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Kinetic Theory of Gases

A gas in a co...

Question

A gas in a container $A$ is in thermal equilibrium with another gas of the same mass in container $B$ . If we denote the corresponding pressures and volumes by the suffices $A$ and $B$ , then which of the following statement is most likely to be true?

P_{A} = P_{B}, V_{A} \neq V_{B}

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P_{A} \neq P_{B}, V_{A} = V_{B}

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\frac{P_{A}}{V_{B}} \propto \frac{P_{B}}{V_{A}}

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\frac{P_{A}}{V_{A}} \propto \frac{P_{B}}{V_{B}}

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Solution

The correct option is D $\frac{P_{A}}{V_{B}} \propto \frac{P_{B}}{V_{A}}$
For an ideal gas $P V = n R T = \frac{W}{M} R T$
$P V M = W R T$
For gas A, $P_{A} V_{A} M_{A} = W_{A} R T$ ....(1)
For gas B, $P_{B} V_{B} M_{B} = W_{B} R T$ .....(2)
But $W_{A} = W_{B}$ as the masses of two gases are constant. Also $T_{A} = T_{B}$ as thermal equilibrium is present.
Hence, $W_{A} R T = W_{B} R T$
So equation (1) $=$ equation (2).
$P_{A} V_{A} M_{A} = P_{B} V_{B} M_{B}$
$\frac{P_{A} V_{A}}{P_{B} V_{B}} = \frac{M_{B}}{M_{A}} =$ constant.
$\frac{P_{A}}{V_{B}} \propto \frac{P_{B}}{V_{A}}$

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A gas in a container A is in thermal equilibrium with another gas of the same mass in container B. If we denote the corresponding pressures and volumes by the suffices A and B, then which of the following statement is most likely to be true?

A gas in a container $A$ is in thermal equilibrium with another gas of the same mass in container $B$ . If we denote the corresponding pressures and volumes by the suffices $A$ and $B$ , then which of the following statement is most likely to be true?