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

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In summary, the equation on the line above "L.H.S bracket is not a function of V thus" is used to derive an equation for dS, which is an exact differential because there are no other sources of variation in S.
  • #1
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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  • #2
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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  • #3
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.
 
  • #4
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.
 
  • #5
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.
 
  • #6
That's how I see it, but talk to other people about this, it might be important.
 

What is the process for deriving an equation?

The process for deriving an equation involves using logical reasoning and mathematical principles to arrive at a mathematical expression that describes a relationship between variables. This process may involve experimentation, data collection, and analysis.

What is the purpose of deriving an equation?

The purpose of deriving an equation is to describe a relationship between variables in a concise and mathematical form. This allows scientists to make predictions, test hypotheses, and gain a better understanding of the phenomenon being studied.

How do scientists ensure the accuracy of a derived equation?

Scientists ensure the accuracy of a derived equation by following the scientific method, which includes making observations, conducting experiments, and analyzing data. Additionally, equations are often peer-reviewed by other scientists to ensure the validity of the results.

Can an equation be derived from any type of data?

Yes, equations can be derived from any type of data, as long as there is a clear relationship between the variables being studied. However, the process of deriving an equation may vary depending on the type of data and the complexity of the relationship between variables.

How do scientists know if an equation is valid?

Scientists determine the validity of an equation by testing it against experimental data and comparing the predicted results to the actual results. If the equation accurately describes the relationship between variables and predicts future outcomes, it is considered valid.

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