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\end{equation}  To evaluate the last term on the RHS let's assum we are dealing with a monatomic ideal gas. Then  \begin{equation}  P = \left(\gamma - 1\right)\epsilon \rho = RT  \rho. \end{equation}  where R is the ideal gas constant.  The first law of thermodynamics says that the change in internal energy $\epsilon$ plus the work done on the gas is equal to the heat added,  \begin{equation}  dQ = d\epsilon + PdV. 
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For an ideal gas  \begin{align}  d\epsilon &= \frac{R}{\gamma -1}dT \\  PdV &= -\left(\gamma -1\right)\epsilon \frac{d \rho}{\rho} = RT  \end{align}