# 21.1: Entropy Increases With Increasing Temperature

## 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= δqrev + δwrev
δqrev= TdS
δwrev= -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 21-1), e.g. T and V. If fact we can derive some interesting relationships if we do:

 S&McQ 854 1. We write U out in T and V
2. We write U in its natural variables
3. We rearrange 2) to find an expression for dS
4. We substitute 1) into 3) and rearrange
5. This is the definition of Cv
6. 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.