Question

Derive the following equations via Legendre Transformations:

dH = TdS + VdP

dA= -SdT- PdV

dG=-SdT + VdP

Answer 1

In thermodynamics Legendre Transformation is used to transform a thermodynamic function of one variable to a new conjugate function that depends on new variables.

We can derive thermodynamic equation for dH, dA and dG, from the known thermdynamic function

dU = TdS - PdV

Now we can derive dH, dA and dG from the known thermodynamic function, f = U whose derivative is

dU = TdS - PdV

(1) We need to derive, a new conjugate function g = H, whose derivative is dH = TdS + VdP from f = U

Here function, f = U

g = H

variable, x = S (variable not to be switched )

y = V (variable to be switched)

Similarly we can derive 2nd and 3rd equation