For a monoatomic ideal gas, if ##c_v## is constant then the internal energy is worth ##U=\frac{3nRT}{2}##. This is a state equation.
From the fundamental equation ##S(U,V,n)= \frac {nS_0}{n_0}+nR \ln \left [ \left ( \frac {U}{U_0} \right ) ^{3/2} \left ( \frac{V}{V_0} \right ) \left ( \frac {n}{n_0} \right ) ^{-5/2} \right ]##, one could get ##U(S,V,n)## which would also be a fundamental equation.
But ##U(S,V,n)## is not equal to ##U(T)=\frac{3nRT}{2}##. In other words, U(S,V,n) is not the internal energy of the gas. Then what is it? The total energy? I thought that the total energy of the gas was the internal energy... I'm confused.
From the fundamental equation ##S(U,V,n)= \frac {nS_0}{n_0}+nR \ln \left [ \left ( \frac {U}{U_0} \right ) ^{3/2} \left ( \frac{V}{V_0} \right ) \left ( \frac {n}{n_0} \right ) ^{-5/2} \right ]##, one could get ##U(S,V,n)## which would also be a fundamental equation.
But ##U(S,V,n)## is not equal to ##U(T)=\frac{3nRT}{2}##. In other words, U(S,V,n) is not the internal energy of the gas. Then what is it? The total energy? I thought that the total energy of the gas was the internal energy... I'm confused.