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

Introduction

• Thermodynamic potentials are important functions in statistical mechanics that connect the behavior of tiny particles to the overall properties of larger systems. 
• These potentials help us find various thermodynamic quantities and allow us to predict how systems will act under different conditions. 
• The main types of thermodynamic potentials include
• Each potential has its own natural variablesand uses, such as: 
  • Finding equilibrium conditions
  • Predicting spontaneous processes
  • Understanding phase transitions

Internal Energy

• The total energy in a thermodynamic system is represented as U.
• This energy includes the kinetic and potential energies of all the particles within the system.
• U is essential for understanding the First Law of Thermodynamics, which is expressed as: dU = δQ - δW
• The energy Udepends on certain extensive properties, such as:
  • Entropy
  • Volume
  • Number of particles
• The natural variablesthat relate to this energy are:
  • Entropy (S)
  • Volume (V)

Enthalpy

• H = U + PV, where P represents pressure and V stands for volume.
• It measures the total heat content of a system.
• This concept is especially useful for processes that happen at constant pressure, known as isobaric processes.
• Changes in enthalpy (ΔH) show the heat that is either absorbed or released during chemical reactions.
• Two important variables related to enthalpy are entropy (S) and pressure (P).

Helmholtz free energy

• A is expressed as A = U - TS, where T stands for temperature and S means entropy.
• This equation shows the useful work that can be obtained from a closed system when the temperature is constant.
• The value of A is at its lowest point when the system is in equilibrium, especially in situations with a fixed temperature and volume.
• The equation helps in figuring out the direction of spontaneous processes under conditions of constant temperature and constant volume.
• The natural variables in this context are temperature (T) and volume (V).

Gibbs free energy

• G is defined as G = H - TS or G = U + PV - TS
• G measures the maximum work that can be obtained from a system in a reversible way.
• G is at its lowest point when a system is in equilibrium at constant temperature and pressure.
• It is important for understanding chemical equilibrium and whether reactions will occur spontaneously.
• The key variables that affect G are temperature (T) and pressure (P).

Mathematical formulations

Legendre transformations

• Mathematical method used to change between different thermodynamic potentials.
• Facilitates the conversion between conjugate variables, which include both extensive and intensive properties.
• Maintains the information content while altering the independent variables.
• General formula:
• F(x,y) = f(x,y) - y ∂f/∂y
• This method allows for the derivation of new potentials from existing ones, such as transforming U (internal energy) to H (enthalpy) and A (Helmholtz free energy) to G (Gibbs free energy).

Partial derivatives relationships

Various thermodynamic properties can be linked through mathematical equations. 

• Fundamental Equation:
  • dU: This represents the change in internal energy. 
  • T: This is the temperature. 
  • dS: This indicates the change in entropy. 
  • P: This refers to pressure. 
  • dV: This signifies the change in volume. 
  • dN: This denotes the change in the number of particles. 
  • The fundamental equation can be expressed as: dU = TdS - PdV + μdN
• Derived Relationships:
  • (∂U/∂S)_V,N = T
  • (∂U/∂V)_S,N = -P
  • (∂U/∂N)_S,V = μ

These equations allow for the calculation of one thermodynamic quantity based on measurements of other quantities.

Maxwell Relations

Maxwell relations are extremely important for two reasons: 

• 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.
• 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 From first law of thermodynamic
dU = TdS - PdV
Hence U,V, and S are perfect differential.
Then

Similarly
Equating equation (1) and (2),

Maxwell first relation:- Put x = T , y = V

Maxwell Second Relation:-  Put x = T ,y = P
Maxwell Third Relation:- Put x = s , y = v

Maxwell Fourth Relation:- Put x = S, y = P

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
then M and N will satisfy the condition

Internal Energy

U and second from the first laws of thermodynamics dU = TdS - PdVFrom Legendre transformationfrom given relation one can derive Maxwell relation

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,

From above relation one can derive Maxwell relation 

△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

From above relation one can derive Maxwell relation