Question

Before we tackle a constant volume situation for an ideal gas let us think about an equivalent problem, where we keep the pressure constant, while volume is allowed to increase. If $\gamma$ is the coefficient of volume expansion at a temperature $T$, which of the following is true?

• A. $\gamma$ is more or less constant, and independent of $T$.
• B. Insufficient data. Need to know the final temperature
• C. $\gamma$ = $T^{-1}$
• D. $\gamma$ = $\frac{1}{T^2}$

Answer 1

The correct option is C

$\gamma$ = $T^{-1}$

The new volume upon expansion, $V^{\prime }$, is given as-

$V^{\prime }$ = $V(1+\gamma \Delta T)$

$\Rightarrow$ $\frac{nR{T}^{\prime }}{V}$ = $\frac{nRT}{V}(1+\gamma \Delta T)$

(where we have used the ideal gas equation, $PV$ = $nRT$, for constant pressure $P$)

$\Rightarrow$ $T^{\prime }$ = $T(1+\gamma \Delta T)$

$\Rightarrow$ $T^{\prime }$ = $T+T\gamma \Delta T$

$\Rightarrow$ $(T^{\prime }-T)$ = $T\gamma \Delta T$

$\Rightarrow$ $T\gamma \Delta T$ = $\Delta T$

$\Rightarrow$ $\gamma$ = $T^{-1}$.
Awesome! This suggests that, for an ideal gas at a temperature $T$, the volume expansion coefficient is just $T^{-1}$. See if you can do a similar analysis for the coefficient of pressure expansion at constant volume, in the earlier problem.