MAXWELL’S THERMODYNAMIC RELATIONSHIPS AND THEIR APPLICATIONS Submitted By Sarvpreet Kaur Associate Professor Department of Physics GCG-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 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
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.
By Legendre transformation (T,V): from (S,V) to Helmholtz free energy (T,P): Gibbs free energy
Product rule easy check: =:H (enthalpy) Enthalpy H=H(S,P) is a thermodynamic potential
Now dU, dF, dG and dH are exact differentials e.g Using these exact differentials we derive maxwell’s relations .
Maxwell’s Thermodynamic Relations
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.
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.
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 Relations P 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
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.
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
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..
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
3. Change of internal energy with Volume, Using the third Maxwell’s relation Since
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.
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
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
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 This equation is known as Clausius-Clapeyron’s latent heat Equation.
Thank You