1. MAXWELL’S THERMODYNAMIC RELATIONSHIPS AND THEIR APPLICATIONS
Submitted By
Sarvpreet Kaur
Associate Professor
Department of Physics
GCG-11, Chandigarh

2. James Clerk Maxwell (1831-1879)
Born in Edinburgh, Scotland
Physicist well-known for his work in electromagnetism and field theory
Also known for his work in thermodynamics and kinetic theory of gases

3. Why Use Maxwell Relations?
Certain variables in thermodynamics are hard to measure experimentally such as entropy
Some variables like Pressure, Temperature are easily measureable
Maxwell relations provide a way to exchange variables

4. Maxwell relations derived by the method based on Thermodynamic Potentials
Why are thermodynamic potentials useful
Thermodynamic potentials give the complete knowledge about any thermodynamic system at equilibrium
e.g. U=U(T,V) does not give complete knowledge of the system and
requires in addition
P=P(T,V)
equation of state
U=U(T,V) and P=P(T,V)
complete knowledge of equilibrium properties
U(T,V) is not a thermodynamic potential
However
complete knowledge of equilibrium properties
We are going to show: U=U(S,V)
U(S,V): thermodynamic potential

5. The thermodynamic potential U=U(S,V)
2nd law
Now Consider first law in differential notation
expressed by
inexact differentials
exact differentials
Note: exact refers here to the
coordinate differentials dS and dV.
TdS and PdV are inexact
So dU is an exact potental.

6. By Legendre transformation
(T,V):
from (S,V) to
Helmholtz free energy
(T,P):
Gibbs free energy

7. Product
rule
easy check:
=:H (enthalpy)
Enthalpy
H=H(S,P)
is a thermodynamic potential

8. Now dU, dF, dG and dH are exact differentials e.g
Using these exact differentials we derive maxwell’s relations .

9. Maxwell’s Thermodynamic Relations

10. Deriving Maxwell Relations Using thermodynamic Potentials
First, start with a known equation of state such as that of internal energy
Next, take the total derivative of with respect to the natural variables. For example, the natural of internal energy are entropy and volume.

11. Deriving Maxwell Relations Continued
Now that we have the total derivative with respect to its natural variables, we can refer back to the original equation of state and define, in this example, T and P.

12. Deriving Maxwell Relations Continued
We must now take into account a rule in partial derivatives
When taking the partial derivative again, we can set both sides equal and thus, we have derived a Maxwell Relation

13. Similarily using dF,dG and dH other Maxwell Relations are

14. Mnemonic Device for Obtaining Maxwell RelationsP V S
Write T,V,S,P in a clockwise manner by Remembering the line
TV Special Programme..
Four relations are obtained by starting either from T or S Clockwise or anticlockwise direction. A negative Sign must appear in the resulting equation

15. Using Maxwell Relations
Maxwell Relations can be derived from basic equations of state, and by using Maxwell Relations, working equations can be derived and used when dealing with experimental data.

16. Application of Maxwell’s Relations
The Four Maxwell relations have a very wide range
of applications . They apply to all kind of substances
(solids,liquids,gases)under all type of conditions of
Pressure, volume and temperature. Before Discussing applications
We define some thermodynamic terms e.g i)
Specific Heat at Constant Volume
ii)
Specific Heat at Constant Pressure
iii)
Pressure and Volume Coffecient of Expansion
1. Cooling Produced By Adiabatic Expansion of Any Substance

17. 1. Cooling Produced By Adiabatic Expansion of Any Substance
In adiabatic process entropy S remains constant. Therefore by considering
the Thermodynamic relation
We can prove
Most of the substances expand on heating , they have +ve beta value Will be –ve
i.e all the substances will cool down. A few substances like rubber have –ve beta value.
They will get heat up..

18. 2. Adiabatic Compression of A Substance
By considering the Thermodynamic relation
We can prove
Above result shows that if is +ve, then adiabatic increase in pressure
causes the temprature to rise.
** Similarily using other maxwell’s equations we can explain the stretching
of wires and thin films

19. 3. Change of internal energy with Volume,
Using the third Maxwell’s relation
Since

20. For Ideal Gas
This result helps to show that the internal energy of an ideal gas does not depend upon specific volume. This is known as Joule’s Law.
For Vander Waal’s / real gases
Thus Vander Waal’s gas expands isothermally as its
internal energy increases.

21. 4. Cp – Cv = R for ideal gases.
Other relations for the specific heats are given below.
where  is the volume expansivity and  is the isothermal compressibility, defined as

22. The difference Cp – Cv is equal to R for ideal gases and to zero for incom­pressible substances (v = constant).
5. Variation of Cv with specific volume.
5. Variation of Cv of an ideal gas does not depend upon specific volume.
For an ideal gas

23. Therefore, the specific heat at constant volume of an ideal gas is independent of specific volume.
For Vander Waal’s gas also it is independent of volume.

24. 6. Change of state and clapeyron’s equation
In ordinary phase transition of matter(solid phase to liquid phase,
liquid to vapour, and solid to vapour) take place under constant
Temperature and pressure. During the transition a certain amount of
heat, known as latent heat must be supplied to the substance for a change
Of phase. During this change temperature remains constant. Therefore
using maxwell relation
This equation is known as Clausius-Clapeyron’s latent heat
Equation.

25. Thank You