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Ans: See Q. No. 4(b) ( P. 2010).(c) From the kinetic theory of gas, prove that P-pel,where p is the density of the gas and is the root mean squarevelocity of the gas molecules. Show that ē /T, where T is theabsolute temperature.

Answer»

Consider a molecule moving in the x-direction towards face A with velocity u1. On collision with face A the molecule will experience a change of momentum equal to 2mu1. (Figure 3)It will then travel back across the box, collide with the opposite face and hit face A again after a time t, where t = 2L/u1.

The number of impacts per second on face A will therefore be 1/t = u1/2L.

Therefore rate of change of momentum = [mu12]/L = force on face A due to one molecule.

But the area of face A = L2, so pressure on face A = [mu12]/L3

But there are N molecules in the box and if they were all travelling along the x-direction then

Total pressure on face A = [m/L3](u12+ u22+...+ uN2)

But on average only one-third of the molecules will be travelling along the x-direction.

Therefore:pressure = 1/3 [m/L3](u12+ u22+...+ uN2)


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