Discussion
In each of the following choices, there are two expressions given. Select the choice that gives, first, the defining expression of volume expansivity and second, the expression of volume expansivity for ideal gases
Answer: A
The correct pair is \(\dfrac{1}{v}\left(\dfrac{\partial v}{\partial T}\right)_{P}\) and \(\dfrac{1}{T}\).
Volume expansivity \(\beta\) is the fractional change in volume per unit temperature rise at constant pressure. For an ideal gas \(v = RT/P\), so
\[ \begin{aligned} \beta &= \frac{1}{v}\left(\frac{\partial v}{\partial T}\right)_{P} \\ &= \frac{1}{v}\cdot\frac{R}{P} = \frac{1}{T} \end{aligned} \]
\(-\frac{1}{v}(\partial v/\partial P)_{T}\) is the isothermal compressibility, which equals \(1/P\) for an ideal gas; the options built on it describe compressibility, not expansivity. The option with \(1/P\) as the ideal-gas expansivity mixes the two.
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