Consider a simple thermodynamic system in which particles can occupy only two states: a lower state, whose energy we define as 0 , and an upper state, energy$E_{u\circ }$

(a) Cany out the sum (with only two states, integration is certainly not valid) giving the average particle energy E. and plot your result as a function of temperature.

(b) Explain qualitatively why it should behave as it does,

(c) This system can be used as a model of paramagnetic, where individual atoms' magnetic moments can either be aligned or anti aligned with an external magnetic field, giving a low or high energy, respectively. Describe how the average alignment or antialignment depends on temperature. Does it make sense'?

The expression for average energy is given by,

$\overline{E}=\frac{{\sum }_n{E}_{\text{n}}{e}^{-\frac{E_{\text{n}}}{k_{\text{B}}T}}}{{\sum }_n{e}^{-\frac{E_{\text{n}}}{k_{\text{B}}T}}}$

It is given,

The lower state energy is 0.

The upper state energy is$E_{\text{u}}$

Calculation:

The average energy is calculated as,

$\begin{array}{rcl}\overline{E}& =& \frac{{\sum }_n{E}_{\text{n}}{e}^{-\frac{E_{\text{n}}}{k_{\text{B}}T}}}{\sum e^{-\frac{E_{\text{n}}}{k_{\text{B}}T}}}\\ & =& \frac{E_0{e}^{-\frac{E_0}{k_{\text{B}}T}}+E_{\text{n}}{e}^{-\frac{E_{\text{n}}}{k_{\text{B}}T}}}{e^{-\frac{E_0}{k_{\text{B}}T}}+e^{-\frac{E_{\text{n}}}{k_{\text{B}}T}}}\\ & =& \frac{\left(0\right)e^{-\frac{0}{k_{\text{B}}T}}+E_{\text{n}}{e}^{-\frac{E_{\text{n}}}{k_{\text{B}}T}}}{e^{-\frac{0}{k_{\text{B}}T}}+e^{-\frac{E_{\text{n}}}{k_{\text{B}}T}}}\\ & =& \frac{E_{\text{n}}{e}^{-\frac{E_{\text{n}}}{k_{\text{B}}T}}}{1+e^{-\frac{E_{\text{n}}}{k_{\text{B}}T}}}\\ & & \end{array}$

The graph for average energy as a function of temperature is shown below

Consider figure 1, the reason for the behaviour of the graph is that at first, the particles occupy the level, because it is easier to fill lower energy than higher ones. Now, as the temperature increases it becomes easier to fill the higher energy level $E_{\text{u}}$.

The moment when temperature is raised, it is more likely to find the magnetic moment that are anti-aligned with the external magnetic field, because there is more heat energy is available to do so. When the temperature is further increased, the number of aligned and anti-aligned moments becomes nearly equal with a slight preference for the aligned moments.

It does make sense because when the more amount of the heat is added, the more amount of the moment will be randomized that gives nearly equal chance for the alignments and anti-alignments of the moments.

Conclusion:

Therefore, the alignment and anti-alignments are nearly equal when the temperature is very high.