1
$\begingroup$

I'm going through a fluid dynamics text and I'm a little confused by a listed relation. The first adiabatic index is defined as

$$ \Gamma_1 = \left(\frac{\partial \ln P}{\partial \ln \rho}\right)_s, $$

where $P$ is the pressure, $\rho$ is the density, and $S$ is the entropy of the system. It is stated that $\Gamma_1$ is related to the ratio of specific heat capacities by

$$ \Gamma_1 = \chi_\rho \gamma = \left(\frac{\partial \ln P}{\partial \ln \rho}\right)_T \frac{T\left(\frac{\partial S}{\partial T}\right)_P}{T\left(\frac{\partial S}{\partial T}\right)_V}. $$

I've been going through the various Maxwell relations and different applications of Schwarz's theorem, but I can't seem to work out why this is true. Can someone please explain how this equality stands?

$\endgroup$

1 Answer 1

1
$\begingroup$

From the density definition $\rho\equiv m/V$, a constant-mass assumption, and the triple product rule:

\begin{align}\frac{\left.\dfrac{\partial \ln P}{\partial\ln\rho}\right|_S}{\left.\dfrac{\partial \ln P}{\partial\ln\rho}\right|_T}=\frac{\dfrac{\rho}{P}\left.\dfrac{\partial P}{\partial \rho}\right|_S}{\dfrac{\rho}{P}\left.\dfrac{\partial P}{\partial \rho}\right|_T}=\frac{\left.\dfrac{\partial P}{(-m/V^2)\partial V}\right|_S}{\left.\dfrac{\partial P}{(-m/V^2)\partial V}\right|_T}=\frac{-\left(\dfrac{\partial V}{\partial S}\right)^{-1}_P\left(\dfrac{\partial S}{\partial P}\right)^{-1}_V}{-\left(\dfrac{\partial V}{\partial T}\right)^{-1}_P\left(\dfrac{\partial T}{\partial P}\right)^{-1}_V}=\frac{\left.\dfrac{\partial S}{\partial T}\right|_P}{\left.\dfrac{\partial S}{\partial T}\right|_V}=\frac{c_P}{c_V}.\end{align}

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.