Title 13.Kinetic Theory-F.pdf ✅

NEET-2023 Ultimate Crash Course by AL RIZWAN jr College PHYSICS Kinetic Theory of Gases
POINTS TO REMEMBER
1. A mole is the amount of substance which contains the same number of elementary entities as there are in 12g of C-12. Experiment shows this to be about 23 6.022 10  , a value denoted by NA and called Avogadro’s constant or Avogadro’s number, i.e., 23 N 6.022 10 A =  /mol. It should be noted that the mole is a measure of number of entities and not of mass 2. The ideal gas equation is often expressed in terms of total number of molecules N. Clearly, N nN = A or A N n N = B A A N R PV RT N T Nk T N N     = = =         where B A R k N = (Boltzmann constant) 23 23 8.31J / molK 1.38 10 J / K 6.022 10 / mol − = =   B V B N P k T n k T V   = =     where V n = number of molecules per unit volume (called number density) 3. One mole of an ideal gas at STP has a volume of 22.4 litres ( ) 3 3 22.4 10 m . − −  This value can be obtained by substituting 5 2 P 1.013 10 N / m , =  R 8.31J / molK = , T = 273 K and n = 1 in nRT V P = . 4. If the pressure is expressed in atmospheres and the volume in litres ( ) 3 3 1L 10 m− = , then ( )( ) ( )( ) PV 1atm 22.4L R 0.0821Latm / molK nT 1mol 273K = = = 5. As the density of a gas at a given pressure and temperature varies inversely as its volume, from 1 1 2 2 1 2 P V P V , T T = 1 2 1 1 2 2 P P T T =   If the temperature does not change, T T 1 2 = and 1 2 1 2 P P =   or 1 1 2 2 P P  =  or  P Hence, the density of a gas at constant temperature, varies directly as the pressure. Again, if the pressure remains constant, i.e., P P 1 2 = ,  =  1 1 2 2 T T or 1 2 2 1 T T  =  or 1 T  Hence, the density of a gas at constant temperature, varies inversely as the absolute temperature. 6. As V T = constant, t 0 0 V T 273.15 t t 1 V T 273.15 273.15 +   = = +    
or t 0 t V V 1 273.15   = +     (Charles’s law) As P T = constant, t 0 0 P T t 1 P T 273.15   = = +     or t 0 t P P 1 273.15   = +     (Gay-Lussac’s law) 7. We have taken into account only collisions between the gas molecules and the walls of the container. However, the above analysis does not change when we include the possibility of intermolecular collisions which change the molecular momenta without affecting the walls. This is a consequence of assumption (g). Even though many intermolecular collisions occur, the net effect of all of them is to leave the system as a whole unaffected. 8. If a molecule strikes any other face of the vessel on its way from face F to face F', the x-component of its velocity does not change, nor does the transit time t . 9. We have found the pressure on one face of the vessel but it is the same on all its faces. 10. We could have chosen a vessel of any shape but a cubical vessel simplifies calculations. 11. The derivation is a simplified version of a more complex piece of theory which makes similar assumptions and arrives at the same result. The mean square speed 2 v is not the same as the square of the mean speed. Thus, if 5 molecules have speeds 1, 2, 3, 4, 5 units, their mean speed is (1 + 2 + 3 +4 + 5)/5 = 3 and its square is 9. On the other hand, the mean square speed is : ( ) 2 2 2 2 2 1 2 3 4 5 / 5 11 + + + + = 12. 2 2 k 1 Nmv 1 Nm 2 2 2 E P v 3 V 3 V 3 V       = = =           ; where 2 k 1 E Nmv 2 = = average KE of translation of the gas. In the above equation reveals that pressure exerted by a gas is equal to two-third of the average KE of translation per unit volume of the gas. 13. Further, k 2 PV E 3 = which is the relation between P, V and Ek and not containing temperature E U k = , where U is the internal energy of an ideal gas Thus, 2 PV U 3 = 14. 2 1 Nmv 2 N 1 2 N 2 P mv K 3 V 3 V 2 3 V     = = =         or 2 PV NK 3 = where 1 2 K mv 2 = is the average KE of translation of a molecule. Thus, the pressure exerted by a gas molecule is directly proportional to the average KE of translation of the molecule.