MAXWELL’S THERMODYNAMIC

RELATIONSHIPS AND THEIR APPLICATIONS

Dinesh kumar K2nd year

Mech

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

http://en.wikipedia.org/wiki/James_Clerk_Maxwell

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

Maxwell relations derived by the method

based on Thermodynamic Potentials

Why are thermodynamic potentials usefulThermodynamic 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

However

U(T,V) is not a thermodynamic potential

We are going to show: U=U(S,V) complete knowledge of equilibrium properties

U(S,V): thermodynamic potential

The thermodynamic potential U=U(S,V)

Now Consider first law in differential notation WdQddU

inexact differentials

Wd

Qd

expressed by exact differentials

PdVWd

TdSQd

2nd law

PdVTdSdU

Note: exact refers here to the coordinate differentials dS and dV.TdS and PdV are inexact

So dU is an exact potental.

By Legendre transformation

from (S,V) to

(T,V): PdVTdSdU PdVSdT)TS(d

PdVSdT)TSU(d

F: Helmholtz free energy

(T,P): PdVSdTdF VdP)PV(dSdT

VdPSdT)PVF(d

G: Gibbs free energy

TSHPVTSUPVFG

VdPTdSPVddU

VdPPVdTdSdU

easy check: PdVVdPPdVVdPVdPPVd

Productrule

VdPTdSPVUd

=:H (enthalpy)

H=H(S,P) is a thermodynamic potential

VdPTdSdH

Enthalpy

Using these exact differentials we derive maxwell’s relations .

VdPTdSdH Enthalpy

VdPSdTPVFddG )(Function sGibb'

PdVSdTTSUddF )(Energy Free Helmholtz

PdVTdSdU Energy Internal

Now dU, dF, dG and dH are exact differentials e.g

Maxwell’s Thermodynamic Relations

VS S

P

V

T

VT T

P

V

S

PS S

V

P

T

PT T

V

P

S

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.

dVV

UdS

S

UdU

SV

Deriving Maxwell Relations Continued

TS

U

V

P

V

U

S

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.

dVV

UdS

S

UdU

SV

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

Similarily using dF,dG and dH other Maxwell

Relations are

Mnemonic Device for Obtaining Maxwell

RelationsT

VP

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

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.

The Four Maxwell relations have a very wide rangeof applications . They apply to all kind of substances (solids,liquids,gases)under all type of conditions ofPressure, volume and temperature. Before Discussing applications We define some thermodynamic terms e.g

Application of Maxwell’s Relations

1. Cooling Produced By Adiabatic Expansion of Any Substance

i) Specific Heat at Constant Volume

Specific Heat at Constant Pressureii)

p

PT

UC

v

v

T

UC

iii) Pressure and Volume Coffecient of Expansion

1. Cooling Produced By Adiabatic Expansion of Any Substance

In adiabatic process entropy S remains constant. Therefore by consideringthe Thermodynamic relation

VS S

p

V

T

vmC

TpT

We can prove

Most of the substances expand on heating , they have +ve beta value. Will be –vei.e all the substances will cool down. A few substances like rubber have –ve beta value. They will get heat up..

T

2. Adiabatic Compression of A Substance

By considering the Thermodynamic relation

pS S

V

p

T

pC

TVT

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

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3. Change of internal energy with Volume,

Using the third Maxwell’s relation

T v

T v

s P

v T

u PT P

v T

Since

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For Vander Waal’s / real gases

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.

RTbVV

ap ))(

2(

VV

a

TV

U

2

For Ideal Gas

Thus Vander Waal’s gas expands isothermally as its internal energy increases.

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Other relations for the specific heats are given below.

where is the volume expansivity and is the isothermal compressibility, defined as

4. Cp – Cv = R for ideal gases.

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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.

For an ideal gas

5. Variation of Cv of an ideal gas does not depend upon specific volume.

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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.

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

V

T

p

TV

S

have Weheat,latent theL,ΔU using

)( if VVT

L

VT

p

This equation is known as Clausius-Clapeyron’s latent heatEquation.

Thank You