What is the derivation of the equation for (d/dv)(ds) in lecture notes?

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Homework Help Overview

The discussion revolves around the derivation of an equation related to the differential of entropy (dS) in the context of thermodynamics, specifically focusing on its dependence on volume (V) and temperature (T) as presented in lecture notes.

Discussion Character

  • Conceptual clarification, Assumption checking, Exploratory

Approaches and Questions Raised

  • Participants explore the conditions under which dS is an exact differential and the implications of this for physical systems. There are attempts to clarify the reasoning behind the derivation and the necessity of certain conditions for dS to be considered an exact differential.

Discussion Status

Some participants express confusion regarding the conditions for dS to be an exact differential and the underlying physical reasoning. Others suggest that the relationship between the terms in the total differential is crucial but may require further exploration. Guidance is offered to accept certain theoretical aspects for the time being while recognizing the need for deeper understanding.

Contextual Notes

There is mention of specific constraints related to the ideal gas assumption and the fixed temperature condition affecting the differentiation process. The discussion also highlights the importance of understanding the mathematical properties of state functions in thermodynamics.

pivoxa15
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In the lecture notes
http://webraft.its.unimelb.edu.au/640322/pub/notes/lectures_common/lecture3.pdf

How did they derive the equation on the line above "L.H.S bracket is not a function of V thus"?

I can see that (d/dv)(ds)= "that equation except with the 'equals sign' replaced by '+' "

(dU/dV)=0 because it is an ideal gas.
 
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The expression for dS has three terms so we can write it as

dS = AdT + BdV + CdV

The condition for dS being an exact differential is

d(A)/dV = d(B+C)/dT ( partial differentiation)

But dB/dT is zero because B is held at fixed T, so

d(A)/dV = d(C)/dT

which is the equation you asked about.
 
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I don't understand why the condition on dS in order for it to be an exact differential. The reasoning for the exact differential on the notes I don't follow either.
 
I'm not clear what's troubling you.

dS has to be an exact differential for physical reasons. It just means there are no other sources of variation in S but these.

The little equation in brackets ( 'dz = ...') is a theorem, which says that if a total differential is the sum of two sources, then the sources have a special relationship. Just accept it for now. It's used to get the next line.
 
Mentz114 said:
I'm not clear what's troubling you.

dS has to be an exact differential for physical reasons. It just means there are no other sources of variation in S but these.

The little equation in brackets ( 'dz = ...') is a theorem, which says that if a total differential is the sum of two sources, then the sources have a special relationship. Just accept it for now. It's used to get the next line.


I see now but I seem to be more interested to see why the exact differential demands this form. You want dS to be an exact differential because S is a function of two variables and is a state function which behaves just like a perfect mathematical function of two variables.
 
That's how I see it, but talk to other people about this, it might be important.
 

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