Average Internal Energy of 2 Paramagnets

• Poop-Loops
In summary, the problem involves two two-state paramagnets with equal number of dipoles (Na = Nb). The energies of each dipole are + or - uB and at thermal equilibrium, the average internal energy of each needs to be found. Using the equation U = -N(muB)^2/kT, it can be deduced that the average internal energy would be (Ua + Ub)/2, assuming Ua = Ub. If Ua > Ub and both are negative, Ua would give energy to Ub due to higher temperature. Similarly, if Ua < Ub and both are positive, both would tend towards infinite temperature to have higher entropy. However, it is necessary to solve for temperature first and then
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Homework Statement

The problem is that there are two , two-state paramagnets with Na = Nb (number of dipoles in each). uB = |kT|, energies of each dipole are + or - uB. Different internal energies to start, but they are brought together at thermal equilibrium and I need to find the average internal energy of each after that happens.

Homework Equations

$$U = -N \frac{(\mu B)^2}{kT}$$

The Attempt at a Solution

Since N=N, my only thought is Ua + Ub divided by 2. If Ua = Ub, then that makes sense. If Ua > Ub, but both < 0, then Ua would give energy to Ub, since the temperature is higher, yes? If Ua < Ub but both Ub > 0, Ua <0, then it depends on the exact values, but they would both tend towards having infinite temperature, since they would both want to have higher entropy, right? It just seems like I'm missing something really big here.

My book is Schroeder's Thermal Physics, which doesn't have any answers in the back, so I can't even tell if I'm doing the problems correctly.

EDIT:

My other idea is that both temperatures have to be equal to have thermal equilibrium, so I should solve for temperature first. Then solve it so that Ta = Tb and see what happens then?

EDIT2: That doesn't seem to work, either, since I get Ua/Ub = Ta/Tb, but have no idea what to do with that...

Last edited:
Hate to bump, but I have to. I have no idea whether I'm right or not, or whether I'm even on the right track.

1. What is the definition of average internal energy?

The average internal energy of a system is the total energy stored within the system, including the kinetic and potential energies of its particles. It is a measure of the system's overall thermal energy.

2. How is average internal energy calculated for 2 paramagnets?

The average internal energy of 2 paramagnets is calculated by adding the individual internal energies of each paramagnet and dividing by 2. This can be represented by the equation: E_avg = (E_1 + E_2)/2, where E_avg is the average internal energy and E_1 and E_2 are the internal energies of each paramagnet.

3. What factors affect the average internal energy of 2 paramagnets?

The average internal energy of 2 paramagnets is affected by the strength of the magnetic field, the distance between the paramagnets, and the temperature of the system. A stronger magnetic field and closer distance between paramagnets will result in a higher average internal energy, while a lower temperature will decrease the average internal energy.

4. How is the average internal energy related to the temperature of 2 paramagnets?

The average internal energy of 2 paramagnets is directly proportional to the temperature of the system. As the temperature increases, the average internal energy also increases, and vice versa. This relationship is described by the equation:E_avg = kT, where k is the Boltzmann constant and T is the temperature in Kelvin.

5. Can the average internal energy of 2 paramagnets be negative?

No, the average internal energy of 2 paramagnets cannot be negative. Internal energy is a measure of the total energy within a system, and it cannot have a negative value. However, the individual internal energies of each paramagnet may be negative, but when added together and divided by 2, the average internal energy will always be positive.

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