1. Chapter 5 Simple Applications of Macroscopic Thermodynamics

2. Preliminary Discussion
Classical, Macroscopic, Thermodynamics
Now, we drop the statistical mechanics notation for average quantities. So that now,
All Variables are Averages Only!
We’ll discuss relationships between macroscopic variables using
The Laws of Thermodynamics

3. Internal Energy = E, Entropy = S (V = volume, p = pressure)
Some Thermodynamic Variables of Interest:
Internal Energy = E, Entropy = S
Temperature = T
Mostly for Gases:
(but also true for any substance):
External Parameter = V
Generalized Force = p
(V = volume, p = pressure)
For a General System:
External Parameter = x
Generalized Force = X

4. 1st & 2nd Laws of Thermodynamics
Assume that the External Parameter = Volume V in order to have a specific case to discuss. For systems with another external parameter x, the infinitesimal work done đW = Xdx. In this case, in what follows, replace p by X & dV by dx.
For infinitesimal, quasi-static processes:
1st & 2nd Laws of Thermodynamics
1st Law: đQ = dE + pdV
2nd Law: đQ = TdS
Combined 1st & 2nd Laws
TdS = dE + pdV

5. Any 3 of these can always be expressed as functions of any 2 others.
Combined 1st & 2nd Laws
TdS = dE + pdV
Note that, in this relation, there are
5 Variables: T, S, E, p, V
It can be shown that:
Any 3 of these can always be expressed as functions of any 2 others.
That is, there are always 2 independent variables & 3 dependent variables. Which 2 are chosen as independent is arbitrary.

6. Brief, Pure Math Discussion dz  (∂z/∂x)ydx + (∂z/∂y)xdy (a)
Consider 3 variables: x, y, z. Suppose we know that x & y are Independent Variables. Then, It Must Be Possible to express z as a function of x & y. That is,
There Must be a Function z = z(x,y).
From calculus, the total differential of z(x,y) has the form:
dz  (∂z/∂x)ydx + (∂z/∂y)xdy (a)

7. dx  (∂x/∂y)zdy + (∂x/∂z)ydz (b)
Suppose that, in this example of 3 variables: x, y, z, we want to take y & z as independent variables instead of x & y. Then,
There Must be a Function x = x(y,z).
From calculus, the total differential of x(y,z) is:
dx  (∂x/∂y)zdy + (∂x/∂z)ydz (b)
Using (a) from the previous slide
[dz  (∂z/∂x)ydx + (∂z/∂y)xdy (a)]
& (b) together, the partial derivatives in (a) & those in (b) can be related to each other.
We always assume that all functions are analytic.
So, the 2nd cross derivatives are equal
Such as: (∂2z/∂x∂y)  (∂2z/∂y∂x), etc.

8. Combined 1st & 2nd Laws of Thermodynamics TdS = dE + pdV
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:
Most Ch. 5 applications use this with the
Combined 1st & 2nd Laws of Thermodynamics TdS = dE + pdV

9. Some Methods & Useful Math Tools for Transforming Derivatives
Derivative Inversion
Triple Product (xyz–1 rule)
Chain Rule Expansion to Add Another Variable
Maxwell Reciprocity Relationship 9

10. Pure Math: Jacobian Transformations
A Jacobian Transformation is often used to
transform from one set of independent
variables to another.
For functions of 2 variables f(x,y) & g(x,y) it is:
Determinant! 10

11. Jacobian Transformations Have Several Useful Properties
Transposition
Inversion
Chain Rule Expansion 11

12. Suppose that we are only interested in the first partial derivative of a function f(z,g) with respect to z at constant g:
This expression can be simplified using the chain rule expansion & the inversion property 12

13. Properties of the Internal Energy E
dE = TdS – pdV (1)
First, choose S & V as independent variables:
E  E(S,V) ∂E ∂E dE
(2)
Comparison of (1) & (2) clearly shows that ∂E ∂E
and
Applying the general result with 2nd cross derivatives gives:
Maxwell Relation I! 13

14. H  H(S,p)  E + pV  Enthalpy
If S & p are chosen 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
(1)
But, also:
(2)
Comparison of (1) & (2) clearly shows that
and
Applying the general result for the 2nd cross derivatives gives:
Maxwell Relation II! 14

15. F  F(T,V)  E - TS Helmholtz Free Energy
If T & V are chosen as independent variables, it is convenient to define the following energy:
F  F(T,V)  E - TS Helmholtz Free Energy
Use the combined 1st & 2nd Laws. Rewrite them in terms of dF:
dE = TdS – pdV = [d(TS) – SdT] – pdV or
dF = -SdT – pdV (1)
But, also: dF ≡ (F/T)VdT + (F/V)TdV (2)
Comparison of (1) & (2) clearly shows that
(F/T)V ≡ -S and (F/V)T ≡ -p
Applying the general result for the 2nd cross derivatives gives:
Maxwell Relation III! 15

16. G  G(T,p)  E –TS + pV  Gibbs Free Energy
If T & p are chosen as independent variables, it is convenient to define the following energy:
G  G(T,p)  E –TS + pV  Gibbs Free Energy
Use the combined 1st & 2nd Laws. Rewrite them in terms of dH: dE = TdS – pdV = d(TS) - SdT – [d(pV) – Vdp] or
dG = -SdT + Vdp (1)
But, also: dG ≡ (G/T)pdT + (G/p)Tdp (2)
Comparison of (1) & (2) clearly shows that
(G/T)p ≡ -S and (G/p)T ≡ V
Applying the general result for the 2nd cross derivatives gives:
Maxwell Relation IV! 16

17. Summary: 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. Gibbs Free Energy: G = G(T,p)  E – TS + pV
1. dE = TdS – pdV
2. dH = TdS + Vdp
3. dF = - SdT – pdV
4. dG = - SdT + Vdp
Combined 1st
& 2nd Laws 17

18. Another Summary: Maxwell’s Relations
(a) ΔE = Q + W
(b) ΔS = (Qres/T)
(c) H = E + pV
(d) F = E – TS
(e) G = H - TS 1. 2. 3. 4.
1. dE = TdS – pdV
2. dH = TdS + Vdp
3. dF = -SdT - pdV
4. dG = -SdT + Vdp 18

19. Maxwell Relations: “The Magic Square”? Thermodynamic Variables
Each side is labeled with an
Energy (E, H, F, G).
The corners are labeled with
Thermodynamic Variables
(p, V, T, S). Get the
Maxwell Relations
by “walking” around the
square. Partial derivatives
are obtained from the sides.
The Maxwell Relations
are obtained from the corners. F V T E G S P H 19

20. The 4 Most Common Maxwell Relations:
Summary
The 4 Most Common
Maxwell Relations: 20

21. Maxwell Relations: Table (E → U)21

22. Maxwell Relations from dE, dF, dH, & dG
Internal Energy
Helmholtz Free Energy
Enthalpy
Gibbs Free Energy 22

23. Some Common Measureable Properties
Heat Capacity at Constant Volume: ∂E
Heat Capacity at Constant Pressure:

24. More Common Measureable Properties
Volume Expansion Coefficient:
Note!! Reif’s
notation for
this is α
Isothermal Compressibility:
The Bulk Modulus is the
inverse of the Isothermal
Compressibility!
B  (κ)-1

25. Some Sometimes Useful Relationships
Summary of Results
Derivations are in the text and/or are left to the student!
Entropy:
Enthalpy:
Gibbs Free
Energy:

26. Typical Example
Given the entropy S as a function of temperature T & volume V, S = S(T,V), find a convenient expression for (S/T)P, in terms of some measureable properties.
Start with the exact differential:
Use the triple product rule & definitions:

27. Use a Maxwell Relation:
Combining these expressions gives:
Converting this result to a partial derivative gives:

28. This can be rewritten as:
The triple product rule is:
Substituting gives:

29. Note again the definitions:
Volume Expansion Coefficient
β V-1(V/T)p
Isothermal Compressibility
κ  -V-1(V/p)T
Note again!! Reif’s notation for the
Volume Expansion Coefficient is α

30. A GENERAL RELATIONSHIP
Using these in the previous expression finally gives the desired result:
Using this result as a starting point,
A GENERAL RELATIONSHIP
between the
Heat Capacity at Constant Volume CV
& the
Heat Capacity at Constant Pressure Cp
can be found as follows:

31. Using the definitions of the isothermal compressibility κ and the volume expansion coefficient , this becomes
General Relationship
between Cv & Cp

32. Simplest Possible Example: The Ideal Gas
For an Ideal Gas, it’s easily shown (Reif) that the Equation of State (relation between pressure P, volume V, temperature T) is (in per mole units!): Pν = RT. ν = (V/n)
With this, it is simple to show that the volume expansion coefficient β & the isothermal compressibility κ are:
and

33. So, for an Ideal Gas, the volume expansion coefficient & the isothermal compressibility have the simple forms:
and
We just found in general that the heat capacities at constant volume & at constant pressure are related as
So, for an Ideal Gas, the specific heats per mole have the very simple relationship:

34. Other, Sometimes Useful, Expressions

35. More Applications: Using the Combined 1st & 2nd Laws (“The TdS Equations”)
Calorimetry Again!
Consider Two Identical Objects, each of mass m, & specific heat per kilogram cP. See figure next page.
Object 1 is at initial temperature T1.
Object 2 is at initial temperature T2.
Assume T2 > T1.
When placed in contact, by the 2nd Law, heat Q flows from the hotter (Object 2) to the cooler (Object 1), until they come to a common temperature, Tf.

36. Two Identical Objects, of mass m, & specific heat per kilogram cP
Two Identical Objects, of mass m, & specific heat per kilogram cP. Object 1 is at initial temperature T1. Object 2 is at initial temperature T2.
T2 > T1. When placed in contact, by the 2nd Law, heat Q flows from the hotter (Object 2) to the cooler (Object 1), until they come to a common temperature, Tf.
Q 
Heat Flows
For some time
after initial
contact:
Object 2
Initially at T2
Object 1
Initially at T1
After a long enough time, the two objects are at the same temperature Tf. Since the 2 objects are identical, for this case,

37. The Entropy Change ΔS for this process can also be easily calculated:
Of course, by the 2nd Law, the entropy change ΔS must be positive!! This requires that the temperatures satisfy:

38. Some Useful “TdS Equations”
NOTE: In the following, various quantities are written in per mole units! Work with the
Combined 1st & 2nd Laws:
Definitions:
υ  Number of moles of a substance.
ν  (V/υ)  Volume per mole.
u  (U/υ)  Internal energy per mole.
h  (H/υ)  Enthalpy per mole.
s  (S/υ)  Entropy per mole.
cv  (Cv/υ)  const. volume specific heat per mole.
cP  (CP/υ)  const. pressure specific heat per mole.

39. Given these definitions, it can be shown that the Combined 1st & 2nd Laws (TdS) can be written in at least the following ways:

40. Internal Energy Enthalpy h(T,P):
Student exercise to show that, starting with the previous expressions & using the definitions (per mole) of internal energy u & enthalpy h gives:
Internal Energy
u(T,ν):
Enthalpy
h(T,P):

41. Student exercise also to show that similar manipulations give at least the following different expressions for the molar entropy s:
Entropy s(T,ν):