Question about the first law of thermodynamics

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Discussion Overview

The discussion revolves around the first law of thermodynamics, specifically the relationships between internal energy, pressure, and heat capacities. Participants explore the mathematical derivations and implications of these relationships, questioning the validity of certain assumptions and equations within the context of thermodynamic principles.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant presents a derivation involving internal energy (U) as a function of volume (V) and temperature (T), leading to a comparison of expressions for dU, which raises questions about the validity of the results.
  • Another participant states the first law of thermodynamics as dU = TdS - pdV and discusses changing independent parameters, suggesting a different approach to the derivation.
  • A third participant agrees with the correct expression for dU but questions the specific error in the initial derivation, proposing that the confusion may stem from treating C as a path function rather than a state function.
  • Another participant introduces the Helmholtz free energy (F) and its relation to the first law, suggesting an alternative perspective on the equations presented.

Areas of Agreement / Disagreement

Participants express differing views on the validity of certain derivations and the treatment of state versus path functions. There is no consensus on the specific errors in the initial derivation or the implications of the equations discussed.

Contextual Notes

Participants highlight the distinction between state functions and path functions, indicating that this may affect the validity of certain derivations. The discussion also touches on the implications of changing independent variables in thermodynamic equations.

Who May Find This Useful

This discussion may be of interest to students and professionals in thermodynamics, particularly those exploring the nuances of the first law and the relationships between different thermodynamic variables.

RedCliff
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In thermodynamics, the internal energy (U) is the function of the volume (V) and temperature (T), U = U (T).
Therefore, according to the mathematics, dU = π dV + Cv dT, where π is internal pressure and Cv is the constant volume heat capacity.
Meanwhile, according to the 1st law, dU = δ w + δq, when under reversible process, dU = - p dV + C dT.
If we compare both, dU = π dV + Cv dT = - p dV + C dT. Since both V and T are free variables, we can get,
π = -p and Cv = C.
Obvious this is not right result. Based on Maxwell equation:
π = T (∂p/∂T)V -p
Which part is wrong in my derivation?

Thanks in advance!
 
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Hi.
1st law of thermodynamics is
dU=TdS-pdV
changing independent parameter pair from (S,V) to (T,S)
dU=TdS-pdV=T(\frac{\partial S}{\partial T})_V dT + T(\frac{\partial S}{\partial V})_T dV - pdV
 
Hello,
Thank you for your replying! I fully agree with you on this point. If we express dU with all state functions, that is, dU = TdS - pdV, yes, your derivation is correct.

But my question is: what part is wrong in my above derivation as I use dU = -pdV + CdT?

However, you post might provide me a clue to the answer to my question:
I guess it is because C is not a state function but a path function. I remember I once read a paragraph saying the derivation/integration related to path functions are different from those related to state functions. In science, we mostly deal with state functions. Therefore, from dU = π dV + Cv dT = -pdV + C dT, we cannot conclude that
π = -p and Cv = C.
 
Hi.
Referring to the eauation of my post, the equation you used is correct in case
T(\frac{\partial S}{\partial V})_T=0
I do not think this stands. We should have double energy or entropy for double volume or two similar systems regarded as united one.You may be interested in Helmholtz free energy F=U-TS.
Introducing F, the first law of thermodynanmics is
dF=dU-TdS-SdT=-pdV-SdT
Is the RHS the form you expect?
 
Last edited:

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