Purchase Solution

Thermodynamics: Maxwell's Relations

Not what you're looking for?

Ask Custom Question

Derive Maxwell's Relations from First and Second Laws of Thermodynamics and Thermodynamic Functions like Internal Energy, Helmholtz's Function, Enthalpy and Gibbs Free Energy.

And also explain how they are satisfied by an ideal monatomic gas?

Purchase this Solution

Solution Summary

This solution is comprised of detailed step-by-step derivations and analysis of the Maxwell's Relations and provides students with a clear perspective of the underlying concepts.

Solution Preview

According to the first and second laws of thermodynamics, we have respectively
dQ=dU+p dV
dQ=T dS
Combining the above two equations we get
dU=T dS-p dV [1]
Let us consider two independent variables, which determine the thermodynamic state of the system of a given gas, as x and y. Now, all thermodynamic variables, like U,S and V, will be functions of x and y.
Partial Differentiation of these thermodynamic variables with respect to x and y gives,
dU=∂U/∂x dx+∂U/∂y dy [2]
dS=∂S/∂x dx+∂S/∂y dy [3]
dV=∂V/∂x dx+∂V/∂y dy [4]

Putting [2], [3] and [4] in [1]:
∂U/∂x dx+∂U/∂y dy=T(∂S/∂x dx+∂S/∂y dy)-p(∂V/∂x dx+∂V/∂y dy)
Equating the coefficients of dx and dy from both sides of the equation, we get
∂U/∂x=T ∂S/∂x-p ∂V/∂x [5]
∂U/∂y=T ∂S/∂y-p ∂V/∂y [6]
Differentiating [5] with respect to y and [6] with respect to x, we get
(∂^2 U)/∂y∂x=∂T/∂y ∂S/∂x+T (∂^2 S)/∂y∂x-∂p/∂y ∂V/∂x-p (∂^2 V)/∂y∂x [7]
(∂^2 ...

Purchase this Solution


Free BrainMass Quizzes
Variables in Science Experiments

How well do you understand variables? Test your knowledge of independent (manipulated), dependent (responding), and controlled variables with this 10 question quiz.

Basic Physics

This quiz will test your knowledge about basic Physics.

The Moon

Test your knowledge of moon phases and movement.

Classical Mechanics

This quiz is designed to test and improve your knowledge on Classical Mechanics.

Intro to the Physics Waves

Some short-answer questions involving the basic vocabulary of string, sound, and water waves.