I Availability at fixed pressure and temperature

[laser1]

We define $dA=dU+P_0dV-T_0dS \leq 0$. In my notes it says if you fix pressure and entropy, $dA=dH$. I don't get this, because at constant T and S, I get $dA=dU+P_0V$. It seems that somehow, $P_0=P$. Is this correct, or am I missing something?

Second question about this:

If $T_0=T$ and $P_0=P$ then $dG=0$ which is probably false. But the above image confuses me also. Why can we write $dH$ as $dU+P_0dV$ rather than $dU+PdV$? Thanks

 

[Chestermiller]

At fixed pressure, P=Po

 

[laser1]

At fixed pressure, P=Po

Cool, thanks, that explains the first part. What about the second? At fixed pressure, P=Po. At fixed temperature, T=To. This means that dG = 0, which doesn't make sense, because why not just write that rather than $dU-T_0dS+P_0dV=dG$?

 

[laser1]

After thinking for a bit, can I say that P=Po and T=To are only true when processes are reversible, i.e., when at constant entropy? I am not sure why that would be correct, but it would resolve all my problems in my OP!

 

[Chestermiller]

See Chater 7 of Fundamentals of Engineering Thermodynamics by Moran et al, on Exergy.

 

[laser1]

See Chater 7 of Fundamentals of Engineering Thermodynamics by Moran et al, on Exergy.

Cheers. I have looked briefly at it, but is my following argument below correct?

$P = P_o$ always at fixed $P$.
$T$ is not always $T_0$ at constant $T$. Counterexample being ice melting. $T$ is not $T_o$ yet $T$ is constant.

$G=H-TS$
$dG = dU + PdV - TdS$ at constant $T$ and $P$. However, there is a correction term in $dG$, which is $(T-To)dS$.

Hence,
$$dG = dU + PdV - TdS + (T-To)dS$$
and substituting $P = P_0$ always,
I get the equation in my textbook,
$$dG = dU + P_0dV-T_0dS$$

 

[Chestermiller]

What are you trying to prove?

 

[laser1]

What are you trying to prove?

In the post #1 in the image after "Second question about this:" it has an expression for $dG$ which is $$dG=dU-T_0dS+P_0dV$$ I want to prove that.

 

[Chestermiller]

In the post #1 in the image after "Second question about this:" it has an expression for $dG$ which is $$dG=dU-T_0dS+P_0dV$$ I want to prove that.

What is the significance of the word "availability" in your thread title?

 

[laser1]

What is the significance of the word "availability" in your thread title?

I have attached the definition of it from Blundell and Blundell

 

[Chestermiller]

I have attached the definition of it from Blundell and Blundell
View attachment 353194

I'm sorry. I can't help you with this. I totally disagree with the author's approach to teaching this material. I strongly oppose the use of differentials for a system that undergoes an irreversible process. If you want to get a better picture of how A and G change under certain well-specified constraints on T and P, see Chapter 1 of Principles of Chemical Equilibrium by Denbigh.