Complete the following for the Gibbs free energy. See the notes for examples with other state

functions.

a$.$ Derive the thermodynamic potential $($or differential form$)$ starting from the definition

for the Gibbs free energy. G $=$ H$-$TS$.$ Show all steps, substitutions, and cancellations.

b$.$ Give the natural variables.

c$.$ Write the total derivative based on the natural variables.

d$.$ Compare the total derivative to the thermodynamic potential. What are the

thermodynamic driving forces in terms of measurable quantities and partial derivative

quantities?

e$.$ Can we say anything definitive about the sign of $(\frac{\text{d}\text{e}\text{l}\text{G}}{\text{d}\text{e}\text{l}\text{T}}{)}_P?$ Why or why not?

f$.$ The Maxwell relation for the Gibbs free energy is $(\frac{\text{d}\text{e}\text{l}\text{V}}{\text{d}\text{e}\text{l}\text{T}}{)}_P=-(\frac{\text{d}\text{e}\text{l}\text{S}}{\text{d}\text{e}\text{l}\text{P}}{)}_T.$ Derive this equation

by setting the mixed $2^{nd}$ derivatives equal to each other.Find a vector equation and parametric equations for the line segment that joins P to Q$.$ Find a vector equation and parametric equations for the line segment that joins P to Q$.$

P$(0,-1,4),$ Q$(1/2,1/3,1/4)$