Etoos Tips and Formulas Kinetic Theory of Gases

KINETIC THEORY OF GASES It related the macroscopic properties of gases to the microscopic properties of gas molecules. 1. 2. 3. Basic postulates of Kinetic theory of gases (a) Every gas consists of extremely small particles known as molecules. The molecules of a given gas are all identical but are different than those another gas. (b) The molecules of a gas are identical, spherical, rigid and perfectly elastic point masses. (c) The size is negligible in comparision to inter molecular distance (10–9 m) Assumptions regarding motion : (a) Molecules of a gas keep on moving randomly in all possible direction with all possible velocities. (b) The speed of gas molecules lie between zero and infinity (very high speed). (c) The number of molecules moving with most probable speed is maximum. Assumptions regarding collision: The gas molecules keep colliding among themselves as well as with the walls of containing vessel. These collision are perfectly elastic. (ie., the total energy before collision = total energy after the collisions.) 4. 5. Assumptions regarding force: (a) No attractive or repulsive force acts between gas molecules. (b) Gravitational attraction among the molecules is ineffective due to extremely small masses and very high speed of molecules. Assumptions regarding pressure: Molecules constantly collide with the walls of container due to which their momentum changes. This change in momentum is transferred to the walls of the container. Consequently pressure is exerted by gas molecules on the walls of container. 6. Assumptions regarding density: The density of gas is constant at all points of the container. 7. 1 PV  mNv 2rms 3 Kinetic interpretation of pressure : [ m = mass of a molecule, N = no. of molecules] 8. Ideal gas equation 9. GAS LAWS PV = RT  P= μRT N A kT  N      kT = nkt V V V (a) Boyle's Law : For a given mass of an ideal gas at constant temperature. V  1 P (b) Charle's Law : For a given mass of an ideal gas at constant pressure. V  T (c) Gay–Lussac's Law : For a given mass of an ideal gas at constant volume. P  T (d) Avogadro's Law : P,V and T are same then number of molecules. N1 = N 2 THERMAL PROPERTIES OF MATTER 1 (e) Graham’s Law : At constant P and T, Rate of diffusion  p (f) Dalton’s law : P = P1 + P2 + ....... Total pressure = Sum of partial pressure 10. 2   Real gas equation (Vander Waal’s equation)  P+ μ 2a  (V–μb) = μRT    v  where a & b are under waal’s constant and depend on the nature of gas. 11. Critical temperature Tc = 8a 27Rb The maximum temperature below which a gas can be liquefied by pressure above. 12. Critical volume Note : For a real gas 14. Different speeds of molecules v rms = 15. 13. Vc = 3b Critical pressure Pc = a 27b 2 Pc Vc 3 = RTc 8 3RT 3kT = Mw m vmp = 2RT 2kT = Mw m vav = 8RT 8kT = πM w πm Kinetic Interpretation of Temperature Temperature of an ideal gas is proportional to the average KE of molecules, PV = 16. 1 1 3 2 2 mNVrms & PV = µRT = mv rms  kT 3 2 2 Degree of Freedom (F) (a) Number of minimum coordinates required to specify the dynamical state of system. (b) For monoatomic gas (He, Ar etc.) f = 3 (Only tarnslational) (c) For diatomic gas (H2, O2 etc.) f = 5 (3 translational + 2 rotational) (d) At higher temperature, diatomic molecules have two degree of freedom due to vibrational motion (one for KE + one for PE) (e) At higher temperature diatomic gas has f = 7 17. Maxwell’s Law of equipartition of energy Kinetic energy associated with each degree of freedom of particles of an ideal gas is equal to (a) Average KE of a particle having f degree of freedom = (b) Translational KE of a molecule = 3 kT 2 f kT 2 1 kT 2 (c) Translational KE of a mole = 3 RT 2 f (d) Internal energy of an ideal gas : U = µRT 2 18. Specific heats (Cp and Cv) (a) dQ Molar specific heat of a gas C = µRT (b)  dQ  dU Cv =  =   µdT  v = constant µdT (c)  dQ  Cp =  = Cv  R  Mayer’s equation   µdT dP =0 Atomicity 19. Trans lational Rotational Total (f) Cp Cv  Cv 5 = 1.67 3 7 =1.4 5 M onoatomic [He, A r, Ne] 3 0 3 Diatomic [H 2 , N 2 ] 3 2 5 Triatomic (Linear CO 2 ) 3 2 5 7 =1.4 5 Triatomic (Non-Linear NH 3 ) & Polyatomic 3 3 6 4 =1.33 3 f R Cp = Cv + R 2 3 R 2 5 R 2 5 R 2 5 R 2 7 R 2 3R 4R 7 R 2 Mean free path Average distance between two consecutive collisions m  where d = diameter of molecule, n = molecular density = 20.   1 2 d 2 n N V For mixture of non-reacting gases (a) Molecular weight M Wmix  (b) Specific heat at constant V CVmix  (c) Specific heat at constant P CPmix  γ mix  µ1M W1  µ2 M W2  ........ µ1 +µ2 +...... µ1CV1  µ2 CV2  ........ µ1 +µ2 +...... µ1CP1  µ 2 CP2  ........ µ1 +µ2 +...... C Pmix C Vmix  µ1CP1  µ2 CP2  ........ µ1CV1  µ 2 CV2  ........ 220 THERMAL PROPERTIES OF MATTER 21. 1 mN  2 3 Kinetic energy per unit volume E v =   vmix = P 2 V  2 22. At absolute zero, the motion of all molecules of the gas stops. 23. At higher temperature and low pressure or at higher temperature and low density, a real gas behaves as an ideal gas. 24. For any general process (a) Internal energy change ΔU=nC v dT (b) Heat supplied to a gas ΔQ=nCdT where C for any polytropic process PVx = constant is C = C v + (c) Work done for any process W = PV It can be calculated as area under P-V curve (d) Work done Q  U  nR dT 1 x For any polytropic process PVx = constant R 1–x