# Measurement of $C_p$ and $C_v$

I have read somewhere but wanted to confirm that how and why $$C_p$$ is measured experimentally but $$C_v$$ is measured theoretically only. Or there is a way where we really actually measure $$C_{v}$$ too experimentally.

Measurement of the constant-pressure heat capacity $$C_P$$ is relatively straightforward; one operates in the familiar environment of constant pressure, and one minimizes heating losses (with a low-stiffness, low-thermal-conductivity container, for instance—perhaps a foam or vacuum) to ensure that all heating is isolated to the sample; then, the ratio of heating to the measured temperature increase provides $$C_P$$.

Measurement of the constant-volume heat capacity $$C_V$$ is more complex; one must employ some confining mechanical mechanism to avoid thermal expansion or (less often) thermal contraction, and this mechanism, with its necessary high stiffness (e.g., metal or ceramic containment), probably conducts heat relatively well and requires constant careful temperature matching to avoid heating loss that would degrade the measurement accuracy.

Fortunately, $$C_V$$ can be obtained from $$C_P$$ from thermodynamic arguments by

$$C_V=C_P-VT\frac{\alpha^2}{\beta_T},$$

with constant-pressure thermal expansion coefficient $$\alpha$$ and constant-temperature compressibility $$\beta_T$$, and these are also relatively easily obtained—we live in an environment of nearly constant pressure and temperature.

As a side note, $$\alpha$$ is small for condensed matter, so the approximation $$C_V\approx C_P$$ is frequently employed.

Note that (in contrast to this answer) we always obtain $$C_P>C_V$$, even for materials that contract with heating, because the thermal expansion coefficient is squared and because the temperature, volume, and compressibility are always positive.