2.3.2 Definition of often-used thermodynamic coefficients

Definition of specific heats:

$$c_p \equiv \left(\frac{\partial h}{\partial T}\right)_{p} = ... ...ight)_{p} = \left(\frac{\partial s}{\partial \ln T}\right)_{p}$$ (23)

$$c_v \equiv \left(\frac{\partial e}{\partial T}\right)_{\rho}... ...{\rho} = \left(\frac{\partial s}{\partial \ln T}\right)_{\rho}$$ (24)

Definitions of further thermodynamic coefficients:

$$\chi_{T} \equiv \left(\frac{\partial \ln p}{\partial \ln T}\right)_{\rho}$$ (25)

$$\chi_{\rho} \equiv \left(\frac{\partial \ln p}{\partial \ln \rho}\right)_{T} \equiv (K p)^{-1}$$ (26)

$$\delta \equiv -\left(\frac{\partial \ln \rho}{\partial \ln T}\right)_{p} \equiv \alpha T = \frac{\chi_{T}}{\chi_{\rho}}$$ (27)

It can be shown that

$$c_p - c_v = \alpha^2 T / (K \rho) = \frac{p}{\rho T} \delt... ...\delta \chi_{T} = \frac{p}{\rho T} \chi_{T}^2/\chi_{\rho}$$ (28)

Definition of adiabatic exponents:

$$\Gamma_1 \equiv \left(\frac{\partial \ln p}{\partial \ln \rho}\right)_{s}$$ (29)

$$\Gamma_3 \equiv \left(\frac{\partial \ln T}{\partial \ln \rho}\right)_{s} + 1$$ (30)

$$\nabla_{\rm ad} \equiv \left(\frac{\partial \ln T}{\partial \ln p}\right)_{s} \equiv \frac{\Gamma_2-1}{\Gamma_2}$$ (31)