# Is $TdS = dU + PdV$ true at very high strain rates? (Thermodynamic equilibrium)

I have read some reports of scientists who are interested in equations of state for unreacted explosives. They want these equations of state to describe the mechanical response of an explosive as it is compressed to high pressure by the detonation shock wave, prior to ignition. Some scientists, e.g. Menikoff have developed "complete" equations of state for these materials, by using the fact that:

$$TdS = dU + PdV \tag{a}$$

See equation (1)in the report below:

Of course a consequence of (a), above is that since $$dS$$ must be an exact differential we end up with the following PDE in T:
$$\frac{\partial}{\partial V} \left ( \frac{1}{T} \right )_{U}-\frac{\partial}{\partial U} \left ( \frac{P}{T} \right )_{V}=0$$