Statistical mechanics/Thermodynamics two spin-1/2 subsystem

[S_Flaherty]

Homework Statement

Consider two spin-1/2 subsystems with identical magnetic moments (μ) in equal fields (B). The first subsystem has a total of N_A spins with initially "a" having magnetic moments pointing against the field and (N_A - a) pointing along the field, so that its initial energy is U_iA = μB(a - (N_A - a) = μB(2a - N_A). The second subsystem has a total of N_B spins with "b" having moments initially pointing against the field so that its initial energy is U_iB = μB(2b - N_B). Now suppose that the two subsystems are brought together so that they can exchange energy. Assume that B = constant and a, b, N_A, N_B >> 1. Show that in equilibrium, a₀/N_A = b₀/N_B, and this implies that the two subsystems will have the same "magnetization," i.e. total magnetic moment/spin.

Homework Equations

I'm not really sure what equations are useful in this case because I'm having trouble understanding what I need to be doing. I think I need to use multiplicity so
Ω(N,n) = N!/(n!(N-n)!)

The Attempt at a Solution

I think that I have to first figure out the most probable macrostate because that is where the systems would be in equilibrium(?). So do I go along the lines of solving Ω(N_A, a) and Ω(N_B, b)? I don't even know if that makes sense or what a₀ and b₀ represent in this question.

 

[DrClaude]

You're going in the right direction. Have you seen entropy yet? If so, can you express the conditions for equilibrium in terms of S and U?

My guess is that a0 and b0 are the number of spins pointing against the field at equilibrium.

 

[BruceW]

I think that I have to first figure out the most probable macrostate because that is where the systems would be in equilibrium(?). So do I go along the lines of solving Ω(N_A, a) and Ω(N_B, b)? I don't even know if that makes sense or what a₀ and b₀ represent in this question.

exactly. But also, don't forget to constrain the total energy! And yeah, as DrClaude says, the a₀ and b₀ seem to simply be a and b once equilibrium is achieved.