Availability at fixed pressure and temperature

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Discussion Overview

The discussion revolves around the concept of availability in thermodynamics, specifically focusing on the relationships between differentials of thermodynamic potentials under fixed pressure and temperature conditions. Participants explore the implications of these relationships, particularly in the context of reversible and irreversible processes.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant questions the validity of the expression ##dA=dH## at constant pressure and entropy, suggesting confusion over the terms used and their implications.
  • Another participant asserts that at fixed pressure, ##P = P_0##, which is reiterated by multiple contributors.
  • Concerns are raised about the expression ##dG=0## under fixed pressure and temperature, with participants debating the conditions under which this holds true.
  • A participant proposes that the equality ##P = P_0## and ##T = T_0## may only be valid for reversible processes, indicating uncertainty about this assertion.
  • One participant presents a mathematical argument involving the correction term in the expression for ##dG##, suggesting that ##T## may not always equal ##T_0## even at constant temperature.
  • Several participants inquire about the significance of the term "availability" in the context of the discussion, indicating a need for clarification on terminology.
  • A participant expresses disagreement with the approach of using differentials for irreversible processes, suggesting an alternative reference for understanding the changes in availability and Gibbs free energy.

Areas of Agreement / Disagreement

Participants exhibit a mix of agreement on certain definitions, such as the relationship between pressure and its fixed counterpart, while disagreements persist regarding the implications of these relationships, particularly in irreversible processes. The discussion remains unresolved with multiple competing views on the validity of the expressions and their applications.

Contextual Notes

There are limitations regarding the assumptions made about reversibility and the definitions of terms used in the discussion. Some mathematical steps and the conditions under which certain expressions hold true are not fully resolved.

laser1
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1730534149507.png

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:
1730534394097.png

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
 
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At fixed pressure, P=Po
 
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Chestermiller said:
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##?
 
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!
 
See Chater 7 of Fundamentals of Engineering Thermodynamics by Moran et al, on Exergy.
 
Chestermiller said:
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,
$$dG = dU + P_0dV-T_0dS$$
 
What are you trying to prove?
 
Chestermiller said:
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.
 
laser1 said:
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?
 
  • #10
Chestermiller said:
What is the significance of the word "availability" in your thread title?
I have attached the definition of it from Blundell and Blundell
1730893136297.png
 
  • #11
laser1 said:
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.
 
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