1. Ch. 5: Applications Using Maxwell Relations & Measurable Properties
“But ma, it’s not my fault… the universe
wants my room like this!”

2. Some Measureable Properties
Heat Capacity at Constant Volume: ∂E
Choose T & V
as Independent
Variables

3. Some Measureable Properties
Heat Capacity at Constant Volume: ∂E
Choose T & V
as Independent
Variables
Heat Capacity at Constant Pressure:
Choose P & V
as Independent
Variables

4. More Measureable Properties
Volume Expansion Coefficient:
Note!! Reif’s
notation for
this is α

5. More 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

6. Some Sometimes Useful Relationships
Summary of Results
Derivations are in the text and/or are left to the student!
Entropy S = S(T,P):
Enthalpy
H = H(T,P):
Gibbs Free
Energy
G = G(T,P):

7. in terms of some measureable properties.
A 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.

8. in terms of some measureable properties.
A 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:

9. in terms of some measureable properties.
A 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:

10. Use a Maxwell Relation:

11. Use a Maxwell Relation:
Combining these expressions gives:

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

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

15. Note again the definitions:
Volume Expansion Coefficient
β V-1(V/T)p

16. Note again the definitions:
Volume Expansion Coefficient
β V-1(V/T)p
Isothermal Compressibility
κ  -V-1(V/p)T

17. 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

18. 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:

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

20. 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

21. 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:

22. Other, Sometimes Useful, Expressions

23. 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.

24. 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,

25. 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:

26. 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.

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

28. 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):

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