Problem involving Negative Temperature

[haruspex]

how do I find the equilibrium temperature of these two ?

Despite my having been able to figure out the first part, the subject is something I've never studied, so just guessing here... won't equilibrium be when A and B have the same total energies?

 

[tanaygupta2000]

Despite my having been able to figure out the first part, the subject is something I've never studied, so just guessing here... won't equilibrium be when A and B have the same total energies?

Sir but A is having positive temperature while B is having negative temperature.

 

[haruspex]

Sir but A is having positive temperature while B is having negative temperature.

Yes... whatever that really means. But I still don't see how they can be in equilibrium unless the energy is shared equally. Then again, it's not a subject I claim to know anything about.

 

[TSny]

(2.) If A and B are two subsystems of the given system, each having N/2 particles, and A having energy E(A) = Nε/8 and B having energy E(B) = 5Nε/8, which of these has positive/negative temperature?

If B has N/2 particles, then the maximum possible energy for system B is when all N/2 particles have the energy ε. Thus, the maximum energy for B is Nε/2. But, E(B) is given to be 5Nε/8 which is greater than the maximum.

I wonder if the problem meant E(A) = N_Aε/2 and E(B) = 5N_Bε/8, where N_A = N_B = N/2.

Anyway, I agree with @haruspex that the energy of the two subsystems should be equal when the subsystems are in equilibrium.

 

[tanaygupta2000]

If B has N/2 particles, then the maximum possible energy for system B is when all N/2 particles have the energy ε. Thus, the maximum energy for B is Nε/2. But, E(B) is given to be 5Nε/8 which is greater than the maximum.

I wonder if the problem meant E(A) = N_Aε/2 and E(B) = 5N_Bε/8, where N_A = N_B = N/2.

Anyway, I agree with @haruspex that the energy of the two subsystems should be equal when the subsystems are in equilibrium.

Sir but how do I solve the third part of the question? What can I say about the equilibrium temperature?

 

[TSny]

Sir but how do I solve the third part of the question? What can I say about the equilibrium temperature?

See your post #5. You can get T from E.

 

[tanaygupta2000]

See your post #5. You can get T from E.

We deduced, Nε/2 < E < Nε
Since E = Nε/(1+e^(ε/kT))
=> Nε/2 < Nε/(1+e^(ε/kT)) < Nε
=> 1/2 < 1/(1+e^(ε/kT)) < 1
=> 2 > 1+e^(ε/kT) > 1
=> 1 > e^(ε/kT) > 0
=> 0 > ε/kT > -∞
=> 0 > 1/T > -∞
Is this correct ?

 

[TSny]

Once the two subsystems A and B are in equilibrium with each other, you can treat the whole system as one system in a state of thermal equilibrium. So, you can apply the relation E = Nε/(1+e(ε/kT)) to the whole system. You know the value of E for the whole system. Therefore, you can solve the relation for T to get the equilibrium temperature of the system.

 

[tanaygupta2000]

Once the two subsystems A and B are in equilibrium with each other, you can treat the whole system as one system in a state of thermal equilibrium. So, you can apply the relation E = Nε/(1+e(ε/kT)) to the whole system. You know the value of E for the whole system. Therefore, you can solve the relation for T to get the equilibrium temperature of the system.

Using E = Nε/(1+e^(ε/kT)), I am getting T = (ε/k) /ln(Nε/E - 1)

Substituting E = Nε/8 + 5Nε/8 = 3Nε/4
=> T = (ε/k) /ln(1/3) = (ε/k) /(-1.098) = -0.91ε/k

Hence the system as a whole is having NEGATIVE TEMPERATURE.

 

[TSny]

Using E = Nε/(1+e^(ε/kT)), I am getting T = (ε/k) /ln(Nε/E - 1)

Substituting E = Nε/8 + 5Nε/8 = 3Nε/4
=> T = (ε/k) /ln(1/3) = (ε/k) /(-1.098) = -0.91ε/k

Hence the system as a whole is having NEGATIVE TEMPERATURE.

Yes. That looks right if the total energy is 3Nε/4.

However, as I mentioned earlier, I don't believe it's possible for system B to have an initial energy of 5Nε/8. The maximum possible energy of B (or A) is (N/2)ε, since B has only N/2 particles.