Chapter 3: Many Variables; Section 1: Differentiation; page 4
Change of Variables, Exercise 5
Question
As a continuation of the example (3.1.4.31/33)
Obtain the total differential of the thermodynamic function
called "Helmholtz free energy" by the use of the Legendre transformation!
Obtain the total differential of the thermodynamic function
called "Gibbs free energy" by the use of the Legendre transformation!
From the total differentials, deduce the corresponding thermodynamic Maxwell equations!
Reminder
The Legendre transformation.... uses a physical function, in order to transform a derivative into a new variable. The transformation obeys the rules of physical dimensions, and modifies the function.
For using u as a variable instead of x, we'll define a new function g as
(3.1.4.28)
With the assistance of (3.1.4.27), one obtains
(3.1.4.29)
with
(3.1.4.30)
Notice that f and g are different functions, but have the same physical dimensions. The procedure can be similarly applied for the transformation .
As a physical example of the Legendre transformation, we'll use again the main law of thermodynamics (3.1.2.12/14), where the internal energy U is a function of two variables: the entropy S and the volume V, and the total differential is
(3.1.4.31)
where T is temperature and p is pressure.
The following application of the Legendre transformation (3.1.4.28/30) yields a new function , which is called Enthalpy:
(3.1.4.32)
Since the differentiation of is independent of the order, we obtain in addition to (3.1.3.11), another Maxwell equation of thermodynamics:
(3.1.4.33)
Parts 1-2
Solution of question 1
In order to obtain
from (3.1.4.31),
one should define
As a consequence
Parts 3-4
Solution of question 2
One can obtain
from
by defining
,
but this is not a unique way, one could consider also the two other alternatives
yielding finally the same expression of G .
Following the first definition of part 3, one obtains
Parts 5-8
Solution of question 3
The total differential of part 2 yields
From the mixed second derivative
, we obtain one of the Maxwell equations
The total differential of part 4 yields
From the mixed second derivative
, we obtain another one of the Maxwell equations
Score
By parts.
Parts 1,2,3,4,5,7 are worth 1 point each.
Parts 6,8 are worth 2 points each.
By questions.
Questions 1 and 2 are worth 2 points each.
Question 3 is worth 6 points.