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Asked Feb 6, 2020
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It's a thermodynamics question.

Consider taking the Legendre transform of the energy, E(S,V,N), with respect to S, V, and N, to get a new
thermodynamic potential, X(T.p,u).
(a) Show that X(T.p,µ) is identically zero.
(b) What does this imply about the variables T, p, and µ? Where have you already seen this result before?
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Consider taking the Legendre transform of the energy, E(S,V,N), with respect to S, V, and N, to get a new thermodynamic potential, X(T.p,u). (a) Show that X(T.p,µ) is identically zero. (b) What does this imply about the variables T, p, and µ? Where have you already seen this result before?

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Step 1

Write the expression for the total differential of the energy E given as a function of entropy S, volume V, and number of particles N.

 

Advanced Physics homework question answer, step 1, image 1

Step 2

Here T is the temperature, p is the pressure and μ is the chemical potential.

In the above expression, the conjugate pair of variables are (T, S), (p, V) and (μ, N).

To obtain the function X (T, p, μ) through Legendre transformation, consider an arbitrary function Y that assumes the following form.

Advanced Physics homework question answer, step 2, image 1

 

Step 3

Take the total differential of above function and simplify the expression.

Advanced Physics homework question answer, step 3, image 1

...

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