1. Chapter 5 Simple Applications of Macroscopic Thermodynamics

2. Preliminary Discussion
Classical, Macroscopic, Thermodynamics: Drop the statistical mechanics notation for average quantities. We know that all variables are averages only.
We’ll discuss relationships between macroscopic variables using the Laws of Thermodynamics.
Internal Energy = E
Entropy = S
Temperature = T
Gases: External parameter = Volume
Volume = V
Pressure = p
General system: External parameter = x
Parameter = x
Generalized Force = X

3. TdS = dE + pdV Combined 1st & 2nd Laws
Assume external parameter = V in order to have a specific case to discuss.
1st & 2nd Laws of Thermodynamics
1st Law: đQ = dE + pdV
2nd Law: đQ = TdS
Combined 1st & 2nd Laws
TdS = dE + pdV
5 Variables: T, S, E, p, V
It can be shown that:
3 of these can always be expressed as functions of 2 others.
That is, there are 2 independent variables & 3 dependent variables. Which 2 are chosen independent is arbitrary.

4. Brief, Pure Math Discussion
Consider 3 variables: x, y, z. Suppose that we know that x & y are independent variables. Then, there must be a function z = z(x,y).
The total differential of z has the form:
dz  (∂z/∂x)ydx + (∂z/∂y)xdy (a)
Suppose instead that we want to take y & z as independent variables. Then, there must be a function x = x(y,z).
The total differential of x has the form:
dx  (∂x/∂y)zdy + (∂x/∂z)ydz (b)
Using (a) & (b) together, the partial derivatives in (a) & those in (b) can be related to each other.
Assume all functions are analytic. So, the 2nd cross derivatives are equal: Such as (∂2z/∂x∂y) = (∂2z/∂y∂x) etc.

5. It’s exact differential is df  y1dx1 + y2dx2 By definition:
Mathematics Summary
Consider a function of 2 independent variables:
f  f(x1,x2).
It’s exact differential is df  y1dx1 + y2dx2
By definition:
Because f(x1,x2) is an analytic function, it is always true that
The applications all use this with the 1st & 2nd Laws of Thermodynamics: TdS = dE + pdV

6. dE = TdS – pdV (1) Properties of the Internal Energy E
Choose S & V as independent variables:
E = E(S,V) ∂E ∂E dE
(2)
Comparison of (1) & (2) clearly shows that ∂E ∂E
Applying the general result with 2nd cross derivatives gives:
Maxwell Relation I! 6

7. H  H(S,p)  E + pV  Enthalpy
If we choose S & p as independent variables, it is convenient to define the following energy:
H  H(S,p)  E + pV  Enthalpy
Use the combined 1st & 2nd Laws. Rewrite them in terms of dH: dE = TdS – pdV = TdS – [d(pV) – Vdp] or
dH = TdS + Vdp
But also:
(1)
(2)
Comparison of (1) & (2) clearly shows that
Applying the general result with 2nd cross derivatives gives:
Maxwell Relation II! 7

8. Summary 2. dH = TdS + Vdp 3. dF = - SdT – pdV 4. dG = - SdT + Vdp
Energy Functions
1. Internal Energy: E  E(S,V)
2. Enthalpy: H = H(S,p)  E + pV
3. Helmholtz Free Energy: F = F (T,V)  E – TS
4. Gibb’s Free Energy: G = G(T,p)  E – TS + pV
Combined 1st & 2nd Laws
1. dE = TdS – pdV
2. dH = TdS + Vdp
3. dF = - SdT – pdV
4. dG = - SdT + Vdp
lll
lll
lll
lll 8 8

9. The 4 Most Common Maxwell Relations
Summary
The 4 Most Common Maxwell Relations 9 9

10. Maxwell’s Relations: Table (E → U)10 10