Solve Exact Differentials: Find G for dG = Vdp-Sdt

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Homework Help Overview

The discussion revolves around finding a function \( G \) such that \( \mathrm{d}G = V \mathrm{d}p - S \mathrm{d}t \), given the relationship \( \mathrm{d}U = T \mathrm{d}S - p \mathrm{d}V \). Participants are exploring the connections between these differential forms in the context of thermodynamics.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants are questioning how the two differential expressions are related and are attempting to manipulate the given equations to find a suitable expression for \( G \). There is a suggestion to consider the function \( U \) and how it might relate to \( G \) through addition or subtraction of terms.

Discussion Status

The discussion is ongoing, with participants actively exploring different manipulations of the differential forms. Some guidance has been offered regarding the expression for \( U \) and its potential role in deriving \( G \).

Contextual Notes

There is a noted uncertainty about the relationship between the differentials and how to approach the problem, indicating a need for further clarification on the underlying concepts.

Froskoy
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Homework Statement


Given that [tex]\mathrm{d}U = T\mathrm{d}S - p\mathrm{d}V[/tex]

find a function [itex]G[/itex] such that [tex]\mathrm{d}G = V \mathrm{d} p - S \mathrm{d} t[/tex].

I'm not sure where to start - how are the two related? Could someone please give me a clue of how to start this off?

3. Attempt at the solution
I was thinking this looks too much like the quotient rule to be a coincidence...

With very many thanks,

Froskoy.
 
Last edited:
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[tex]dG = VdP - SdT[/tex]

adding a few terms gives
[tex]dG = (VdP - SdT) + (VdP-VdP)+(TdS-TdS)[/tex]

rearranging
[tex]dG = (VdP+PdV)-(SdT-TdS)- (PdV-TdS)[/tex]
 
lanedance said:
[tex]dG = VdP - SdT[/tex]

adding a few terms gives
[tex]dG = (VdP - SdT) + (VdP-VdP)+(TdS-TdS)[/tex]

rearranging
[tex]dG = (VdP+PdV)-(SdT-TdS)- (PdV-TdS)[/tex]

So, what is the function G?
 
Do you know an expression for the function U in terms of T,S,P,V?

If so, think about what you could add or subtract to U in order to get the differentials to work for G.
 

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