Derive all the Maxwell relations.

Name: Problem 3, Chapter 13: Thermodynamics: A complete undergraduate course
Uploaded: 2022-03-19T09:27:43-08:00
Duration: 5 min 21 s
Channel: Mohammad Mehran
Description: Derive all the Maxwell relations.

Key Concepts

Exact Differentials

Exact differentials are expressions that can be expressed as the total derivative of a state function. In thermodynamics, they ensure that the integral of a differential around a closed path is zero, which implies that the order of differentiation is irrelevant. This property is crucial because it allows one to equate mixed partial derivatives of thermodynamic potentials and is the mathematical underpinning of Maxwell relations.

Thermodynamic Potentials

Thermodynamic potentials, such as the internal energy, enthalpy, Helmholtz free energy, and Gibbs free energy, are functions that describe the state of a system in terms of its natural variables. Each potential has a differential form that captures how the potential changes with its associated variables. These differential forms are the starting point for deriving Maxwell relations by exploiting the condition of exact differentials.

Legendre Transformations

Legendre transformations are mathematical operations used to switch between different thermodynamic potentials by changing the natural variables of a function. They are essential in thermodynamics because they allow one to derive potentials that are more convenient for a given set of constraints, thereby facilitating the use of Maxwell relations in different scenarios.

Schwarz's Theorem (Symmetry of Second Derivatives)

Schwarz's theorem states that if a function's second partial derivatives are continuous, then the order of differentiation does not matter. This symmetry is foundational for Maxwell relations since it justifies setting the mixed partial derivatives of thermodynamic potentials equal to each other, linking different thermodynamic variables.

Maxwell Relations

Maxwell relations are a set of equations derived from the equality of mixed partial derivatives of thermodynamic potentials. They provide useful relationships between different thermodynamic properties, such as entropy, volume, pressure, and temperature. These relations are instrumental in both the theoretical framework and practical calculations in thermodynamics.

Transcript

00:01 All right guys, to determine the maxwell equation, we will use the cyclic relation.

00:07 And the first maxwell equation, we start by substituting the first maxwell equation, maxwell equation into the cyclic relation.

00:18 So we've got partial derivative b over partial derivative s into partial derivative s over partial derivative b, multiplied by partial derivative v over partial derivative b minus 1 so we have partial derivative t over partial derivative b is equal to minus partial derivative p divided by partial derivative s so minus partial derivative t over partial derivative b into partial derivative s over partial derivative s over partial derivative v into partial derivative b over partial derivative p is equal to minus 1 we can now rearrange the last equation by grouping the first and the last term into the second max maxwell equation so we have partial derivative t divided by partial derivative p into partial derivative s over partial derivative b is equal to one so partial derivative t over partial derivative p is equal to partial derivative b over partial derivative s for the third maxwell equation we repeat the same process but we use different properties in the cyclic relation so we have partial derivative t over partial derivative v into partial derivative v over partial derivative s into partial derivative s over partial derivative t is equal to minus 1 so we have minus partial derivative b over partial derivative s into partial derivative v over partial derivative s into partial derivative s over partial derivative t is equal to minus 1.

02:57 Again, we group the first and the last term in the last equation and rearrange it to match the third maxwell equation.

03:06 So we have got partial derivative b over partial derivative t into partial derivative v over partial derivative s is equal to 1.

03:18 So we have partial derivative s over partial derivative b is equal to partial derivative p over partial derivative t.

03:33 So for the fourth maxwell equation we again take the same steps but use again different properties in the cyclic relation...

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