Physlib.Thermodynamics.IdealGas.Basic

The entropy $S$ of a monophase ideal gas is defined as a function of the internal energy $U$, the volume $V$, and the amount of substance $N$. Given the heat capacity parameter $c$, the gas constant $R$, and reference values for entropy $s_0$, internal energy $U_0$, volume $V_0$, and amount of substance $N_0$, the entropy is given by: \[ S(U, V, N) = N s_0 + N R \left( c \ln \frac{U}{U_0} + \ln \frac{V}{V_0} - (c+1) \ln \frac{N}{N_0} \right) \] where $\ln$ denotes the natural logarithm.

Let $s_0, U_0, V_0, N_0, c, R$ be parameters for a monophase ideal gas, and let $U_a, U_b, V_a, V_b, N$ be real numbers. Suppose that $U_a, U_b, V_a, V_b, N, U_0, V_0, R > 0$. If the entropy $S$ is equal for the states $(U_a, V_a, N)$ and $(U_b, V_b, N)$, such that \[ S(U_a, V_a, N) = S(U_b, V_b, N) \] then the following logarithmic adiabatic relation holds: \[ c \ln \left( \frac{U_a}{U_b} \right) + \ln \left( \frac{V_a}{V_b} \right) = 0 \] where $\ln$ denotes the natural logarithm.

Let $s_0, U_0, V_0, N_0, c, R$ be the parameters for a monophase ideal gas, where $U_0, V_0, R > 0$. Let $U_a, U_b, V_a, V_b$ and $N$ be positive real numbers representing the internal energies and volumes of two states $a$ and $b$ at a fixed amount of substance $N$. If the entropy $S$ is equal for both states, such that \[ S(U_a, V_a, N) = S(U_b, V_b, N) \] where the entropy function is defined as \[ S(U, V, N) = N s_0 + N R \left( c \ln \frac{U}{U_0} + \ln \frac{V}{V_0} - (c+1) \ln \frac{N}{N_0} \right) \] then the following adiabatic relation in product form holds: \[ \left( \frac{U_a}{U_b} \right)^c \left( \frac{V_a}{V_b} \right) = 1 \]