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

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The discussion focuses on deriving extremum principles for internal energy (U), Helmholtz energy (A), Gibbs energy (G), and enthalpy (H) based on thermodynamic laws. It establishes that under specific conditions, such as constant entropy, volume, and number of species, the change in internal energy (dU) must be less than or equal to zero, indicating a tendency towards equilibrium. Similar reasoning applies to Helmholtz and Gibbs energies, where their differentials also suggest that spontaneous processes lead to minimized thermodynamic potentials. The calculations presented aim to confirm these principles, emphasizing that at equilibrium, the system's thermodynamic potentials reach extremum values. The overall conclusion is that these results reflect the conditions for spontaneous processes and equilibrium in thermodynamic systems.
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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.
 
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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.
 
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