Availability at fixed pressure and temperature

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SUMMARY

The discussion centers on the thermodynamic relationships involving availability, specifically the equations for differential changes in availability (dA), enthalpy (dH), and Gibbs free energy (dG) under fixed pressure and temperature conditions. The participants clarify that at constant pressure (P = P0) and temperature (T = T0), the expression for dG simplifies to dG = dU + P0dV - T0dS. They also emphasize that while P is always equal to P0 at fixed pressure, T may not always equal T0, particularly in irreversible processes. The significance of these equations is further explored in the context of reversible processes and the implications for thermodynamic systems.

PREREQUISITES
  • Understanding of thermodynamic principles, particularly availability and Gibbs free energy.
  • Familiarity with differential calculus as applied to thermodynamic equations.
  • Knowledge of the concepts of reversible and irreversible processes in thermodynamics.
  • Access to "Fundamentals of Engineering Thermodynamics" by Moran et al. for reference.
NEXT STEPS
  • Study the derivation of the Gibbs free energy equation, focusing on the terms involved at constant temperature and pressure.
  • Examine Chapter 1 of "Principles of Chemical Equilibrium" by Denbigh for insights on thermodynamic changes in irreversible processes.
  • Explore the concept of availability in thermodynamics as defined by Blundell and Blundell.
  • Review the implications of entropy changes in thermodynamic systems, particularly during phase transitions like melting.
USEFUL FOR

This discussion is beneficial for students and professionals in thermodynamics, particularly those studying chemical engineering, mechanical engineering, or physical chemistry, who seek a deeper understanding of thermodynamic potentials and their applications in real-world systems.

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?
 
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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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