1. Peter Atkins • Julio de Paula Atkins’ Physical Chemistry
Eighth Edition
Chapter 3 – Lecture 3
The Second Law
Copyright © 2006 by Peter Atkins and Julio de Paula

2. Concentrating on the System
Consider a system in thermal equilibrium with its
surroundings at constant T
When heat is transferred between system and surroundings,
the Clausius inequality reads:
We can use this to develop expressions for sponteneity
under constant volume and under constant pressure

3. Constant volume conditions
dqV = dU
so becomes
which rearranges to
Constant pressure conditions
dqP = dH
so becomes

4. Concentrating on the System
We introduce two new thermo functions:
Helmholtz energy (constant V)
A ≡ U – TS → dA = dU - TdS → ΔAT,V = ΔU - TΔS
Gibbs energy (constant P)
G ≡ H – TS → dG = dH - TdS → ΔGT,P = ΔH - TΔS

5. The Gibbs Energy Criterion for a spontaneous process:
dGT,P ≤ O based on the system alone
“Free energy”:
max additional (non-PV) work obtainable
dwadd,max = dG or
wadd,max = ΔG

6. Standard Molar Gibbs Energies
where:
and:
Or:

8. Combining First and Second Laws
Major objectives:
Find relationships between various thermo properties
Derive expressions for G(T,P)
The Fundamental Equation
From 1st Law: dU = dq + dw
From 2nd Law:
Substituting: dw = -PΔV and dq = TdS
Gives:
dU = TdS - PdV

9. Properties of the Internal Energy
dU = TdS – PdV indicates that U = f (S,T)
So the full differential of U:
Indicates that:
The first partial is a pure thermo definition of temperature

10. A Math Lesson An infinitesimal change in the function f(x,y) is:
df = M dx + N dy
where M and N are functions of x and y
Criterion for df to be exact is:
Euler Reciprocity relationship (p 968)

11. A Math Lesson (cont’d)
dU = TdS – PdV must pass this test if it indeed is exact
Applying the Euler Reciprocity:
This equation is one of the four Maxwell relations
df = M dx + N dy

13. The Maxwell Relations Table 3.5
Natural Variables
Relation
U(S,V)
H(S,P)
A(V,T)
G(P,T)

14. Variation of Internal Energy with Volume
Internal pressure
Beginning with dU = TdS – PdV
Dividing both sides by dV and imposing constant T
Maxwell #3 here...
A thermo equation of state

15. Properties of Gibbs Energy
dG = dH – TdS
So: G(H,T,S) and expanding:
dG = dH – d(TS)
= dH – TdS – SdT
= dU + d(PV) – TdS – SdT
= dU + PdV + VdP – TdS – SdT
= TdS - PdV + PdV + VdP – TdS – SdT
dG = VdP - SdT

16. Fig 3.18 Variation of Gibbs energy with T and with P
dG = VdP - SdT

17. Fig 3.19 Variation of Gibbs energy with T depends on S

18. Fig 3.20 Variation of Gibbs energy with P depends on V