In my course they're using the equality [itex]U = \frac{p}{\alpha \rho}[/itex] with alpha some constant (U = internal energy per mass, p = pressure, rho = density). They explicitly derive it for an ideal gas yet later apply it to a liquid (in the context of deriving the Navier-Stokes energy equation). Seems pretty unfounded... However, is there perhaps a reason we should expect such an equation to hold in more general cases?
NB: to see it follows from the ideal gas law, note that [itex]p = \rho \beta T[/itex] for some constant beta, and that [itex]U = \gamma T[/itex] (note that U is energy per mass, i.e. up to a constant energy per particle [itex]\propto k_B T[/itex])
NB: to see it follows from the ideal gas law, note that [itex]p = \rho \beta T[/itex] for some constant beta, and that [itex]U = \gamma T[/itex] (note that U is energy per mass, i.e. up to a constant energy per particle [itex]\propto k_B T[/itex])