In thermodynamics as I understand entropy is a state function.
A state function is a property whose value does not depend on the path taken to reach that specific value. In contrast, functions that depend on the path from two values are called path functions. Both path and state functions are often encountered in thermodynamics.
Second Law of Thermodynamics (Variational statement of the second law of thermodynamics): : Stated in words, at equilibrium, any small change to the state of the system that induces a small change in the entropy while the internal energy and volume remain constant must lower the entropy of the system.
My problem is that since entropy is a state function, if the internal energy and volume remain constant then how can the entropy change, would this not contradict the definition of a state function above, in particular "property whose value does not depend on the path taken to reach that specific value..."?
Can anyone see where I am going wrong with my reasoning? Thanks.
Answer
The answer to your question is that entropy is a state function only at equilibrium.
Consider a system with two parts, A and B. Each part is in equilibrium. Then the entropy of part $A$ is $S_A=S(U_A,V_A,N_A)$ and the entropy fo part $B$ is $S_Β=S(U_Β,V_Β,N_Β)$.
Now consider the two parts as a single system. The entropy of the combined system is $$ S_{A+B} = S_A+S_Β=S(U_A,V_A,N_A)+S(U_Β,V_Β,N_Β) $$ In general, $S_{A+B}$ is not a state function of the combined system because it depends on the six variables, $$ U_A, U_B, V_A, V_B, N_A, N_B . $$ In the special case that $S_{A+B}$ happens to be a function of $$ U_A+U_B, V_A+V_B, N_A+ N_B $$ then entropy is indeed a state function for the combined system. It turns out that the conditions for this to happen are: $$ T_A = T_B, P_A=P_B, \mu_A = \mu_B $$ which are of course the equilibrium conditions.
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