Question

for a small positive coefficient of expansion in case of solid : then comment on Cp and Cv

Answer 1

Cp is slightly greater than Cv

Answer 2

For a solid, the relationship between the molar heat capacities at constant pressure ($C_p$) and constant volume ($C_v$) is given by the thermodynamic relation:

$C_p - C_v = \frac{\alpha^2 V T}{\beta_T}$

where: $\alpha$ is the coefficient of volume expansion. $V$ is the molar volume. $T$ is the absolute temperature. $\beta_T$ is the isothermal compressibility, defined as $\beta_T = -\frac{1}{V} \left(\frac{\partial V}{\partial P}\right)_T$.

For a typical solid at a positive absolute temperature $T > 0$, the molar volume $V > 0$, and the isothermal compressibility $\beta_T > 0$ (solids are compressible, and increasing pressure decreases volume, so $(\partial V / \partial P)_T$ is negative).

The question states that the solid has a small positive coefficient of expansion. This means $\alpha$ is positive ($\alpha > 0$) and small. Since $\alpha$ is positive, $\alpha^2$ is also positive ($\alpha^2 > 0$). Given that $V > 0$, $T > 0$, and $\beta_T > 0$, the term $\frac{\alpha^2 V T}{\beta_T}$ is positive. Therefore, $C_p - C_v > 0$, which implies $C_p > C_v$.

Furthermore, the question states that the coefficient of expansion $\alpha$ is small. The difference $C_p - C_v$ is proportional to $\alpha^2$. If $\alpha$ is small, then $\alpha^2$ is very small. Thus, the difference $C_p - C_v$ is small.

Combining these two points, $C_p$ is greater than $C_v$, and the difference is small. So, $C_p$ is slightly greater than $C_v$.

This is consistent with the physical understanding that the difference $C_p -$C_v arises from the work done during expansion against the external pressure when heat is supplied at constant pressure. For solids, the expansion upon heating is generally small compared to gases, leading to a smaller difference between $C_p$ and $C_v$. If the coefficient of expansion is explicitly stated to be small, this difference will be particularly small.

The comment on $C_p$ and $C_v$ is that $C_p$ is slightly greater than $C_v$.