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Basic Physics for IIT JAM

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The Maxwell Relations

Modeling the dependence of the Gibbs and Helmholtz functions behave with varying temperature, pressure, and volume is fundamentally useful. But in order to do that, a little bit more development is necessary. To see the power and utility of these functions, it is useful to combine the First and Second Laws into a single mathematical statement. In order to do that, one notes that since

 Physics Notes | EduRev                                           (1)
for a reversible change, it follows that 
dq = Tds                                                 (2)
And since
dw = Tds - pdV                                     (3)
for a reversible expansion in which only p-V works is done, it also follows that (since dU = dq + dw):

dU = Tds - pdV                                     (4)
This is an extraordinarily powerful result. This differential for dU can be used to simplify the differentials for H, A, and G. But even more useful are the constraints it places on the variables T, S, p, and V due to the mathematics of exact differentials!
Maxwell Relations
The above result suggests that the natural variables of internal energy are S and V (or the function can be considered as U(S,V)). So the total differential (dU) can be expressed:
 Physics Notes | EduRev                         (5)
Also, by inspection (comparing the two expressions for dU) it is apparent that:
 Physics Notes | EduRev                                                      (6)
and  Physics Notes | EduRev                                             (7)
But the value doesn’t stop there! Since dU is an exact differential, the Euler relation must hold that
 Physics Notes | EduRev                  (8)
or  Physics Notes | EduRev                                       (9)
This is an example of a Maxwell Relation. These are very powerful relationship that allows one to substitute partial derivatives when one is more convenient (perhaps it can be expressed entirely in terms of α and/or kT for example.)
A similar result can be derived based on the definition of H.
H = U + pV                                                                 (10)
Differentiating (and using the chain rule on (pV)) yields
 Physics Notes | EduRev                                     (11)
Making the substitution using the combined first and second laws (dU=TdS–pdV) for a reversible change involving on expansion (p-V) work
 Physics Notes | EduRev                  (12)
This expression can be simplified by canceling the pdV terms. 
 Physics Notes | EduRev                                            (13)

And much as in the case of internal energy, this suggests that the natural variables of H are S and p. Or  Physics Notes | EduRev           (14)
Comparing equation 13 and 14 show that
 Physics Notes | EduRev                                                      (15)
and  Physics Notes | EduRev                                              (16)
It is worth noting at this point that both (equation 6)
 Physics Notes | EduRev                                                             (17)
and (Equation 15).
 Physics Notes | EduRev                                                             (18)
are equation to T. So they are equation to each other
 Physics Notes | EduRev                                      (19)
Morevoer, the Euler Relation must also hold

 Physics Notes | EduRev          (20)
This is the Maxwell relation on H. Maxwell relations can also be developed based on A and G. The results of those derivations are summarized in Table
 Physics Notes | EduRev
The Maxwell relations are extraordinarily useful in deriving the dependence of thermodynamic variables on the state variables of p, T, and V.
Show that  Physics Notes | EduRev                    (21)

Solution: Start with the combined first and second laws:
dU = TdS - pdV                                              (22)
Divide both sides by dV and constraint to constant T:
 Physics Notes | EduRev                                (23)
Noting that
 Physics Notes | EduRev                                           (24)
 Physics Notes | EduRev                                           (25)
 Physics Notes | EduRev                                                        (26)
The result is 
 Physics Notes | EduRev
Now, employ the Maxwell relation on Table
 Physics Notes | EduRev
to get  Physics Notes | EduRev
and since   Physics Notes | EduRev
It is apparent that  Physics Notes | EduRev

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