KINETIC THEORY. Contents

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1 KINETIC THEORY This brief paper on inetic theory deals with three topics: the hypotheses on which the theory is founded, the calculation of pressure and absolute teperature of an ideal gas and the principal consequences of the theory. The level of this treatise is accessible by people who now algebra, since calculus is only arginally eployed, but to understand the atter thoroughly soe nowledge of Physics is required, such as the eaning of absolute teperature, energy, average value of a quantity etc; however, several quantities and physical constants involved in the developent of the theory are introduced at the right oent. Contents Hypotheses of inetic theory a qualitative discussion of the preises of the theory with references to facts and considerations confiring each hypothesis. Calculation of the pressure - a statistical forula for the pressure of an ideal gas is given by starting fro the hypotheses of the theory through the analysis of the olecular collisions. Absolute teperature is a easure of olecular energy - by coparing the forula of the pressure and the state equation of an ideal gas it is proved that absolute teperature is a easure of the average inetic energy of the olecules. Polyatoic gases extension of the theory to polyatoic ideal gases. Results of inetic theory this part exaines the iplications of inetic theory: extension of the theory to real gases and solids; calculation of the root ean square of the olecular velocity for an ideal gas; Graha s and Dalton s laws; internal energy of ideal gases and solids; olar heats of ideal gases at constant volue and of onatoic solids; calculation of exponent γ in adiabatic processes. 1

2 The developent of Therodynaics in the first decades of 19th century too place by integrating sets of experiental observations and theoretical preises which leaded to the odern heat theory, expressed by the first and the second law of Therodynaics. Although First Law copletely expresses the equivalence of wor and heat by stating that heat is a for of energy, it has an essentially epirical nature, since this law doesn t offer any further explication of heat-wor equivalence. This blan, due to lac of a coplete theory on the behaviour of the olecules, is filled just by inetic theory. The conceptual eaning of inetic theory consists of the providing a echanical and statistical interpretation of Therodynaics, by calculating pressure P and absolute teperature T as average values of echanical quantities (lie ass, velocity, energy etc.). By calculating the pressure of a perfect (or ideal) gas 1 ; and coparing the result with the state equation it s possible to express absolute teperature as a function of olecular otion. Hypotheses of inetic theory Kinetic theory is based on a set of hypotheses on the icroscopic structure of atter founded on epirical observations. Soe hypotheses apply to atter in general (also to liquid and solid phases), the others only to perfect gases. 1. Matter is coposed of inial units ( olecules ) which can be considered as assive but diensionless aterial points. In reality olecules have a coplex structure, but according to the base version of the theory they are devoid of structure and volue. This eans that olecules are aniated only by translational oveents, so ignoring rotations and vibrations. Although physically incorrect, this hypothesis enables us to build an easily treatable odel; the corrections necessary to tae into account all the degrees of freedo of the olecules and their total volue will be introduced later.. Molecules are aniated by a continuous rando otion (said otion of theral agitation). This applies also to states of atter such as liquid and solid phases which apparently see especially this last one to consist of otionless particles; according to the theory in the solid phase olecules could only perfor oscillations around fixed equilibriu positions. Molecular otion is evident in the case of gases and solutions, since these substances tend to diffuse, that is to fill the whole available volue. Also the pressure of a gas filled in a container is due to the tendency to tae up the whole space available, so one can easily deduce that pressure originates fro the olecules nocs on the internal walls of the container and that it depends on olecular otion, that is on ass and average speed of the olecules. This hypothesis is confired by Brownian otion, so naed after botanist R. Brown who observed it in 187. Brown discovered that pollen particles (which are extreely light) in suspension in a water drop see to be aniated by an unceasing rando otion which can be described as a succession of extreely short segents. Also particles of different ind but with very little ass and diensions anifest this behaviour when floating in a fluid. The explication consists in aditting that in the fluids olecules theselves ove continuously and randoly and transfer their oveent to the pollen particles by nocing on the.. The laws of Classical Mechanics apply also to olecular otion. This hypothesis is necessary for proceeding in calculation and can be verified only a posteriori, i.e. by coparing the results predicted by the theory with the available experiental data. The following hypotheses apply only to ideal gases. Let s reeber that a gas behaves in perfect or ideal way, that s it satisfies the state equation with a good approxiation, when it s very

3 rarefied and therefore pressure is very low and teperature sufficiently high, that is rather higher than condensation point. 4. Molecules don t collide on and don t interact each other (it is a siplification; in a ore correct theory one has to consider wea interolecular forces, nown as an der Waals forces), so they persist in linear unifor otion until collide against the walls of the container. This hypothesis is equivalent to aditting that the diensions of the olecules are extreely sall with respect to the average distance between each other, so the less dense a gas is, the nearer to reality it is. The epirical grounding of this hypothesis is the possibility to strongly copress a gas, especially if it s very rarefied. By increasing pressure it s possible to copress a gas till a very little fraction of the initial volue (for instance, 1/10,000) unless condensation occurs; this eans that the total volue of all the olecules of a gas (called covolue) is very sall with respect to the volue of the container. In practise, the volue of one single olecule is negligible, because it s the ratio between covolue and nuber of olecules contained in a given gas aount; in one ole there are about particles. Therefore the space apparently fitted with an ideal gas is alost copletely epty and can be strongly reduced by increasing pressure. 5. The collisions of the olecules against internal walls of the container are elastic. Perhaps this is the ore probleatic hypothesis, but it s necessary to adit it because otherwise the pressure of a gas at constant volue and teperature could not reain constant over tie, as coon experience teaches us. In fact, collisions occurring at acroscopic scale never are perfectly elastic, while can be totally inelastic; actually every single collision iplies a though inial energy dispersion in the for of heat. Therefore we are considering the hypothesis ore difficult to accept while aing reference to phenoena occurring at acroscopic scale. But just coherence of theory with observations copels us to adit that collisions are elastic at icroscopic scale, because otherwise we should coe to conclusions contrary to experience itself. 1 Bearing in ind that by definition a collision is elastic if inetic energy v is conserved, let s suppose ex absurdo that the speed of the olecules slows down at every collision against the walls of the container. It follows that also intensity and frequency (nuber of collisions per second) of the collisions diinish quicly with tie. But pressure of a gas is a growing function of both these factors and also it should tend to zero. A balloon fitted with a gas at constant teperature should soon becoe floppy itself. Therefore we have to adit that olecular nocs against the walls of the container should be elastic; this conclusion ust be extended to eventual collisions between olecules, but as we have seen these can be neglected in the case of very rarefied gases. Elasticity of collisions iplies that olecules don t undergo peranent deforations after collisions, coherently with the hypothesis according to which olecules are devoid of structure: in fact, we are reasoning as they were stiff, extreely little balls. Calculation of the pressure Let s start calculating the pressure P exerted by an ideal gas. In principle, the shape of the container should have no iportance, because it s expected that pressure depends only on ass and velocity of the olecules; however, in order to proceed with calculation it s better to consider a parallelepiped with faces perpendicular to axes x, y and z respectively. In particular, let s indicate with d the distance between the two faces perpendicular to x-axis and with S the area of their surface. Since pressure is the ratio between the noral force acting on a surface and the area of this surface, we have to calculate the force acting on the walls of the container, with particular reference to a

4 face perpendicular to the x-axis. By assuing N are the olecules, total force (i.e. the su of the forces of collision exerted by all the olecules) F will be given by F N f in which f is the intensity of the force that a olecule exerts during a collision. Since a force of collision depends on velocity of the single olecule (which isn t the sae for all), it s better to substitute f with its average value on all the olecules < f > and interpret F as total average force, whose value, because of the extreely high nuber of particles, is constant over tie unless therodynaic variables change. Therefore we can write F N < f > Besides we have to consider that olecules have different velocities. Since velocity is a vector, denoting by v x, v y and v z the three coponents of v, by Pythagoras theore in the space we have v v x + v y + v z. We are interested to average values calculated on all the olecules; since in a chaotic otion there aren t privileged directions of otion, the ean values of v x, v y and v z will be equal on the whole set of the olecules, thus we ll have <v x > <v y > <v z > [ <X> ean value of X ]. The ean value <v > of the square of v will be the su of three ters equal each other, so we get <v > <v x >. Moreover in order to siplify the calculation let s suppose that all the olecules have sae ass (this is a siplification; consider as a ean value ). To calculate f we have to apply the second law of Dynaics in the for q f t in which q is the variation of linear oentu of a single olecule due to one collision. We have to eploy the hypothesis according to which collisions are elastic. Let s treat olecules as very little, rigid spheres which when touching the walls of the container undergo a force f noral to the wall and directed toward inside. Kinetic energy is conserved and absolute values of v x, v y and v z don t vary. Since acceleration also is noral to the hit surface, only v x varies by changing sign. Hence the variation of linear oentu q is equal to the only coponent q x : q q x v x - v x 1 where is the ass of the olecule and 1 and refer respectively to the instants iediately previous and following the collision. Since v x 1 - v x by neglecting the sign we get q v x 4

5 Elastic collision against a wall: the olecule is reflected by the wall and fors equal angles of incidence and reflection. To find the force f we have to divide q per the tie t between two consecutive collisions against the sae wall. Tie is distance per speed; in our case the space is the double of the distance d between the walls perpendicular to the x-axis (the olecule, after a collision against a wall, will undergo soe nocs on the others before returning to the sae wall; the distance covered along the x-axis is d), and speed is v x. Therefore for a single olecule with velocity v x we get t d v x f v d v x x vx d We have to consider the ean value < f >, therefore substitute v x with <v x >. By ultiplying < f > by the nuber N of all the olecules we get the total force F exerted by an ideal gas on a wall: N v F < > x d To calculate the pressure P it s enough to divide F per the area S of the wall, so we have d S (volue of the container) at the denoinator. One reaches the sae result by considering the other faces and the y and z axes: pressure doesn t depend on the axis that we have choose to execute the calculations, as one can rightly expect since pressure ust be unifor in all the directions. Therefore < v i > P N in which the index i can be x, y or z. Since < v > < v i > we have P 1 N < v > This forula contains echanical quantities ( and v ) and has statistical character, because iplies the average value of the square of velocity on all the olecules. 5

6 Absolute teperature is a easure of olecular energy Now we can pass to the eaning of absolute teperature according to inetic theory. Fro the state equation of an ideal gas P n R T in which n are the oles and R is the ideal gas constant we can express P in function of the others variables: nrt P By aing equal the two expressions of the pressure we get 1 N < v > nrt The total nuber of the olecules N is equal to nuber of oles n ties Avogadro s nuber N A (i.e. the nuber of olecules in one ole); finally, we obtain the following echanical and statistical expression for absolute teperature: N A T < v > R This forula confirs the founding idea of the whole inetic theory, according to which absolute teperature is a statistical easure of chaotic olecular otion. To generalize this forula to non-hoogeneous gas ixtures we have to substitute <v > with < v >. It s better to find a relation between teperature and average olecular inetic energy < E c >. Since < v > < E c > we get N A R T < Ec > < E c > T. R N The ratio between the gas constant R and the Avogadro s constant N A is Boltzann s constant, a very iportant quantity in Therodynaics, equal to about 1, J. K A Finally we get < E c > T So, according to the siplest version of inetic theory, ean inetic energy of one olecule of an ideal gas is proportional to absolute teperature of the gas. This equation applies also to gas ixtures. Though not specified in the forulas, velocity and ean velocity-squared are always relative to the container, which is supposed in rest. The unifor linear otion of all olecules doesn t influence teperature, which depends only by rando otion. This principle applies also to liquids: systeatic otions lie rotations with sae angular velocity don t iply increase of teperature. 6

7 Polyatoic gases The final equation < E c > T applies only to a perfect gas by ignoring rotations and vibrations of the olecules. Actually, olecules are often coposed of two or ore atos; therefore one has to consider also those otions. Since to ignore rotations and vibrations eans to consider only the translational otion, it follows that the previous equation gives us the ean value of the only translational inetic energy, which is the total inetic energy only in the case of a onatoic olecule. To extend the theory to a polyatoic ideal gas we have to apply the principle of energy equipartition, according to which average energy is distributed uniforly aong all the degrees of freedo of the olecule; the degrees of freedo are all the independent coordinates defining position and orientation of one object. For a aterial point the degrees of freedo are the position coordinates, e.g. the three Cartesian coordinates x y z in a given frae of reference. Therefore ean inetic energy associated with one degree of freedo is 1 < E c 1 > T that we can consider as the fundaental relation between energy and teperature. The degrees of freedo of a diatoic olecule are five (three are translational and two rotational), since one of the two atos while aditting that the cheical bond is rigid can rotate around the another, so its relative position is identified by two angles ϑ and ϕ ; for a -atoic or polyatoic olecule the rotational degrees are three (the third ato can rotate around the axis connecting the first two at fixed distances fro each of the). While ignoring vibrational degrees, ean inetic energies of diatoic and -atoic olecules will be respectively Results of inetic theory 7 5 T and T. Real gases Kinetic theory can apply to real gases, liquids and solids by aditting that attractive and repulsive forces act between the olecules. In the case of a real gas one has to consider also the gas covolue, that is the su of the volues of all the olecules, so the state equation of a perfect gas P n R T ust be replaced with equations ore coplex as the an der Waals one 4. Solids - In the case of solids it s supposed olecules are oscillating around fixed equilibriu positions, so ean total energy of a olecule is the su of potential energy and inetic energy. By supposing that, on the average, potential energy is equal to inetic energy, the total energy of one olecule of a substance at the solid state is the double of the inetic energy. By considering onatoic solids such as etals the average total energy of one olecule is < E > T Root ean square velocity of the olecules Let s consider a cheically hoogeneous perfect gas (that is, all the olecules have sae ass). We can calculate the root ean square velocity (v rs ) of the olecules, i.e. the square root of the average value of the translational velocity of the olecules: v rs < v > v1 + v v N N

8 where N is the nuber of olecules, and v i refers to translational velocity of the i-th particle. Starting fro 1 < v > T we have < v > T v rs Fro this icroscopic forula we can get an expression containing acroscopic quantities by ultiplying nuerator and denoinator by the nuber N of olecules. Since N n ( N A ) nr where n is the nuber of oles, and N n (N A ) nm where M is the olar ass, we get v rs RT M T or also, since for one ole RT P and M 1 where ρ is the density, ρ which is the forula we looed for. v rs P ρ Graha s law of diffusion according to this experiental law, average velocities of the olecules of two gases at the sae pressure are inversely proportional to the square root of the respective densities, that is va ρb v ρ b where ρ is the density (ratio between ass and volue) and a and b denote the two gases. It eans that, pressure being equal, lighter gases diffuse ore rapidly. It s a consequence of the law of the root ean square velocity calculated above, since in Statistics π it s proved that v rs v, where v is average speed. So the ratios between r..s. velocities 8 and between average velocities are equal. By aditting that pressures are equal, we have va ( vrs ) a ρb v ( v ) ρ b rs Soeties physicists refers to Graha s law as the law according to which velocity of diffusion of a gas is inversely proportional to the square root of the olar ass at a given teperature, as we can see by the forula v rs Dalton s law of partials pressures This law affirs that [total] pressure of a ixture of ideal gases is the su of the partial pressures exerted by each [cheically hoogeneous] gas. Let s start fro 1 < v > P N valid also for a ixture with N olecules. 8 b a RT M a

9 If the gas we denote by the index contains N olecules of ass and average velocity squared < v >, we can write N < v > 1 N < v > < v > P P N which is just Dalton s law. n RT One can get this law directly by state equation: for each gas of the ixture P and for the whole ixture P n R T. Now, P RT n with n n 9 P nrt P. Internal energy of an ideal gas It s well nown that internal energy U of an ideal gas is a function only of its teperature. Kinetic theory iplies just that; indeed internal energy of a perfect gas is equal to total inetic energy of the olecules of the gas, therefore U N < E c > NT where is the nuber of degrees of freedo and the Boltzann s constant. Since N n N A, where n are the oles and N A is Avogadro s nuber, and N A R where R is the constant of the ideal gases, we get U RT This forula is the equivalent, at acroscopic level, of the forula < E c > T valid for olecules with degrees of freedo. Internal energy of a onatoic solid (etals, etc.) On the basis of the sae ethod we applied to an ideal gas, the internal energy of one ole of onatoic solid substance is U R T (reeber that the particles of solids oscillate about their equilibriu positions, so their total energy is the su of inetic energy and potential energy, which on the average are equal over tie). Molar heats Preised that olar heat of a substance is the heat capacity of one ole of this substance, we have Q n C M Τ. in which Q is the heat transferred into one ole of a substance not undergoing a phase transition. In the case of ideal gases at constant volue Q U since the whole heat absorbed by the gas is converted into internal energy. By denoting with C the olar heat at constant volue we get C T U R T and finally C R

10 In the case of onatoic solids (etals etc.) the olar heat is given by C M R cal ( R ca. 6 ), in agreeent with the experiental Dulong-Petit s law (which is correct ol K only if the teperature is higher than a critical threshold depending on the cheical coposition of the substance). Calculation of the exponent in adiabatic transforations - The equation of an adiabatic process for an ideal gas is P γ Κ where γ is an exponent depending on the olecular structure of the gas and K is a constant. Fro the first principle written in differential for, put dq 0, we get du + Pd 0 RdT d dt + RT 0 T d T By substituting into the state equation P nrt we get P nr + P nr, fro which we deduce γ +. Notes 1. A gas is said perfect or ideal if its behaviour agrees by a good approxiation with Boyle s and Gay-Lussac s laws, which occurs if its teperature is uch higher than its point of condensation and if the gas is very rarefied. The state equation of perfect gases P n R T puts in relation pressure, volue, absolute teperature and nuber of oles. The constant of ideal gases is R ca 8.1 J. ol K. Even if a gas were cheically pure, nevertheless its olecules should not have sae ass, since eleents have several isotopes.. The nuber of olecules in one ole is always equal to Avogadro s nuber (or Avogadro s constant), that is ca for any substance. Hence ole, not ass, ust be considered as easureent unit of the aount of atter or substance. 4. According to an der Waals s equation, pressure P is given by P nrt nb 10 an - where n are the oles and a and b are positive constants characteristics of each gas. In particular, b is covolue of one ole. Turin, August 009 Author: Ezio Fornero Copyright by Superzeo.net You ay, for your non-coercial use, reproduce, in whole or in part and in any for or anner, unliited copies of this paper. However, for any other use, especially reproducing in a publication or on web, please provide that credit is given to the original source.