Cl^– are equal to s.

s is thus equal to √K_s. For CaF₂, three ions are formed during the process.

The constant of solubility K_s is

In this second case, there are two anions F^– in solution for each dissolved CaF₂. Its concentration won’t be s but 2s.

As a general expression,

With m and n the charge of the ions, or their stoichiometric coefficients (it is equivalent).

The value of K_s gives us precious information on the solubility of a salt, and on how much solid we can put in solution without waste. Remember that K_s depends on the temperature and that it is sometimes necessary to heat a solution to dissolve a precipitate, to reach a given concentration of ions.

This relation is very similar to the expression (Clausius-Clapeyron) for the vapour pressure in the system at the equilibrium p° which was seen at the beginning of this chapter

It is thus again possible to plot the evolution of Ks or of s with regard to the temperature

The 2 under the fraction bar is the exponent of s in the relation between K_s and s.

Exercises

1.  What is the temperature of ebullition of water at 0.7atm? At 2 atm? (ΔH_w^vap=40kJ/mol)

2.  Determine the vapour pressure of nitrogen at T=-120°C knowing that its usual ebullition point is -196°C and that ΔH^vap=5.6kJ/mol.

3.

The following table shows some experimental measurements of the vapour pressure of the benzene as a function of the temperature.

Determine the enthalpy of vaporisation of the benzene. If a sample of benzene at T=50°C and p=150mm Hg is cooled down to 40°C in a closed recipient, will it condense? At 30°C? Estimate the temperature at which the gas stats to condensate.

4.  In one mole of ice, there are two moles of H bonds. Determine the energy of the H bonds in one mole of ice if you know that at the triple point the enthalpies of fusion and vaporisation are respectively 333.9J/g and 2493.7J/g. Can you estimate the amount of H bonds in the water?

5.  In the following systems, how many phases, independent components and variance?

1. water in equilibrium with its vapour at 1 atm
2. water in equilibrium with its vapour

6.  1l of naphthalene C₁₀H₈ in benzene C₆H₆ has 67g of solute in it. Give the mass fraction, molar fraction, molarity and the molality in naphthalene if the density of the solution is equal to 0.850.

7.  Which mass of decahydrated carbonate of sodium do you have to weight to prepare 100g of a solution of Na₂CO₃ 10%, knowing that the density of the solution will be 1.10. Determine the molarity, molality and molar fraction in Na₂CO₃.

8.  We mix together 30ml of n-heptane (ρ=0.684) and 20ml of n-octane (ρ =0.703). Their vapour tensions at 100°C are respectively 829 and 344torr. Find out the molar fraction of the solution and of the vapour if the temperature is 100°C, Is this an ideal solution? Draw the corresponding isotherm (T=100°C) diagram of phases of this melange.

9.  The temperature of ebullition of the benzene and of the toluene are respectively 80.1°C and 110.6°C. Their enthalpies of vaporisation are 33.9kJ/mol and 37.4kJ/mol. Determine the vapour pressure of those two compounds at 90°C and 100°C.

Determine the composition of the solution of benzene and toluene that starts to boil at those temperatures if the pressure is 1 atm.

What is the composition of the vapour at those temperatures?

10.  How much ethylene glycol (CH₂OH)₂ has to be dissolved in 2l of water to prevent it to freeze at -10°C considering a constant of freezing K_f=1.86K kg mol^-1 for the water)? Same question with CaCl₂ instead of the ethylene glycol, considering a total dissociation of the salt.

11.  The osmotic pressure of 100ml of a solution containing 0.947g of a protein corresponds to a column of water of 12mm at 25°C. What is the molar mass of the protein?

Answers

1.

2.

3.  If we plot ln p⁰ as a function of 1/T, the slope α equals –ΔH^vap/R, so ΔH^vap=- αR

The sample is not at the equilibrium but we have the values of the vapour pressure at the equilibrium. At 40°C, the vapour pressure is larger than the pressure in the flask, so the gas doesn’t condense. At 30°C however, the vapour pressure from the table is smaller than the pressure in the flask and the gas condense under the effect of the cold.

To estimate the temperature at which the gas starts to condensate, we chose one value of the table and apply the formula

to determine the temperature at which the condensation begins:

4.  At the triple point, the chemical potential are equals. It means that

This enthalpy is the energy required to break the bonds between the molecules, i.e. the hydrogen bonds. As there are two H bonds per mol, the energy of one mol of H bond is 25,57kJ.

We can estimate the amount of H bonds in the water from the enthalpy of fusion. During the fusion, a given amount of H bonds are broken to make the solid ice become liquid.

We know that 25.57kJ are required to break 1 mol of H bond, so we can estimate that 6.01/25.57=0.235 mol are broken during the fusion. There are thus 1.765 mol of H bond/mol of water.

5.  The expression for the variance is V=2+c-φ-n where c is the amount of constituents, φ is the amount of phases and n is the amount of fixed parameters.

1. V=2+1-2-1=0
2. V=2+1-2-0=1

6.  The density is the mass by unit of volume, so the litre of solution weights 850g. We have 67g of naphthalene and 783g of benzene. The mass fractions are thus

Knowing the masses, we can find the quantity of moles for both species

7.  As the mass of the solution is 100g, we want to have 10g of Na₂CO₃. To determine the mass of decahydrated carbonate, we first need to know the amount of moles of Na₂CO₃ in solution.

The volume of solution is 100g/1.1g ml^-1=90.90..90..ml

In the vapour, the molar fractions are

This melange is an ideal solution because the n-heptane and the n-octane are both nonpolar species without dipolar moment and unable to form H bonds. As a result, they don’t have any specific interactions together.

The information we have to draw the phases diagram are that the pure solutions have a vapour tension of 0.4526atm (χ₇=0) and 1.0908atm (χ₇=1). We also know that at p=0.8512atm, the molar fraction is 0.625 for the liquid and 0.8 for the gas.

8.  The partial pressures can be found with the equations of Clausus-Clapeyron

Either we find the constant in the first one with the values we know from the wording to be able to solve the equations at the given temperatures. Or we can use the second equation that doesn’t need to determine the value of the constant.

First method:

Second method:

We do the same for the tow compounds and the two temperatures to find

To determine the molar fractions at a given temperature, we use this expression

At 90°C, we have

We can plot all those points on the phase diagram.

9.

10.

The osmotic pressure is normally given as a pressure. To obtain it in atm, we transform the height of water in height of mercury and then in atm. The ratio between the density of mercury and water is 13.59

We can now determine the molar mass of the protein

The post Chapter 4e: Phase diagram appeared first on BORZUYA UNIVERSITY.

In a solid, there is no translation and no rotation. The atoms may vibe around their equilibrium position. The vibration involves a potential energy and a kinetic energy, each term participating for ½R. As a solid has 3 dimensions, C_v=3R

However, this equality is not true at low temperatures. It is related to the radiation of the black body. In this case, the heat capacity is given by

A black body is an idealised body that absorbs all electromagnetic radiation (of any frequency). It only emits a radiation called black-body radiation that depends on the temperature alone. A model for the black body is a cavity with a tiny opening.

The radiation penetrating in the cavity are trapped inside because of tis geometry and/or the absorption of the radiation by the walls of the cavity.

At a fixed temperature and considering the thermal equilibrium, the hole may allow some radiation to escape from the black body. It is the black-body radiation and it only depends on the temperature. Because of the size of the hole with regard to the body, the escaping radiation has negligible effect upon the equilibrium of the radiation inside the cavity.

The reticular energy U₀, or energy of network is the energy of the solid at T=0K. The energy of the solid at a given temperature is

We cannot measure U₀ but we can approximate its value from the enthalpy of sublimation: to determine the energy of the solid, we break all the liaisons to obtain the gas.

The last term comes from the dilatation of the solid into the gas. The volume of the solid is negligible with regard to the volume of the gas. If we consider a perfect gas, then pV=RT. As a result,

We can also find the reticular energy with the Born-Haber cycle explained in a previous section.

For a solid with several elements, such as an ionic solid, the calculation is more complex: there are two or more gases and electronic interactions (ionic liaisons, repulsion).

Born–Haber cycle

Atoms form solids because it leads to a decrease of their energy. One measure to quantify this stabilisation is the lattice energy ΔH⁰_L, i.e. the energy required to form a solid from the gaseous ions.

This energy is negative and cannot be determined experimentally. The usual way to determine it is to build the so-called Born–Haber cycle. This cycle is made from the basic transformations of the constituents, taken them apart one from each other and starting from their normal state at sctp. Several transformations can be considered:

• Sublimation: transformation from a solid to a gas
• Vaporisation: transformation from a liquid to a gas
• Electroaffinity: the addition of an electron to a gas particle
• Dissociation: the separation of two atoms
• Ionisation: the liberation of an electron from a substrate

Let’s build the Born–Haber cycle for NaCl. We want to determine the enthalpy of this reaction:

To do so, we start from a reaction with a known enthalpy. We know that the enthalpy of reaction to form the solid from Na(s) and Cl₂(g) is

The transformation from Na (s) to Na⁺(g) is made in two steps: first the sublimation of the solid into a gas

And then the ionisation of the gas

There are two steps as well to obtain Cl^–(g) from Cl₂(g). The first step is the dissociation of the two atoms of Cl₂. As we only need one atom of chlorine, the enthalpy of dissociation is divided by 2.

The second step is to add an electron to Cl(g) (electroaffinity)

This energy is heavily negative because the chlorine reaches the octet. Now we have the two gaseous ions. We just need to add the enthalpy of all the steps together to obtain the lattice energy. The Born–Haber cycle is often represented this way:

The post Chapter 4d: Solids – lattice energy appeared first on BORZUYA UNIVERSITY.

There are different types of structure for solids:

Amorphous: disordered arrangement of the atoms

Crystalline: ordered arrangement of atoms. In a monocrystal, the whole solid grew from one point and there is a single arrangement. However, it is difficult to obtain a monocrystal because several growth point can develop simultaneously crystals with different orientations. Several crystals can grow in different directions and coexist in one solid.

Ceramic: some crystals in an amorphous network. It is a melange of the two previous types of structure.

The structure of a solid can be determined through the diffraction of X rays.

The solids can be made of different types of liaisons:

Metallic: electrons can move along the solid. They are very compact and show identical properties in all the directions

Ionic: anions are surrounded by cations and vice versa. In the case of spherical ions, the properties are identical in the three directions.

Covalent: electrons are localised between the nuclei, in the liaisons between atoms. The atoms cannot move freely. It is less dense than ionic or metallic solids because of the imposed liaison length and orientation.

Molecular: there is no specific rule to bind. In molecular solids, the molecules can move easily.

Crystal lattices

The structure of crystals can be defined through the smallest repeatable constituent of the network. NaCl is an ionic solid. Each Na is surrounded by 4 Cl and vice versa. The smallest repeatable constituent of the network is a cube made of 4 Na and 4 Cl, the atoms being in the corners of the cube.

There are different base structures for solids. In general, one compound means one crystalline structure. Crystals of salt NaCl have a cubic structure.  Carbon can have two different structures: the graphite (hexagonal) and the diamond (face-centred cubic). In the graphite, there are weak interactions between the sheets of carbon, making them easy to separate.

While the atoms are the same, the properties of the diamond and of the graphite are very different. Diamonds don’t conduct electricity, don’t break and are translucent. Their colour and opacity depend on impurities. Graphite  conducts electricity (the electron of the orbital p are not bound and can move along the planes), break (planes can slide on each other) and are opaque.

3 parameters are to be considered to sort the crystals: the lengths of the sides (a, b and c) of the structure, the angles between the sides (α, β and γ) and the position of the atoms in the structure. The geometric parameters give 7 lattice systems, given bellow while the position of the atoms sub-sorts them in 14 Bravais lattices

The different types of structures are:

Cubic

In the cubic structure, the lengths of the sides are equal, a=b=c and α=β=γ=90°. If one atom is in the middle of the cube, we call this structure body centred. If one atom is in the middle of each face, we call it face centred.

Tetragonal

When one side has a different length than the two others (a=b≠c), the structure is called tetragonal. The body-centred also exists for this structure.

Orthorhombic

The angles are still 90° but none of the three characteristic lengths are equal (a≠b≠c). In the case of orthorhombic structures, an additional structure exists: the base-centred structure. In this structure, two opposite sides exhibit an atom at their centre. The other two sides do not.

Hexagonal

This structure is called hexagonal but none of its face is hexagonal. However, if you assemble three of them you obtain a hexagonal prism. The base and top sides are diamond-shaped (a=b) with an angle γ=120°. The other faces of the structure are rectangular (a=b≠c) or cubic (a=b=c).

Rhombohedral

As for cubic structures, a=b=c and the angles are equal but in a rhombohedral structure they are not right: α=β=γ≠90°.

Monoclinic

In this case, two angles are equal to 90° but the third angle is not right. The sides have different lengths a≠b≠c.

Triclinic

The angles α≠β≠γ≠90° are different and there is no right angle. Also, the lengths are different a≠b≠c.

Atomic Packing factor

It is determined by the ratio of the volume of the atoms in the crystal and the total volume

Considering spherical atoms, a portion of the cube is inevitably empty. The figures above are schematic and in a real solid, the liaisons are not so long. We would see something like this instead:

This proportion varies from one solid to another one. As an example, let’s consider a face-centred cubic structure wherein all the atoms have the same size, the same radius r. The length of the cube is a and is volume is thus a³.

The length of the diagonal is 4r, and from Pythagoras’ theorem, the square of the hypotenuse equals the sum of the square of the other two sides of a triangle. As a result,

We know the total volume of the cube

 We just need to determine how many atoms are in the structure. In each corner of the cube, there is 1/8 of an atom. There are 8 corners, so there is the volume of one atom in total for the corners. At the centre of the faces, there is the half of an atom. There are 6 faces so that makes 3 additional atoms. In total, there are 4 atoms in the cube. The volume of one atom is 4/3 pi r³. The APF for this face-centred cubic is thus

For this particular structure, approximately one fourth of the cube is empty. It is the same for the hexagonal structure, reaches one third for the body-centred cubic and one half for the cubic.

• Hexagonal close-packed: 0.74
• Face-centred cubic: 0.74
• Body-centred cubic: 0.68
• Simple cubic: 0.52

Miller indices

Miller indices form a notation system for planes in crystal lattices. It is mainly used in crystallography but I did not used it a lot myself as a chemist. I won’t develop it intensely here. The Miller indices are a trio of 3 integers, written (hkl). Considering one corner of a crystal structure, let’s say a cubic one, as the base (000) of the notation, the axis are following the sides of the crystal. This base is thus not always orthogonal if one of the angles α, β and γ is not right. In the case of a cubic structure, the base is orthogonal.

Moving in the x direction, we will find an atom after a distance a=2r. It is the same if we move in the y or z directions. For the Miller indices, the coordinates x, y and z are divided by the structure dimensions a, b and c respectively.

The atom on the x axis is on the location [100] (it is not a Miller index, just a coordinate). The atom on the y axis is on [010] and the atom on the y axis is on [001].

The Miller indices (hkl) describe a plane by telling where they cross the axis x, y and z. Considering the following plane

We see that it is parallel to the y and z axis and that it crosses the x axis at x=1. As the plane is parallel to y and z, they will never cross. We consider that it would be done at y=∞ and z=∞. To obtain the Miller indices, we take 1 over these values:

If negative values are used, we overstrike them instead of putting a minus sign.

Some examples of planes are next

Exercises

1.  How do you call these structures?

2.  Give the Miller indices of the following planes:

3.  The density of the solid NaCl is 2.165. What volume is occupied by 1 mol of it?

4.  Calcium has a fcc configuration. The length of one side of the cube measures 0.557nm. What is the radius of the calcium, the filling ratio and what is the density of this crystal?

Answers

1:

2:

For the first one, the showed plane passes by the y axis. Consequently we have to translate it. By default, we translate it following the x axis by one length a to the left.

For the second one, the plane is placed on the middle of the structure, horizontally. Remember that we take the inverse of the position: 1/0.5=2.

As for the first one, the third plane passes by an axis. We translate the plane to obtain the correct answer.

The two last planes have the same Miller indices. They are parallel and merge if we apply a translation in the z direction.

3.

4.  The structure is fcc (face-centred cubic) so our point of reference is the diagonal: its length is 4 times the radius of one atom.

To know the density, we calculate the volume of 1 mol

The post Chapter 4c: Solids – crystallography appeared first on BORZUYA UNIVERSITY.

Liquids are a condensed state: the volume of a liquid does not change by much when a pressure is applied. A liquid is a fluid: it has not its own shape and molecules can move inside it. As said previously, liquids are characterised by a short range order and a long range disorder.

Properties of liquids

Superficial tension

When a droplet of water falls on a metal surface (for example), the droplet does not spread completely. The droplet is not a monolayer of water but has still a characteristic shape and a width.

The shape depends on the interactions between the molecules of the liquid between themselves, their interaction with the molecules of the surface. If the interactions with the surface are stronger than the intern interactions of the liquid, then the droplet spreads on the surface. The surface is said to be hydrophilic if a liquid spreads on it. On the opposite, on a hydrophobic surface, droplets remain in a spherical shape (it is possible to almost have a complete sphere). If the surface is inclined, the droplet will just roll on the surface. With a designed surface, it is possible for a droplet to climb on an inclined surface shaped like a stair (Enhanced Droplet Control by Transition Boiling, Alex Grounds, Richard Still & Kei Takashina, Scientific Reports 2, 720)

The superficial tension is a property of the liquid while the hydrophobicity is a property of the surface.

To increase the surface (by ΔS) between a liquid and a solid, a work W is required

γ is given for a type of surface, so that it only depends on the interaction between the molecules of the liquid. A small superficial tension means that the liquid will easily spread, that the work to increase its surface with the solid is small.  Oil for example spreads easily on a surface. On the contrary, the mercury has very strong metallic bonds and minimises its surface with the solid. It keeps a spherical form.

Capillarity

It is the ability of a liquid to flow in narrow spaces without external forces, or in opposition to external forces. Typically, if you partially plunge a piece of paper in water, water will move upwards on the paper, despite the influence of the gravity. The capillarity is the resultant of several forces: the adhesion force between the liquid and the surface against the weight of the liquid and the cohesion force in the liquid. On the paper, the water will stop to move upwards when those forces are cancelling each other.

In a tube of glass, the adhesion force is important because the glass possesses O^– anions which favour the adhesion. Consequently most of the inorganic liquids climb in a tube, we can see that the surface of the liquid is generally not flat but incurved. It is called the meniscus. The adhesion energy is favourable enough to increase slightly the water-air surface so that the height along the wall is greater than in the middle of the tube. The difference of height can be several millimetres. On a graduated tube, the height to consider is the one in the middle of the tube.

When the adhesion force is smaller than the cohesion force, the level in the tube will be lower than the level of liquid out of it. It is the case for the mercury. Its meniscus is also different from the meniscus of the water as the mercury avoid the contact with the surface.

The post Chapter 4b: Liquids appeared first on BORZUYA UNIVERSITY.

We can consider 3 different states –gas, liquid and solid– and a melange of them. One important difference between the three states of the matter is the volume they occupy. A gaz takes all the available space, a liquid takes the form of its recipient and a solid has its own form. It is the result of the interactions between the molecules composing the matter. The composition matters but it is not the single parameter to considerate. Simply set, the composition of water, ice and vapour of water is the same (H₂O), but the interactions between the molecules of H₂O are different. Sometimes things are not totally a solid nor a liquid nor a gas. An example is the shaving foam: it has not its own shape but does not completely take the shape of its recipient. Generally, in this case we are in presence of a melange of two cohabitating states. The goal of this chapter is to describe the different states of the matter and the transitions between them.

The arrangement of the matter can be defined from the repartition of neighbour atoms between them. If we focus on one specific atom, the probability g(r) to find one or many atoms at a given distance r of the atom differs strongly between liquids, solids and gases.

In a solid, the atoms are organised and don’t move freely (they can vibrate around their position). The repartition depends on the geometry of the solid but it is characterised by high, localized peaks at regular distances with g(r)»0 between them. In a liquid, we can see that there are still layers of neighbours around a given atom but that the order is less important. Moreover, the peaks vanishes after a few layers. A liquid is thus characterised by a short range order and disorder at long range.  The average number of neighbours in a liquid is in general smaller than in a solid. In a perfect gas, the atoms don’t feel each other and have no interaction. The probability to find an atom at a given distance of another one is thus equal at any distance. In real gas however, interactions can exist and g(r) varies slightly.

Gases

We are surrounded by gases. We can displace them when we are moving but they apply a resistance to our moves. Even when we are not moving, there is a pressure of the air on us. We don’t feel it because we are used to it and our body applies a force to counter this pressure. If the void was made, we would expand because of this force. The fact that air applies a pressure has been proved by Otto Von Guericke (XVII^th century). He build two half spheres, fitting each other. He removed the air from them and asked to his lord to try to open it with the help of four horses. They did not success. Putting back air into the sphere, it was possible to open it again.

There are still objects that we often use based on this principle: suction pads for plumbing, the ones to put your GPS on the windscreen. The void that we create is never complete and the strength necessary to remove a suction pad depends on the degree of void that we created. It becomes way easier to remove them once we allowed a bit of air to get in the suction pad.

A pressure is a force applied on a surface. The pressure applied by the air in standard conditions is 1 atmosphere, which equals 101325 Pascals, more usually given in hectoPascals (1013hPa). We also measures the pressure of a gas with torr. 1 atm=760torr. The torr unit is the height, in mm, that mercury reaches in a voided tube when a pressure is applied on the rest of its surface in the following disposition.

There is a relation between the volume of a gas and its pressure. Boyle determined that at a constant temperature, the multiplication of the pressure of a gas by its volume gives a constant.

If the volume decreases, the pressure increases by the same ratio:

Charles Gay Lussac observed on the other hand that the volume of gas depends on the temperature and that, does not matter the gas, its volume trends to 0 when the temperature goes towards 0K (he extended the curves when the gas changes of state).

A third contributor to the understanding of the gases is Avogadro who said that at a given temperature and pressure, the volume of a gas is proportional to the number of atoms it contains:

k is the constant of Boltzmann. Considering all of that, the law of perfect gases was established:

This law is limited to diluted gases and we consider that the particles are punctual, distant and that there is no interaction between them.

If we consider several gases, Dalton showed that the total pressure is given by the sum of their individual pressures.

As a consequence, if we consider a melange of two gases

The individual pressure of one component of the gas is proportional to its concentration in the gas:

Kinetic theory of gases

We can measure the pressure applied by a gas on a surface. As explained before, a pressure is a force applied on a surface. A force is a mass multiplied by an acceleration. An acceleration is the variation of speed over a time.

Note that only the component of the speed perpendicular to the surface affects the pressure. The two other components don’t have an influence on this particular surface.

We make several assumptions about the particles of the gas:

• The particles are small and their volume is negligible compared to the volume of the gas. They are considered as dots.
• They all have the same mass m,
• They have a constant speed and go in random directions. We can define an average speed <v>.
• The particles don’t interact together
• All the collisions are purely elastic: there is no loss of speed or energy

To determine the pressure on a surface S of one of the walls, we consider the particles that are able to hit this surface in a period of time dt. As a result, we only consider the particle present in a small cylinder in front of the surface.

When a particle hits the wall, there is an elastic collision and the particles goes back with the same speed. Over one period of time, the speed of a particles changes from v to –v. The force applied on the surface by one particle is thus

In the cylinder, there is a given number of particles:

N/V is the density of particle in the gas. Amongst them, only the particles going in the x direction are important for us, and amongst them only the particles going in the direction of the surface (not in the opposite direction) induce a pressure on the surface. A 1/6 multiplication is thus applied to determine the number of particles impacting the surface. The pressure is thus

In this equation, we recognise the kinetic energy E_k=mv²/2. Keeping in mind that N=n.N_A, we can write

As a result, we see that the kinetic energy of particles in a gas is directly proportional to the temperature

From there, we also see that the average speed of the particles is related to their mass:

It means that in a gas at a given temperature, particles of oxygen and of nitrogen do not move at the same speed. For example, a balloon of nitrogen (N₂, M=28) deflates faster than a balloon full of oxygen (O₂, M=32)

This property is used to enrich uranium, i.e. obtain a larger proportion of one isotope of the uranium. Considering a volume of gas containing ²³⁵UF₆ and ²³⁸UF₆, a bigger proportion of ²³⁵UF₆ will pass through a small gap because it has a larger speed than ²³⁸UF₆. This process is the effusion: a separation based on the speed of the particles.

The heat capacity (at V=cst) is the energy required to increase the temperature of one mole of gas by 1° is

The heat capacity (at p=cst) is

In a monoatomic gas, all the energy is used to increase the speed of the particles.

Diatomic gases

For molecules of more than one atom, the kinetic energy is split up in the three directions (1/2R by direction).

If a diatomic gas is heated, the energy is used by several processes:

1. translation/acceleration: the speed/kinetic energy of the molecule increases,
2. rotation and vibration

There are two angles of rotation, the energy required to rotate is 2*1/2R

If the molecules vibes, they have a potential energy. The energy obtained from the heating is thus distributed in kinetic energy and in potential energy. The energy required to vibe is thus also 2*1/2R.

The values of Cv and Cp are thus higher than for a monoatomic gas:

So, if we give energy to Ne and N₂, which gas should heat more? N₂ uses the energy of the heating to accelerate but also to vibe and to rotate. Ne uses all the energy as kinetic energy and will thus heat more than N₂.

Distribution of Maxwell-Boltzmann

The calculations we have done assumed that all the particles of a gas were affected identically by the heat. In reality, the heat capacity depends on the temperature but trends to 7/2R for T trending to infinite. The explanation follows:

Particles can have different levels of kinetic energy. When energy is supplied, not all the particles are equally excited. One proportion of their population increases its kinetic energy by one level (or more eventually). At OK (absolute zero), all the particles are at the lowest level and don’t move. They are frozen.

The population on the second level is given by

ΔE₁ is the difference of energy between the level 1 and the level 0. k=R/N_A is the Boltzmann’s constant. In general, the population is given by

The distribution of speed of the particles in the levels of energy are given by the law of distribution of Maxwell-Boltzmann. From above, we can determine that the variation of level with regard of the variation of speed

With A an undetermined constant. We can write this equation differently:

It is possible to integer this equation, knowing that

It would mean that A=(m/2πkT)^1/2. However, the speed has three components. As a result, the distribution is

Similarly to the probability of presence of the electrons for the electronic orbitals, we integer on a sphere and we should multiply this equation by 4πv². We have thus obtained the proportion of particles dN on the total number of particles N with a given speed dv.

One can see on this figure that the average speed increases with the temperature, as it was determined previously in the relation

from the law of perfect gases. The most probable speed is found at the maximum of the curves, when dN/N/dv=0, i.e. at

The average speed that we find is a bit smaller than the one obtained from the law of perfect gases:

Real gases

The law of perfect gases works well for diluted gases. If we increase the pressure or the temperature of gases, we see deviations from the law of perfect gases.

It is because the number of interactions between the particles becomes less negligible. The interactions can be attractive or repulsive but are weak with regards to the energy of a liaison for example. The interactions can be between induced dipoles or between permanent dipoles:

Interaction permanent dipole-permanent dipole

In this case, the molecules composing the gas already possess a dipolar moment. The δ+ and δ- charges of molecules passing near each other interact: an attraction is induced between opposite charges and repulsion between dipoles of same charge.

Interaction induced dipole-induced dipole

The electrons near a nucleus are, statistically, equally distributed around the nucleus. In other words, the cloud of electrons is spherical and the density of electrons is identical in all the directions. However, the electrons are continuously moving and heterogeneities of charge may appear temporarily. In fact the apparition of these dipoles is frequent but in average the dipolar moment is cancelled.

When an atom passes near another atom showing a dipole, the atom adapts its electrons cloud to the dipole to reduce the repulsion between them.

In general, the interactions are favourable between the particles of a gas. The particles tend to remain close of each other, decreasing the pressure with regard of a perfect gas.

On the other hand, the particles have a proper volume and the other particles cannot penetrate this volume. In the law of perfect gases, the particles were considered as dots, with a negligible volume.

As a result, despite the increase of interactions, the pressure increases faster than for a perfect gas.

Instead of the perfect gas law PV=nRT, we have to consider the van der Waals equation for real gases:

In the pressure term, the +a(n/v)^2  expresses the interactions between the particles. a stands for the interaction strength, the value of which depends on the type of interactions, the size of the molecules, the number of electrons, etc. (n/V)^2 stands for the “density” of interactions and should in fact be n/V*(N-1/V) as each particle of the volume  (n) can interact with all the other particles (n-1) but as n is proportional to NA, the difference is negligible.

In the volume term, b stands for the covolume of one particle. The empty volume where particles can move is the total volume minus the volume of the n particles composing the gas.

In the figure of the potential of Lenard Jones, we can relate a and b to the energy of the pit and to the distance of smaller approach σ.

The enthalpy of vaporisation ΔH_v reflect the energy required to transform a liquid into a gas. It requires more energy to vaporise molecules with strong interactions than molecules with only induced dipoles. Sprays of water are sold because of this property: once on our skin, the small droplets vaporise, taking the energy from the skin, decreasing its temperature and giving a feeling of freshness.

The opposite process is the liquefaction. The molecules in the gas are distant from each other. To become liquid, the particles move into the potential pit. However if they have a too large kinetic energy (depends on the temperature), they will get away from the other particle. It takes energy to leave the potential pit. As a result, liquefaction involves a decrease of the temperature.

Exercises

1.  What is the volume of 1,5g of N₂ under a pressure of 1atm, at 20°C?

2.  What is the total pressure of a gas composed of 120g of H₂ and of 80g of He if the temperature is 25°C and the volume is 300l? Express your answer in atm, Pa and torr. What in a volume of 450l or 600l?

3.  What is the required volume of oxygen to burn 1g of butane (C₄H₁₀)? Knowing that all the products are gaseous, what will be the volume after the complete combustion?

4.  Considering that the combustion of 1g of a gas composed of CH₄ and C₂H₆ formed 1568ml of CO₂ and water at 18°C and 742torr, what are the masses of CH₄ and C₂H₆ of the sample?

5.  To produce a neon lamp of 520cm long by 2cm of diameter, the pressure inside the tube has to be equal to 1.5mm Hg at 37°C. Determine the mass of neon required to obtain those conditions.

6.  A recipient of 500ml is filled with CO₂(g) at the pressure of 1 atm at 20°C.  What is the mass of the gas? What is the pressure if we increase the temperature by 40K? What is the pressure if we next compress the gas to a volume of 100ml?

7.  Sort the following systems from the closest to the farthest of a perfect gas:

a)H₂ at SCTP

b)He at 350K and 1 atm

c)H₂ at 273K and 2 atm

d)He at SCTP?

8.  Compare the results of the equations of the perfect gases and of Van der Waals (a=4.17l²/mol², b= 0.0371l/mol) for a volume of 20l, 2l and 0.2l containing 1 mol of NH₃ at 0°C.

9.  Considering that the minimum of the function of Lennard-Jones for the argon is at -996J/mol at a distance of 0.382nm between the atoms, what would be the potential at a distance of 1/2×0.382nm, 2×0.382nm and 3×0.382nm?

10.  What is the kinetic energy of 1 mol of Ne at 20°C and 1 atm. What are its heat capacity at a constant pressure and at a constant volume? What is the difference if instead of neon we had nitrogen?

Answers

1.  We use the law of perfect gases to solve this problem

2.

3.  The first step is to determine how many moles is 1g of butane:

Now we write the chemical reaction to find the stoichiometric coefficients

The required volume of oxygen is thus

9 moles of gas are produced by the reaction per mole of butane, so the volume after the reaction is

4.  The reactions of combustion are

We know that 1g of reactants give 1568ml of CO₂. The molar masses of the reactants are

In 1g we have n₁ moles of CH₄ and n₂ moles of C₂H₆

As we have the volume, the pressure and the temperature of the CO₂, we can determine how many moles of CO₂ are produced by the reaction with the law of perfect gases

The pressure has to be put in the IUPAC units (atm) instead of torr: 1atm=760torrà 740torr=0.974atm and the temperature is put in Kelvin (°C+273.15)

1 mol of CO₂ is produced for each reaction of combustion of CH₄ and 2 are produced by each reaction of C₂H₆. The total amount of CO₂ is thus equal to

We have two equations with two unknowns to solve. To do so, we can insert the expression of x into the first equation

The masses are thus

5.

6.

7.  b-d-a-c

8.

9.

10.  Monoatomic gas:

Diatomic gas:

The post Chapter 4a: The states of matter: Gas appeared first on BORZUYA UNIVERSITY.