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Physics Exam  >  Physics Notes  >  Physics for IIT JAM, UGC - NET, CSIR NET  >  Thermodynamic Potentials - Thermodynamic and Statistical Physics

Thermodynamic Potentials - Thermodynamic and Statistical Physics | Physics for IIT JAM, UGC - NET, CSIR NET PDF Download

Maxwell Relations

F = F ( x , y ) and if it is perfect differential then d F =M dx + NdyThermodynamic Potentials - Thermodynamic and Statistical Physics | Physics for IIT JAM, UGC - NET, CSIR NETthen M and N will satisfy the condition Thermodynamic Potentials - Thermodynamic and Statistical Physics | Physics for IIT JAM, UGC - NET, CSIR NETMaxwell relations are relationship between two derivatives of thermodynamic variables, and energy due to the equivalence of potential second derivative under a change of operation order
Thermodynamic Potentials - Thermodynamic and Statistical Physics | Physics for IIT JAM, UGC - NET, CSIR NETWhere F is thermodynamic potential and x and y are two of its natural independent variables.

Maxwell relations are extremely important for two reasons: 

(i) First they show us that derivative of thermodynamic parameters are not all independent. This can serve as a consistency check in both experiments and in theoretical analysis.
(ii) Maxwell relations provide a method for expressing some derivative in other ways. This enables as to connect difficult to measure quantities to those which are readily accessible experimentally.
The measurement of entropy and chemical potential can not be directly measurable in lab but with the help of Maxwell relation there thermodynamic property can be determine theoretically.
For Maxwell relation,

Let us Legendre the independent variable as x , and y such that
U = U(x,y), S = S(x, y) V = V(x, y)
So Thermodynamic Potentials - Thermodynamic and Statistical Physics | Physics for IIT JAM, UGC - NET, CSIR NETThermodynamic Potentials - Thermodynamic and Statistical Physics | Physics for IIT JAM, UGC - NET, CSIR NETFrom first law of thermodynamic
dU = TdS - PdV
Thermodynamic Potentials - Thermodynamic and Statistical Physics | Physics for IIT JAM, UGC - NET, CSIR NET
Hence U,V, and S are perfect differential.
Then
Thermodynamic Potentials - Thermodynamic and Statistical Physics | Physics for IIT JAM, UGC - NET, CSIR NETThermodynamic Potentials - Thermodynamic and Statistical Physics | Physics for IIT JAM, UGC - NET, CSIR NET

Thermodynamic Potentials - Thermodynamic and Statistical Physics | Physics for IIT JAM, UGC - NET, CSIR NET

SimilarlyThermodynamic Potentials - Thermodynamic and Statistical Physics | Physics for IIT JAM, UGC - NET, CSIR NET
Equating equation (1) and (2)
Thermodynamic Potentials - Thermodynamic and Statistical Physics | Physics for IIT JAM, UGC - NET, CSIR NET

Maxwell first relation:- put x = T , y = V
Thermodynamic Potentials - Thermodynamic and Statistical Physics | Physics for IIT JAM, UGC - NET, CSIR NET

Maxwell Second Relation:- put x = T ,y = P
Thermodynamic Potentials - Thermodynamic and Statistical Physics | Physics for IIT JAM, UGC - NET, CSIR NETMaxwell Third Relation:- put x = s , y = vThermodynamic Potentials - Thermodynamic and Statistical Physics | Physics for IIT JAM, UGC - NET, CSIR NET

Maxwell Fourth Relation:- put x = S, y = P
Thermodynamic Potentials - Thermodynamic and Statistical Physics | Physics for IIT JAM, UGC - NET, CSIR NET

Thermodynamic potential is a scalar function used to represent the thermodynamic state of system. The concept of thermodynamic potentials was introduced by Pierre Duhem in 1886.

One main thermodynamic potential that has a physical interpretation is the internal energy. It is energy of configuration of a given system of conservative forces. Expression for all other thermodynamic energy potentials are drivable via Legendre transformation.

Different Types of Thermodynamic Potential and Maxwell Relation

Thermodynamic potentials are different form of energy which can be used in different thermodynamic process. Thermodynamic potentials are path independent variables so they are perfect differential.
If F is unique thermodynamic potential defined by variables x and y as F = F ( x , y ) and if it is perfect differential then dF = Mdx + Ndy
Thermodynamic Potentials - Thermodynamic and Statistical Physics | Physics for IIT JAM, UGC - NET, CSIR NETthen M and N will satisfy the condition

Thermodynamic Potentials - Thermodynamic and Statistical Physics | Physics for IIT JAM, UGC - NET, CSIR NET

Internal Energy
U and second from the first laws of thermodynamics 
dU = TdS - PdV
from Legendre transformation
Thermodynamic Potentials - Thermodynamic and Statistical Physics | Physics for IIT JAM, UGC - NET, CSIR NETfrom given relation one can derive Maxwell relationThermodynamic Potentials - Thermodynamic and Statistical Physics | Physics for IIT JAM, UGC - NET, CSIR NET
Enthalpy
(H) the enthalpy is defined as H = U + PV 
dH = dU + PdV + VdP 
from Laws of thermodynamics,
 TdS = dU + PdV 
& dH = TdS + VdP 
from Legendre transformation
Thermodynamic Potentials - Thermodynamic and Statistical Physics | Physics for IIT JAM, UGC - NET, CSIR NET
The Enthalpy H is Extensive quantity, which can not be measured directly. Thus change in enthalpy is more useful.
△H is positive in endothermic reaction and negative in exothermic reaction.
From above relation one can derive Maxwell relation Thermodynamic Potentials - Thermodynamic and Statistical Physics | Physics for IIT JAM, UGC - NET, CSIR NET
△H of a system is equal to sum of non-mechanical work done on it and the heat supplied to it.
Helmholtz Free Energy
(F) the Helmholtz free energy is defined F = U - T S
 dF = dU - TdS - SdT 
From laws of thermodynamics, d U = T d S - P d V 
dF = TdS - PdV - TdS - SdT 
dF = - P d V - SdT
 From Legendre transformation
Thermodynamic Potentials - Thermodynamic and Statistical Physics | Physics for IIT JAM, UGC - NET, CSIR NET
From above relation one can derive Maxwell relationThermodynamic Potentials - Thermodynamic and Statistical Physics | Physics for IIT JAM, UGC - NET, CSIR NET
The free Energy F, which is available energy for work in reversible isothermal process.
Gibbs Energy: ‘G’ is defined as G = H - TS .
G = U + P V - T S
dG = dU + PdV - VdP - TdS - SdT 
TdS - PdV + PdV + VdP - TdS - SdT 
dG = VdP - SdT
 from Legendre transformation
Thermodynamic Potentials - Thermodynamic and Statistical Physics | Physics for IIT JAM, UGC - NET, CSIR NET
From above relation one can derive Maxwell relation
Thermodynamic Potentials - Thermodynamic and Statistical Physics | Physics for IIT JAM, UGC - NET, CSIR NET
Gibbs free energy is popularly as free enthalpy.
  • The Gibbs free energy is Maximum amount of nonexpanding work that can be exacted from a closed system.
  • The maximum will activated when the system is in reversible process.
  • Gibbs free energy is also treated as chemical potential.
  • In thermodynamics, chemical potential, as partial molar free energy, is a form of potential energy that can be absorbed or relived during a chemical reaction.
  • The chemical potential of a species in the minute can be defined the slope of the energy at system with respect to a change in the no of moles.Thermodynamic Potentials - Thermodynamic and Statistical Physics | Physics for IIT JAM, UGC - NET, CSIR NET

where Thermodynamic Potentials - Thermodynamic and Statistical Physics | Physics for IIT JAM, UGC - NET, CSIR NET is chemical potential, G is Gibbs energy and N is no of molecules

Example 1: Prove that internal energy U is given by
Thermodynamic Potentials - Thermodynamic and Statistical Physics | Physics for IIT JAM, UGC - NET, CSIR NETSolution: (a) F = U-TS =U = F + TSThermodynamic Potentials - Thermodynamic and Statistical Physics | Physics for IIT JAM, UGC - NET, CSIR NETThermodynamic Potentials - Thermodynamic and Statistical Physics | Physics for IIT JAM, UGC - NET, CSIR NETG = H-TS

Thermodynamic Potentials - Thermodynamic and Statistical Physics | Physics for IIT JAM, UGC - NET, CSIR NETApplication of Maxwell Relation

First T - dS equation

Let T, and \/are independent variable S = S (T, V)

Thermodynamic Potentials - Thermodynamic and Statistical Physics | Physics for IIT JAM, UGC - NET, CSIR NET

Second T - dS Equation
Let T and P are independent variable S = S(T, P).

Thermodynamic Potentials - Thermodynamic and Statistical Physics | Physics for IIT JAM, UGC - NET, CSIR NETFrom Maxwell relation

Thermodynamic Potentials - Thermodynamic and Statistical Physics | Physics for IIT JAM, UGC - NET, CSIR NETThird T-dS Equation

Let P , V are independent variable S = S (P ,V )

Thermodynamic Potentials - Thermodynamic and Statistical Physics | Physics for IIT JAM, UGC - NET, CSIR NET

Thermodynamic Potentials - Thermodynamic and Statistical Physics | Physics for IIT JAM, UGC - NET, CSIR NET

The First Energy Equation
Let T and V are independent variable U = U(T, V)

Thermodynamic Potentials - Thermodynamic and Statistical Physics | Physics for IIT JAM, UGC - NET, CSIR NETFrom first law of thermodynamics,
dU = TdS - PdV

Thermodynamic Potentials - Thermodynamic and Statistical Physics | Physics for IIT JAM, UGC - NET, CSIR NETUsing Maxwell relationThermodynamic Potentials - Thermodynamic and Statistical Physics | Physics for IIT JAM, UGC - NET, CSIR NETThermodynamic Potentials - Thermodynamic and Statistical Physics | Physics for IIT JAM, UGC - NET, CSIR NET

Second Energy Equation

Thermodynamic Potentials - Thermodynamic and Statistical Physics | Physics for IIT JAM, UGC - NET, CSIR NET

Using Maxwell relation,Thermodynamic Potentials - Thermodynamic and Statistical Physics | Physics for IIT JAM, UGC - NET, CSIR NETThis is popularly known as second energy equation.

Application of second energy equation

If U is function of independent variable of T and P.

Thermodynamic Potentials - Thermodynamic and Statistical Physics | Physics for IIT JAM, UGC - NET, CSIR NET


Example 2: From relation dU = TdS - P d V
Derive Maxwell relation Thermodynamic Potentials - Thermodynamic and Statistical Physics | Physics for IIT JAM, UGC - NET, CSIR NET 
Solution: dU = TdS - PdVThermodynamic Potentials - Thermodynamic and Statistical Physics | Physics for IIT JAM, UGC - NET, CSIR NETHence U is exact differentialThermodynamic Potentials - Thermodynamic and Statistical Physics | Physics for IIT JAM, UGC - NET, CSIR NET


Example 3: A real gas which obey van der Waal's equation of state are kept in container which has temperature T0 and volume V0. if volume of container changes to V such that temperature of gas become T what is change in entropy?
Solution: Assume Cv is specific heat of constant volume

 For van der Waal's gasThermodynamic Potentials - Thermodynamic and Statistical Physics | Physics for IIT JAM, UGC - NET, CSIR NETFrom first T -dS equationThermodynamic Potentials - Thermodynamic and Statistical Physics | Physics for IIT JAM, UGC - NET, CSIR NETwhere S0 is integration constant.

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FAQs on Thermodynamic Potentials - Thermodynamic and Statistical Physics - Physics for IIT JAM, UGC - NET, CSIR NET

1. What are thermodynamic potentials?
Thermodynamic potentials are mathematical functions that describe the state of a thermodynamic system. They are derived from fundamental thermodynamic principles and provide information about the system's energy, entropy, and other thermodynamic properties.
2. How are thermodynamic potentials related to thermodynamic variables?
Thermodynamic potentials are related to thermodynamic variables through partial derivatives. For example, the internal energy (U) is related to the temperature (T) through the partial derivative dU/dT. Each thermodynamic potential is associated with a specific set of independent variables, and their derivatives provide valuable information about the system's behavior.
3. What is the significance of thermodynamic potentials in studying equilibrium states?
Thermodynamic potentials play a crucial role in studying equilibrium states of a system. Each potential is minimized at equilibrium with respect to its associated independent variables. For instance, the Gibbs free energy (G) is minimized at constant temperature and pressure, making it a useful tool for predicting whether a reaction or process will occur spontaneously.
4. Can you explain the concept of Legendre transformation in relation to thermodynamic potentials?
A Legendre transformation is a mathematical operation that allows us to express a function in terms of its conjugate variables. In the context of thermodynamic potentials, a Legendre transformation is used to convert one potential into another by replacing one independent variable with its conjugate variable. For example, the enthalpy (H) is obtained from the internal energy (U) by a Legendre transformation with respect to volume.
5. How do thermodynamic potentials help in understanding phase transitions?
Thermodynamic potentials provide insights into phase transitions by analyzing the behavior of the system at different values of the potential. For instance, the Helmholtz free energy (F) is useful in studying transitions between different equilibrium states at constant temperature and volume. By examining the behavior of F during phase transitions, we can understand the conditions under which a substance undergoes a phase change.
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