Response to the following problem:
The Maxwell relations in thermodynamics are obtained by treating a thermodynamic relation as an exact differential equation. Exact differential equations are of the form
dz=(dz/dx)y dx+(dz/dy)x dy=Mdx+Ndy
where z = z(x, y) and M and N correspond to the partial derivatives of z with respect to x and y, respectively. A useful relation in exact differential equations is that
(∂M/∂y)x=(∂N/∂x)y
(Can you prove why this is so?) When x, y, and z are thermodynamic quantities, such as free energy, volume, temperature, or enthalpy, the relationship between the partial differentials of M and N as described above are called Maxwell relations. Use Maxwell relations to derive the Laplace equation for a sphere at constant temperature using the following thermodynamic expression for free energy, G:
dG=VdP-SdT+ydA
where V is the volume of the sphere, P is the pressure, T is temperature, γ is the surface energy, and A is the sphere surface area.