11 - KINETIC THEORY OF GASES Page 1

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1 - KIETIC THEORY OF GASES Page Introduction The constituent partices of the atter ike atos, oecues or ions are in continuous otion. In soids, the partices are very cose and osciate about their ean positions. In gases, at ow density, the partices are far away fro each other and perfor rando otion in a directions. Aso, the interactions aong the are negigibe. In iquids, the partices are sighty ore away than in soids and their otion is ess free than in gases. Pressure, teperature, voue, interna energy associated with gas are known as acroscopic physica quantities which are anifested as an average cobined effect of the icroscopic processes. Macroscopic quantities ike, pressure, teperature, voue can be easured whie interna energy can be cacuated fro the. Description of a syste using these quantities is known as acroscopic description. Macroscopic quantities and their interreationships can be understood fro the processes occurring between the constituent partices at icroscopic eve, e.g., pressure of a gas can be understood fro the transfer of oenta to the was of the container by the coisions of the oecues aking rando otion. Thus, description of the syste in reation to the speed, oentu and kinetic energy of its constituent partices is known as icroscopic description. Kinetic theory of gases is an approach wherein the aws of echanics are appied ( statisticay ) to the constituent partices of the syste ( i.e., gas ) and acroscopic quantities are obtained in ters of its icroscopic quantities with the hep of a atheatica schee.. Laws of idea gas Boye s aw: At constant teperature, voue of a fixed aount of gas, having sufficienty ow density, is inversey proportiona to its pressure. ( for fixed aount and fixed P teperature ) P constant Figure shows P curves for soe rea gas at three different teperatures obtained experientay ( shown by continuous ines ) and theoreticay using Boye s aw ( shown by broken ines ). Fro the graphs, it can be seen that rea gas foows Boye s aw at high teperature and ow pressure.

2 Chares and Gay-Lussac s aw: - KIETIC THEORY OF GASES Page At constant pressure, the voue of a given aount of gas, having ow density, is proportiona to its absoute teperature. T ( for fixed aount and fixed pressure of gas) T constant Equation of state for an idea gas: Cobining Boye s and Chares aws, P constant ( for a given aount of gas ). T Aso, the voue of gas is proportiona to its aount at constant teperature and pressure. P µ R, or P µrt ( ) T where µ represents the nuber of oes and R is universa gas constant 8.4 J ( o ) - K Ca ( o ) - K -. A gas obeying the equation P µrt at a pressures and teperatures is known as an idea gas. o rea gas behaves as an idea gas under a the circustances. However, at high teperature and at ow pressure, i.e., at ow density, rea gas behaves ike an idea gas and obeys the above idea gas aw equation. As the therodynaic state can be fixed using the above equation, it is caed the equation of state for an idea gas. Avogadro s nuber: The nuber of constituent partices ( atos or oecues ) contained in one oe of gas is caed Avogadro s uber ( A ). The vaue of A is the sae in a eeents and is equa to ( o ) -. If M g of gas has oecues and M 0 is the oecuar weight of gas, then uber of oes of gas, µ A M. M 0 Different fors of equation of state of an idea gas: P RT A R T kt ( ) A Here, R A k J ( oecue ) - K - ( Botzann s constant ) P k T n k T, ( )

3 where n - KIETIC THEORY OF GASES Page nuber of oecues per unit voue of the container, which is aso caed the nuber density of the oecues. n P k T [ fro equation ( ) ] M ow, P µrt R T M 0 M ρ M P R T R T ( 4 ), where ρ M 0 M 0 density of gas Equations ( ), ( ), ( ) and ( 4 ) are different fors of the equation of state of an idea gas. Avogadro s hypothesis: At constant teperature and pressure, the nuber of oecues in gases having the sae voue is the sae.. Kinetic theory of gases Macroscopic physica quantities of a gas ike pressure, teperature, etc. can be understood fro the interreationships between its icroscopic quantities. This is discussed in the kinetic theory of gases based on the foowing postuates. Moecuar ode of idea gas: Postuates: ( ) A gas is ade up of icroscopic partices caed oecues which ay be onoatoic or poyatoic. If ony one eeent is present in a gas, a its oecues are sae and cheicay stabe. ( ) The oecues of a gas can be considered as perfecty rigid spheres or partices devoid of interna structure. ( ) The oecues are in continuous rando otion coiding with each other and with the was of the container. ( 4 ) The oecues of a gas foow ewton s aws of otion. ( 5 ) The nuber of oecues in a gas is very arge. This assuption justifies randoness of their otion. ( 6 ) The tota voue of a the oecues of a gas is negigibe as copared to the voue of the vesse containing the gas. ( 7 ) Interoecuar forces act ony when two oecues coe cose to each other or coide. ( 8 ) The coision between the oecues and between the oecues and the wa of the container are eastic. The ipact tie of coision is negigibe as copared to the tie between successive coisions. Kinetic energy is conserved in an eastic coision. During the ipact tie of coisions, kinetic energy before coision is oentariy converted into potentia energy but is again reconverted into the sae aount of kinetic energy after the coision. Hence kinetic energy of the gas can be considered to be its tota echanica energy.

4 . Pressure of an idea gas - KIETIC THEORY OF GASES Page 4 Suppose an idea gas is fied in a cubic container having eastic was having each side of ength,. area of each wa. Let nuber of oecues having veocities v, v, v,,..., v at soe instant and ass of each oecue. ow consider opposite was A and A of the container perpendicuar to X-axis. Let the oecue have veocity X-axis, Y-axis and Z-axis respectivey. v with its coponents, v x, v y and v z aong When this oecue coides easticay with the wa A, its veocity aong X-axis gets reversed and becoes - v x. But y and z coponents of its veocity do not change. The x-coponent of oentu of the oecue before coision is p i v x The x-coponent of oentu of the oecue after coision is p f - v x the change in oentu of the oecue due to this coision is p p f - p i - v x - v x - v x by the aw of conservation of oentu, the wa gains oentu v x in the direction of +ve X-axis. ow, the oecue returning after coiding with A, coides with the wa A and without aking any other coision on its path, coides again with the wa A. Between these two coisions with the wa A, it traves a distance with veocity v x aong X-axis. tie between two successive coisions, t v x nuber of coisions per second v x oentu gained by the wa per second force, F v x v x v x

5 - KIETIC THEORY OF GASES Page 5 tota force on the wa due to a the nuber of oecues n v n F x i [ v ] i pressure on the wa, P i Force Area x i n i n i F v x i v x i n [ v ] i P ρ < v x > ( ) x i ( voue of the container ) and n i vxi where, ρ density of gas < v x > average of the squares of x-coponents of oecues. ow, as the nuber of oecues is very arge and their otion is rando, < v > < v x > + < vy > + < vz > and < vx > < vy > < vz > < v > < v x > and < vx > < v > Putting this vaue of < v x > in equation ( ), P ρ < v > ( ) [ This equation gives the pressure of an idea gas. ] rs speed v rs : The square root of ean speed of oecues, aso known as ean oecuar speed, < v > is caed root ean square speed, v rs. Fro equation ( ), v rs < v > P ρ

6 - KIETIC THEORY OF GASES Page 6.4 Kinetic energy and teperature Pressure of an idea gas is given by the equation, P P ρ < v > ρ < v > M < v > ( as ρ M is the tota ass of gas. ) µ M0 < v >, where µ nuber of oes of gas and M 0 oecuar weight of gas. Coparing this equation with P µrt ( idea gas aw equation ), we get µ M0 < v > µrt M 0 < v > RT M0 < v > RT ( ) which is the ean transationa kinetic energy of oe of gas and is proportiona to the absoute teperature of gas. RT v rs ( ) M 0 Dividing equation ( ) by Avogadro nuber, A, M 0 < v > A R T A < v > k T ( ) [ ass of a oecue, k Botzann s constant. ] This is the ean transationa kinetic energy per oecue of the gas and is proportiona to the absoute teperature of the gas. It does not depend on pressure, voue or type of gas. v rs k T ( 4 ) This equation shows that at a given teperature, the speed of ighter oecues is ore as copared to that of heavier oecues. Daton s aw of partia pressure Suppose a ixture of µ, µ,, oes of different idea gases, utuay inert, is fied in a container of voue at teperature T and pressure P and µ is the tota nuber of oes. P µ RT ( µ + µ + ) RT µ P RT µ RT P + P +, where P, P, are the partia pressure of the gases in the ixture.

7 - KIETIC THEORY OF GASES Page 7 Thus, the tota pressure of the ixture of idea gases, utuay inert, is the su of their partia pressures. The partia pressure of any gas of the ixture is sae as the pressure of that gas at the sae teperature when it aone is fied in the container having the sae voue..5 Maxwe s aw of oecuar speed distribution Gas oecues perfor rando otion with different speeds in different directions. Jaes Cerk Maxwe gave oecuar speed distribution aw for a sape of gas containing oecues as v dv 4 π v - e k T v dv, π k T where, tota nuber of gas oecues v nuber of oecues per unit speed interva v dv nuber of oecues having speed interva, dv ass of a oecue k Botzann s constant T absoute teperature The graph shows the nuber of oecues per unit speed interva, v versus speed, v for oxygen gas at two different teperatures. The tota nuber of oecues, is given by v dv 0 and the average speed of oecues of the gas, each of ass, at teperature T is < v > v v dv 0 v rs < v > 8 k T π k T.59.7 k T k T Most probabe speed ( v p ): The speed possessed by the axiu nuber of oecues is caed the ost probabe speed, v p. When v v p, d dv 4 π π k T v - e k T v vp 0 v p k T.4 k T Thus, v p : < v > : v rs :.8 :.4.

8 .6 Law of equipartition of energy - KIETIC THEORY OF GASES Page 8 The average kinetic energy of each onoatoic oecue of a gas in a container is, < E > < vx > + < vy > + < vz > k T But, < v x > < vy > < vz > < E > < vx > k T < vx k T Thus, the energy associated with each possibe independent otion of a oecue in a container is k T. ow, consider diatoic gas oecues. They perfor rotationa and vibrationa otion besides transationa otion. The rotationa otion of such a oecue is possibe in two different ways, i.e., about two utuay perpendicuar axes both passing through the id-point of a ine joining the oecues and perpendicuar to the ine as shown in the figure. For poyatoic gas oecues, such a otion can occur about three utuay perpendicuar axes. The atos of a diatoic oecue perfor osciations aso due to interatoic forces. Thus, a diatoic oecue possesses tota energy coprising of three different types of energy: ( ) Transationa kinetic energy, E t ( vx + vy + vz ), ( ) Rotationa kinetic energy, E r I ω + I ω and ( ) ibrationa energy, E v µv + kx, where the first and the second ters are the potentia and kinetic energy respectivey of the vibrator, µ is the reduced ass and k is the force constant of the syste. The nuber of quadratic ters for different otions appearing in the expression of tota energy of a oecue are caed degrees of freedo of the syste. It is for a onoatoic oecue, 5 for a non-vibrating diatoic oecue and 7 if it is vibrates. Law of equipartition of energy states that the average energy of a oecue in a gas associated with each degree of freedo is ( / ) k T where k is Botzann s constant and T is the absoute teperature.

9 .7 Mean free path - KIETIC THEORY OF GASES Page 9 The inear distance traveed by a oecue of gas with constant speed between two successive coisions ( with oecues ) is caed free path. Mean free path is the average of such free paths. The path of rando otion of a gas oecue is shown in the figure. The oecue oves on a straight path between two successive coisions, the ength of which is caed free path. But on coision, the direction and agnitude of its veocity changes. Consider that one oecue of a gas oves with average speed v whie other oecues are stationary. Its diaeter is d. During its otion on a straight path, it wi not coide with a oecue which is at a perpendicuar distance d fro the straight path aong which the centre of the oecue oves. Hence, we can iagine a cyinder around its path of radius d or diaeter d such that the oecues outside this cyinder wi not coide with the oving oecue. In tie t, the oecue wi sweep the iaginary cyinder of cross-sectiona area, πd, and ength v t. Thus, it wi pass through the cyinder of voue πd v t in tie t. If n is the nuber of oecues per unit voue, the oving oecue wi undergo n πd v t coisions in tie t. The ean free path is the average distance between two successive coisions. ean free path dis tance tota traveed by the oecue nuber of coisions in tie t. v t n π d v t n π d In this derivation, the other oecues were assued to be stationary. With rigorous anaysis it can be shown that when the otion of a the oecues is considered, then the ean free path works out to be n π d