Why Is Energy Held Fixed in an Open System Equation?

[ehrenfest]

Homework Statement

I am confused about this solution. The question explicitly says that the systems are allowed energy exchange, so why is energy held fixed in the equation after "therefore"?

Homework Equations

The Attempt at a Solution

 

[siddharth]

The solution does not claim that the energy of each system is held fixed. In the equation, the parameter $\zeta$ doesn't represent the the energy of the state. The relation,

$$\zeta_A = \zeta_B$$

is obtained from the fact that at equilibrium, the closed system is at its most probable state (ie, the entropy is maximum).

Refer to section 1.7 in "Elementary Statistical Physics" by Kittel or section 3.4 in Reif's statistical physics text for a discussion.

 

[ehrenfest]

The equation after "since" is always true, correct?

I am just confused about how they went from that to the next equation with the partial derivatives. It seems like they just divided each side by dN_A and then changed the total derivatives to partial derivatives and then added the subscripts V,E which I understand are redundant. I am just confused about how they changed the total derivatives to partial derivatives.

 

[siddharth]

The equation after "since" is always true, correct?

Yes, that's the combined law for an open system.

I am just confused about how they went from that to the next equation with the partial derivatives. It seems like they just divided each side by dN_A and then changed the total derivatives to partial derivatives and then added the subscripts V,E which I understand are redundant. I am just confused about how they changed the total derivatives to partial derivatives.

It's just a differential, right? For example, if S is a function of N,V,E, ie S=S(N,V,E) then

$$dS = \frac{\partial S}{\partial N} dN + \frac{\partial S}{\partial V} dV + \frac{\partial S}{\partial E} dE$$

and at constant V,E

$$dS = \frac{\partial S}{\partial N} dN$$

 

[ehrenfest]

Why would the partial $$\left(\frac{\partial{E_A}}{\partial{N_A}}\right)_{V,E}$$ not be identically 0 because the subscript E means energy is not changing?

 

[kdv]

Why would the partial $$\left(\frac{\partial{E_A}}{\partial{N_A}}\right)_{V,E}$$ not be identically 0 because the subscript E means energy is not changing?

The total energy does not change, but the energy of the two subsytems A and B may vary.

 

[ehrenfest]

It's just a differential, right? For example, if S is a function of N,V,E, ie S=S(N,V,E) then

$$dS = \frac{\partial S}{\partial N} dN + \frac{\partial S}{\partial V} dV + \frac{\partial S}{\partial E} dE$$

and at constant V,E

$$dS = \frac{\partial S}{\partial N} dN$$

I don't see how that explains the equation after "therefore".

 

[siddharth]

I don't see how that explains the equation after "therefore".

Why not? The notation is to tell you that V and E are held constant. I think it follows directly.

 

[cracketabhi]

Why not? The notation is to tell you that V and E are held constant. I think it follows directly.

dudes... the question asked the minimum value of... so . if u've got the answer after manipulation, u must have ASSUMED the minimum value condition in the course of the problem and thus the answer...

@ others - wat say?