How to derive 2nd law extremum principles for U, A, G, and H?

[zenterix]

Homework Statement

In following the book "Physical Chemistry" by Silbey, Alberty, and Bawendi, I did not understand how they derived conditions on differentials of $U, H, A$, and $G$ that lead to extremum principles on certain very specific processes.

Relevant Equations

In what follows, I will show the desired results and also what I came up with to justify the results. My question is if my calculations are correct.

For the internal energy function $U(S,V,\{n_i\})$ we have

$$dU=TdS-pdV+\sum\limits_{i=1}^{N_s}\mu_id n_i\tag{1}$$

where $N_s$ is the number of species in the system.

We also have

$$dU=\delta q+\delta w\tag{2}$$

by the 1st law of thermodynamics. I am using $\delta$ to denote an inexact differential.

The 2nd law of thermodynamics tells us that

$$dS\geq \frac{\delta q}{T}\tag{3}$$

Suppose we have a system at constant $S, V$, and $\{n_i\}$.

Then, somehow, using (1), (2), and (3) we can conclude that

$$(dU)_{S,V,\{n_i\}}\leq 0\tag{4}$$

Why can we conclude (4)?

What I came up with is

$$dU=\delta q+\delta w \leq TdS-pdV+\sum\limits_{i=1}^{N_s}\mu_idn_i$$

and so for $dS=dV=dn_i=0$ we reach $dU\leq 0$.

However, I am not sure this is correct.

For example, consider Helmholtz energy

$$dA=dU-TdS-SdT$$

$$=-SdT-PdV+\sum\limits_{i=1}^{N_s}\mu_i dn_i$$

Somehow, we should be able to conclude that

$$(dA)_{T,V,\{n_i\}}\leq 0$$

for a spontaneous change at fixed $T,V$, and $\{n_i\}$.

What I came up with is

$$dA=\delta q+\delta w-TdS-SdT$$

$$=(\delta q-TdS)-PdV-SdT+\sum\mu_i dn_i$$

and for fixed $T,V$, $n_i$ we have

$$(dA)_{T,V,\{n_i\}}=\delta q-TdS\leq 0$$

The analogous calculations for Gibbs energy are

$$dG=\delta q+\delta w-TdS-SdT+PdV+VdP$$

$$=\delta q +(-PdV+\sum \mu_i dn_i)-TdS-SdT+PdV+VdP$$

$$=(\delta q-TdS)-SdT+VdP+\sum\mu_i dn_i$$

and since $\delta q\leq TdS$ we have that for constant $T,P$, and $\{n_i\}$ we have

$$(dG)_{T,P,\{n_i\}}=\delta q-TdS\leq 0$$

The analogous calculations for enthalpy are

$$dH=dU+PdV+VdP$$

$$=\delta q+\delta w+PdV+VdP$$

$$=\delta q+VdP+\sum\mu_i dn_i$$

At fixed $S,P$, and $\{n_i\}$ we have

$$(dH)_{S,P,\{n_i\}}=\delta q\leq TdS=0$$

My question boils down to if these calculations are correct?

I kinda think it is although for me at this point this is just an exercise in algebra, I don't see the big picture of these calculations very well.

I mean, each result means that for the spontaneous process in question we have a condition on the relevant differential of a thermodynamic potential. In all cases, the result is that at equilibrium we have the minimum of a thermodynamic potential.

These are extremum principles associated with equilibrium after each specific type of process.

 

[DrDu]

Obviously with dS=dV=dn_i=0, from (1), dU=0 and never dU<0. If I remember correctly, the point is that in deriving these extremal principles, you consider situations where either there are other work sources than -pdV, e.g. the aforementioned stirrer, or situations where e.g. p and T are not the values in the system, but those of the surrounding.