A paramagnetic crystal contains N atoms which possess spin 1/2 and magnetic moment m. The crystal is placed in a heat bath at temperature T and in a magnetic field B. Use the fact that the Helmholtz free energy is a minimum in equilibrium to show that the met magnetic moment of the crystal is
M=Nm tanh(mB/kT)

I've been trying this and I can prove the relationship but my derivation doesn't involve the Helmholtz free energy. Could someone please explain exactly what is meant by Helmholtz energy being a minimum at equilibrium and maybe point me in the correct direction, thanks very much :)

Question

A paramagnetic crystal contains N atoms which possess spin 1/2 and magnetic moment m. The crystal is placed in a heat bath at temperature T and in a magnetic field B. Use the fact that the Helmholtz free energy is a minimum in equilibrium to show that the met magnetic moment of the crystal is 
M=Nm tanh(mB/kT)

I've been trying this and I can prove the relationship but my derivation doesn't involve the Helmholtz free energy. Could someone please explain exactly what is meant by Helmholtz energy being a minimum at equilibrium and maybe point me in the correct direction, thanks very much :)

shayaan_mustafa · Answer

OK. I assume that you know Helmholtz energy.
minimum equilibrium implies that external parameters are fixed in thermal contact with a heat bath or heat reservoir at equilibrium.

Does this make sense to you?

anonymous · Answer

So its like its cut of from the surroundings?

shayaan_mustafa · Answer

yes,