21.1: Entropy Increases With Increasing Temperature
 Page ID
 14476
Entropy versus temperature
S&McQ 853 
We can put together the first and the second law for a reversible process with no other work than volume work and obtain:


 dU= δq_{rev} + δw_{rev}



 δq_{rev}= TdS
 δw_{rev}= PdV

Thus:


 dU= TdS PdV for reversible changes

This is a very interesting expression because we no longer have any path functions in it, as U, S and V are all state functions. This means this expression must be an exact differential.
(We can generalize the expression to hold for irreversible processes, but then it become an inequality


 dU≤ TdS  PdV

We will develop this later, because we still have an item on the wish list)
Natural variables
S&McQ 897 
Consider:


 dU= TdS PdV

This equality expresses U in two variables U(S,V). They play a special role and are called the natural variables of U.
Entropy and heat capacity
However, there is nothing to stop us from expressing U in other variables (see example 211), e.g. T and V. If fact we can derive some interesting relationships if we do:
S&McQ 854 
 We write U out in T and V
 We write U in its natural variables
 We rearrange 2) to find an expression for dS
 We substitute 1) into 3) and rearrange
 This is the definition of C_{v}
 We write out S in T and V
Comparing the three bottom formulas it becomes clear that
A less elegant expression for the partial derivative of S versus V is also found.
We can play the same game for our other state function H= U + PV


 As dH= dU +d(PV)= dU + PdV + VdP

we find for the reversible case


 dH = dU + PdV + VdP= TdS PdV + PdV + VP = TdS + VdP

The natural variables of the enthalpy are therefore S and P (not: V).
A similar derivation as above departing from the enthalpy shows that the temperature change of entropy is related to the constant pressure heat capacity:
This means that if we know the heat capacities as a function of temperature we can calculate how he entropy changes with temperature. Usually it is a lot easier to do this on basis of data that were obtained for P constant than for V constant, so that the route with Cp is the more common one.