Question

In the previous section I derived the formula $(\partial F/\partial V{)}_T=-P$. Explain why this formula makes intuitive sense, by discussing graphs of F vs. V with different slopes.

Short answer

The equilibrium condition occurs when the slope of a curve of F verses V is equal.

Step-by-step solution

Given

$P=-{\left(\frac{\partial F}{\partial V}\right)}_{T,N}$

Explanation

From thermodynamics Helmholtz free energy is given by
$P=-{\left(\frac{\partial F}{\partial V}\right)}_{T,N}$
Here, P is the pressure, F is Helmholtz free energy, V s the volume, T is the temperature and N is the number of molecules.

Write the expression for Helmholtz free energy: F=U-T S
Where, U is the internal energy and S is the entropy of the system.
The equilibrium state occurs when the energy of the system is the minimum and when the overall entropy is maximum.

For a given number of molecules and given temperature, the increase in the volume yields an increase in the entropy and decrease in the Helmholtz free energy.
Draw a graph to show the variation of F with V.
Conclusion:
So, the equilibrium condition occurs when the slope of a curve of F verses V is equal