# Consider taking the Legendre transform of the energy, E(S,V,N), with respect to S, V, and N, to get a newthermodynamic 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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It's a thermodynamics question.

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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.

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.

Step 3

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

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