A rigid, perfectly insulated container has a membrane dividing its volume in hal

Question

A rigid, perfectly insulated container has a membrane dividing its volume in half. One side contains a gas at an absolute temperature T0 and pressure p0 while the other half is completely empty. Suddenly a small hole develops in the membrane, allowing gas to leak out into the other half until it eventually occupies twice its original volume. In terms of T0 and p0 what will be the new temperature and pressure of the gas when it is distributed equally in both halves of the container? Explain your reasoning.

Explanation / Answer

given that

The box has a definite volume of V and a definite amount of mass inside (n). If the gas starts off with a volume of V, Pressure of Po, Temp of To, and mass of n,

use the Ideal Gas Law

P*V = n*R*T   where R is ideal gas constant
for case1

Po*V = n*R*To

for case 2

P2*V2 = n*R*T2

given that

Vo = 2*V

so from second casse

P2/T2 = n*R / V2

P2 / T2 = n*R / 2*V

from first case

Po/To = n*R/V

so

P2/T2 = Po/2*To

the ratio of the pressure to the temperature of the final volume of gas will be half that of the original

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