Is there a generic relationship between internal energy and entropy? For instance, I would like to compute the entropy $S$, given that the internal energy is expressed as:
$$ E(V, T)=\sigma V T^{\frac{\alpha+1}{\alpha}} $$
I feel like there has to be a relationship between these two properties but I am unable to find it.
 A: It probably isn't possible to find the function $S(T,V)$, because entropy is integral of $dQ/T$ and from the given function $E(T,V)$ it isn't possible to find out how $dQ$ depends on $dV$ and $dT$. In other words, the given function $E$ isn't enough information about the system.
It is possible to express function $S(T,V)$ by integrating $dQ/T$ for a process where $V$ is constant:
$$
S(T,V) = S(T_0,V) + \int_{T_0}^T\frac{dE}{T}
$$
The integral is a known function of $T,T_0,V$, but the term $S(T_0,V)$ is unknown function of $V$ and not findable from the given information.
Some standard scenarios which allow computing entropy function of other variables are:

*

*energy depends only on temperature, not volume (ideal gas), plus we assumed entropy $S(kU,kV,kN) = kS(U,V,N)$; then one can express $dQ$ as function of $dT$ and calculate $S(T,V,N)$ and then $S(U,V,N)$;


*we have microscopic model of the system detailed enough that we can define $S(E,V)$ by $k_B\log W(E,V)$, where $W$ is measure of number of microscopic states that can realize macroscopic state $E,V$;


*or alternatively, for given $V,T$ we can calculate partition function $$
Z = \sum_k e^{-E_k/k_BT}
$$
where sum runs over all relevant microscopic states for macrostate $V,T$ and then define entropy as
$$
S = k_B\log Z + \frac{E}{T}.
$$
A: The answer is yes. There is a general relationship between energy and entropy. However, it is not like in your example.
The reason why your example does not work has been discussed extensively in another answer. Briefly, from an explicit expression for energy as a function of $T$ and $V$ it is not possible to reconstruct the entropy. The formal reason is that a relation like $E=E(V,T)$ can be seen as a partial differential equation for the entropy $S$ by recalling that $\frac1T=\left.\frac{\partial{S}}{\partial{E}}\right|_{V}$. Therefore the best one can hope to obtain without additional information is $S$ within an arbitrary function of $V$.
What can be positively said is that in general there is an invertible relation between entropy and energy ($S=S(E,V,N)$ or $E=E(S,V,N)$).
A direct approach to finding it, using statistical mechanics, is to use the microcanonical ensemble, where entropy is obtained from the Boltzmann relation
$$
S(E,V,N)=k_B \log \Omega(E,V,N),
$$
where $\Omega$ represents the number of microscopic states consistent with the macroscopic state at fixed $E$, $V$, and $N$.
