maxwell's relations
TRANSCRIPT
MAXWELL’S THERMODYNAMIC
RELATIONSHIPS AND THEIR APPLICATIONS
Submitted BySarvpreet Kaur
Associate ProfessorDepartment of PhysicsGCG-11, Chandigarh
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