From my past studies I have had the presumption that for an ideal gas the equation:
( where $du$ is the density of internal energy, $c_v$ is the specific heat capacity at constant volume and $T$ is temperature)
is only valid when the process is isentropic. However searching the internet I'm doubting if this is really true. For example the wikipedia entry for the subject and this MIT blogpost, mention the equation without any assumptions such as being reversible, adiabatic or isentropic. So my question is if the equation above is valid all the time or just unders certain assumptions? if later then what are those assumptions and what is the general form? and if former is the case how we can prove that?