# Name for thermodynamic derivative $dP/dT|_V$?

While trying to express the isoentropic sound speed as partial derivatives of $V$ and $T$ only I end up, as part of the longer expression with $dP/dT|_V$ (which according to a Maxwell relation is the same as $dS/dV|_T$).

Does such quantity $dP/dT|_V$ have a given name? or it is simply expressible as a combination to other quantities (like $V dP/dT|_V$) with a given name?

(By "given name" I mean, for example that $-V dP/dV|_T$ is the Isothermal Bulk Modulus)

• If has unity $[V]^{-1}$, so it's some specific density. The objects with names, like "Isothermal Bulk Modulus" (DYEL?) are often nondimensional quantities. It's some derivative divided by the canonical variable with the same units. Since you hold $V$ constant, I'd rather can ask for $V\left(\frac{\partial P}{\partial T}\right)_V$, which also happens to be $\left(\frac{\partial V\cdot P}{\partial T}\right)_V$, which is the temperature derivative of a characteristic energy. For ideal with $U=PV$ gas, it's the heat capactiy $C_T$. – Nikolaj-K Dec 17 '13 at 11:03
• @NickKidman, thank you, yes simply related quantities, including dimensionless, with given names, are also welcomed. I added that to the question. – alfC Dec 17 '13 at 11:06

$\left(\frac{\partial P}{\partial T}\right)_V=\frac{\alpha_V}{\beta_T},$
where the two constants are the thermal expansion (dimension $[T]^{-1}$) and the isothermal compressibility (dimension $[P]^{-1}$), respectively.
One derives at it as $|_V$ leads one to consider
$0\overset{!}{=}\mathrm dV(P,T) = \left(\frac{\partial V}{\partial P}\right)_{T}\mathrm dP+\left(\frac{\partial V}{\partial T}\right)_{P}\mathrm dT= (-V\cdot\beta_T)\ \mathrm dP+(V\cdot\alpha_V)\ \mathrm dT.$
Now divide this equation by $(-V\cdot\beta_T)\ \mathrm d T$.