Thermodynamic Potentials

Question 1

Conservative Fields

Answer 1

• a conservative field is one for which the work done is independent of the path
• the curl of a conservative field is 0
• e.g. thermodynamic potentials, gravity & electrostatic fields

Question 2

Non-Conservative Fields

Answer 2

curl ≠0

e.g. time varying magnetic field, friction forces, air drag

Question 3

Extremum Principles in Nature

Description

Answer 3

-several of the most fundamental rules of nature involve some physical quantity that is minimised or maximised

Question 4

Extremum Principles
in Nature
Examples

Answer 4

1) Fermat’s Principle - light takes the shortest optical path between two points
2) Hamilton/Lagrange Formulation - bodies move in accordance with the principle of least action

Question 5

Extremum Principles in Nature

Classical Thermodynamics

Answer 5

-the equilibrium state of a system under different conditions is determined by the extremum of the appropriate thermodynamic potential

Question 6

The Second Law as an Extremum Principle

Answer 6

-a physical process will occur spontaneously is it increases the entropy of the universe, that is if:
dS > 0
-a system will therefore move to a state of higher entropy if a path is available
-the equilibrium state is reached when dS=0
-this describes a reversible process which occurs equally well in both directions

Question 7

The Second Law as an Extremum Principle - Concentrating on the System

Answer 7

• the condition for maximum entropy applies to the whole universe
• the system of interest may be held under a constraint e.g. constant P, isolation etc.
• under each constraint the evolution of the system to equilibrium corresponds to the extremisation of a thermodynamic potential

Question 8

List All 6 Thermodynamic Potentials

Answer 8

• Entropy, S
• Internal Energy, U
• Enthalpy, H
• Helmholtz Free Energy, F
• Gibbs Free Energy, G
• Chemical Potential, μ

Question 9

Thermodynamic Potential

Constant Volume Derivations

Answer 9

-consider a system in thermal equilibrium with its surroundings at temperature T
-Claussius inequality
dS - dQ/T ≥ 0
-first law at constant volume
dU = dQ + dW = dQ
-> dS - dU/dT ≥ 0
-rearrange
TdS ≥ dU
-at constant energy
dS≥0
-at constant entropy
0≥dU

Question 10

Thermodynamic Potentials

Isolated and Constant Volume and Constant Energy

Answer 10

-a system maintained at constant volume and energy (an isolated system) will evolve to maximise entropy

Question 11

Thermodynamic Potentials

Constant Entropy and Volume

Answer 11

• a system at constant entropy and volume will evolve to minimise the energy U
• there is an accompanying increase in the entropy of the surroundings as energy flows out of the system as heat

Question 12

Enthalpy

Definition

Answer 12

H = U + PV

dH = dU + PdV + VdP
-sub in dU = dQ - PdV
dH = dQ -PdV + PdV + VdP
dH = dQ + VdP

Question 13

Thermodynamic Potentials

Isolated, Constant Pressure (and Constant Enthalpy OR Constant Entropy)

Answer 13

dH = dQ + VdP
-at constant pressure, dP=0
dH = dQ = TdS
-sub in to the Claussius inequality
dS - dH/T ≥ 0

-if enthalpy and pressure are constant, entropy is maximised at equilibrium:
dS |H,P ≥ 0

-if entropy and pressure are held constant, enthalpy is minimised at equilibrium:
dH|S,P ≤ 0

Question 14

Claussius Inequality

Answer 14

dS - dQ/T ≥ 0

Question 15

Helmholtz Free Energy

Answer 15

F = U - TS

dF = dU - TdS

Question 16

Gibbs Free Energy

Answer 16

G = H - TS

dG = dH -TdS

Question 17

Thermodynamic Potentials

Constant Temperature and Volume

Answer 17

F = U - TS
dF = dU - TdS

-at constant volume and temperature, helmholtz free energy is minimised at equilibrium
dF|T,V ≤ 0

Question 18

Thermodynamic Potentials

Constant Temperature and Pressure

Answer 18

G = H - TS
dG = dH - TdS

-at constant pressure and temperature, Gibbs free energy is minimised at equiibrium

Question 19

How to derive the differential forms of the thermodynamic potentials

Answer 19

1) differentiate both sides, use product rule where necessary
2) using the first law substitute dU = dQ - PdV
3) sub in dQ = TdS
4) cancel any terms
5) you should have arrived at the differential form

Question 20

Proper Independent Variables

Answer 20

-the differential forms of each potential shows that each potential has a different pair of variables as its proper independent variables:
U = U(S,V)
H = H(S,P)
F = F(T,V)
G = G(T,P)
-if any one of these potentials is known explicitly in terms of its proper variables, then because they are all state functions, we have complete information about the system

Question 21

How to Derive the Maxwell Relations

Answer 21

1) start with the differential form of one of the thermodynamic potentials
2) hold one of the proper independent variables constant to obtain a partial derivative equation
30 hold the other proper independent variable constant to obtain a second partial derivative equation
3) differentiate the first partial derivative equation with respect to the second proper independent variable
4) differentiate the second partial derivative equation with respect to the first proper independent variable
5) the two second order partial derivatives are equal for state functions, so the two expressions can be equated
6) this gives a Maxwell relation

Question 22

When is entropy maximised or minimised?

Answer 22

-at constant volume and constant energy, the system evolves to maximise entropy
OR
-at constant pressure and constant enthalpy, the system evolves to maximise entropy

Question 23

When is internal energy maximised or minimised?

Answer 23

-at constant volume and constant entropy, the system evolves to minimise internal energy

Question 24

When is internal energy maximised or minimised?

Answer 24

-at constant pressure and entropy, the system evolves to minimise enthalpy

Question 25

When is Helmholtz free energy maximised or minimised?

Answer 25

-at constant volume and constant temperature, the system evolves to minimise Helmholtz free energy

Question 26

When is Gibb’s free energy maximised or minimised?

Answer 26

-at constant pressure and constant temperature, the system evolves to minimise Gibb’s free energy