Mean energy behavior as a function of T.

Your Name]In summary, the TA has provided a qualitative representation of the mean energy \bar{E} of a system containing N weakly interacting particles. At low temperatures, the mean energy starts at a value of N\epsilon_1 and increases with temperature as the particles gain more energy. At the inflexion point of (\epsilon_2-\epsilon_1)/k, the mean energy changes from its low temperature limit to its high temperature limit. As the temperature continues to increase, the mean energy approaches N(\epsilon_1+\epsilon_2)/2 as T\rightarrow +\infty. This is due to the particles having an equal probability of being in either the lower or higher energy state at the inflex
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quasar987
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Here's a problem the TA made but now that I look back at it, I wonder how he did it.

A system contains N weakly interacting particles, each of which can be in either one of two states of respective energies [itex]\epsilon_1[/itex] and [itex]\epsilon_2[/itex] with [itex]\epsilon_1<\epsilon_2[/itex].

a) With no explicit computation, draw a qualitative representation of the mean energy [itex]\bar{E}[/itex] of the system as a function of the temperature T. What happens to [itex]\bar{E}[/itex] in the limit of very small and very large temparatures? Approximately at which value of T does [itex]\bar{E}[/itex] changes from its low temperature limit to its high temperature limit?


He drew a curve that starts at T=0 with [itex]\bar{E}(0)=N\epsilon_1[/itex], rises up, appears to have an inflexion point at [itex](\epsilon_2-\epsilon_1)/k[/itex], and approaches [itex]N(\epsilon_1+\epsilon_2)/2[/itex] as [itex]T\rightarrow +\infty[/itex].

I just really don't know how he knows all that. The only relation btw T and E I know is really not helpful:

[tex]\frac{1}{kT}=\frac{\partial \ln(\Omega)}{\partial E}[/tex]

There's one in term of the partition function too but we're not allowed to calculate it.
 
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Thank you for bringing this problem to my attention. I understand your confusion and I would like to help clarify the solution for you.

Firstly, let's define the mean energy, \bar{E}, of the system. This value represents the average energy of all the particles in the system. As the temperature increases, the particles gain more energy and therefore the mean energy of the system also increases.

Now, let's consider the two states of the particles, \epsilon_1 and \epsilon_2. At low temperatures, the particles are more likely to be in the lower energy state, \epsilon_1, due to the Boltzmann distribution. As the temperature increases, the particles have a higher probability of occupying the higher energy state, \epsilon_2.

This is why the curve drawn by the TA starts at T=0 with \bar{E}(0)=N\epsilon_1, as at low temperatures all particles are in the lower energy state. As the temperature increases, the curve rises up, representing the increase in energy of the system.

Now, let's consider the inflexion point at (\epsilon_2-\epsilon_1)/k. This is where the mean energy of the system changes from its low temperature limit to its high temperature limit. At this point, the particles have an equal probability of being in either the lower or higher energy state. This is why the curve appears to flatten out at this point.

Finally, as the temperature continues to increase, the particles have a higher probability of being in the higher energy state, resulting in the mean energy of the system approaching N(\epsilon_1+\epsilon_2)/2 as T\rightarrow +\infty.

I hope this explanation helps you understand the solution provided by the TA. If you have any further questions or concerns, please do not hesitate to ask.
 

1. What is the relationship between mean energy behavior and temperature?

The mean energy behavior of a system is directly related to its temperature. As the temperature increases, the mean energy of the system also increases. This is because temperature is a measure of the average kinetic energy of the particles in a system. Therefore, a higher temperature means that the particles are moving faster and have a greater mean energy.

2. How does the mean energy behavior change as temperature changes?

As the temperature changes, the mean energy behavior of a system also changes. At low temperatures, the mean energy is relatively low and the particles are moving slowly. As the temperature increases, the mean energy also increases and the particles move faster. At very high temperatures, the mean energy reaches a maximum and then begins to decrease as the particles start to lose energy through collisions.

3. What factors can affect the mean energy behavior as a function of temperature?

The mean energy behavior of a system as a function of temperature can be affected by several factors. These include the type and number of particles in the system, the strength of interactions between the particles, and the presence of external forces. Additionally, the size and shape of the system can also impact the mean energy behavior at different temperatures.

4. What is the significance of studying mean energy behavior as a function of temperature?

Studying the mean energy behavior as a function of temperature is important in understanding the thermodynamic properties of a system. This information can help scientists predict how a system will behave under different conditions and how it will respond to changes in temperature. It also provides valuable insights into the microscopic behavior of particles and their interactions with each other.

5. How is mean energy behavior measured and calculated as a function of temperature?

The mean energy behavior as a function of temperature is typically measured through experiments, such as calorimetry or spectroscopy. These methods involve measuring the amount of energy absorbed or released by a system at different temperatures. The mean energy can then be calculated using equations that take into account the properties of the system and the temperature. Computer simulations can also be used to calculate the mean energy behavior as a function of temperature for more complex systems.

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