Chapter 17, Problem 17.58E

Interpretation Introduction

Interpretation:

The values of partition function with the given temperatures are to be calculated. The validation of the fact that leveling off of the value of $q$ as the temperature increases and the interpretation of the values of $q$ are to be stated.

Concept introduction:

Statistical thermodynamics is used to describe all possible configurations in a system at given physical quantities such as pressure, temperature and number of particles in the system. An important quantity in thermodynamics is partition function that is represented as,

$q\equiv {\displaystyle \sum _i{g}_i{e}^{-{\in }_i/\text{kT}}}$

Where,

• $g_i$ represents the degeneracy.

• ${\in }_i$ represents the energy of $i^{th}$ microstate.

• $k$ represents the Boltzmann constant with value $1.38\times {10}^{-23}\text{J}/\text{K}$.

• $T$ represents the temperature $\left(\text{K}\right)$.

It is also called as canonical ensemble partition function.

Answer to Problem 17.58E

The values of partition function at given temperatures $50\text{K}$, $100\text{K}$, $200\text{K}$, $300\text{K}$, $500\text{K}$ and $1000\text{K}$ is $2.530$, $3.431$, $4.898$, $5.894$, $7.080$ and $8.320$ respectively. The fact that leveling off of the value of $q$ as the temperature increases has been validated.

Explanation of Solution

The given values of temperatures are $50\text{K}$, $100\text{K}$, $200\text{K}$, $300\text{K}$, $500\text{K}$ and $1000\text{K}$.

It is given that a system has energy levels at $0$, $1\times {10}^{-21}\text{J}$, $2.5\times {10}^{-21}\text{J}$, $4\times {10}^{-21}\text{J}$ and $6\times {10}^{-21}\text{J}$.

The degeneracy of each level is $2$.

The important quantity in the thermodynamic is partition function that is represented as,

$q\equiv {\displaystyle \sum _i{g}_i{e}^{-{\in }_i/\text{kT}}}$ …(1)

Where,

• $g_i$ represents the degeneracy.

• ${\in }_i$ represents the energy of $i^{th}$ microstate.

• $k$ represents the Boltzmann constant with value $1.38\times {10}^{-23}\text{J}/\text{K}$.

• $T$ represents the temperature $\left(\text{K}\right)$.

The partition function of a system that has five different energy levels with energy ${\in }_0$, ${\in }_1$, ${\in }_2$, ${\in }_3$, ${\in }_4$ and ${\in }_5$ is represented as,

$q=g{e}^{-{\in }_0/\text{kT}}+g{e}^{-{\in }_1/\text{kT}}+g{e}^{-{\in }_3/\text{kT}}+g{e}^{-{\in }_4/\text{kT}}+g{e}^{-{\in }_5/\text{kT}}$

$q=g\left[e^{-{\in }_0/\text{kT}}+e^{-{\in }_1/\text{kT}}+e^{-{\in }_3/\text{kT}}+e^{-{\in }_4/\text{kT}}+e^{-{\in }_5/\text{kT}}\right]$ …(2)

Substitute the value of energies, degeneracies, $k$ and $T=50\text{K}$ in the equation (2).

$\begin{array}{rll}q& =& 2\left[\begin{array}{c}\exp \left(-\frac{0\text{J}}{\left(50\text{K}\right)\left(1.38\times {10}^{-23}\text{J}/\text{K}\right)}\right)+\exp \left(-\frac{1\times {10}^{-21}\text{J}}{\left(50\text{K}\right)\left(1.38\times {10}^{-23}\text{J}/\text{K}\right)}\right)\\ +\exp \left(-\frac{2.5\times {10}^{-21}\text{J}}{\left(50\text{K}\right)\left(1.38\times {10}^{-23}\text{J}/\text{K}\right)}\right)+\exp \left(-\frac{4\times {10}^{-21}\text{J}}{\left(50\text{K}\right)\left(1.38\times {10}^{-23}\text{J}/\text{K}\right)}\right)\\ +\exp \left(-\frac{6\times {10}^{-21}\text{J}}{\left(50\text{K}\right)\left(1.38\times {10}^{-23}\text{J}/\text{K}\right)}\right)\end{array}\right]\\ & =& 2\left[\exp \left(0\right)+\exp \left(-1.449\right)+\exp \left(-3.623\right)+\exp \left(-5.797\right)+\exp \left(-8.696\right)\right]\\ & \approx & 2.530\end{array}$

Therefore, the value of $q$ is $2.530$ at $50\text{K}$.

Substitute the value of energies, degeneracies, $k$ and $T=100\text{K}$ in the equation (2).

$\begin{array}{rll}q& =& 2\left[\begin{array}{c}\exp \left(-\frac{0\text{J}}{\left(100\text{K}\right)\left(1.38\times {10}^{-23}\text{J}/\text{K}\right)}\right)+\exp \left(-\frac{1\times {10}^{-21}\text{J}}{\left(100\text{K}\right)\left(1.38\times {10}^{-23}\text{J}/\text{K}\right)}\right)\\ +\exp \left(-\frac{2.5\times {10}^{-21}\text{J}}{\left(100\text{K}\right)\left(1.38\times {10}^{-23}\text{J}/\text{K}\right)}\right)+\exp \left(-\frac{4\times {10}^{-21}\text{J}}{\left(100\text{K}\right)\left(1.38\times {10}^{-23}\text{J}/\text{K}\right)}\right)\\ +\exp \left(-\frac{6\times {10}^{-21}\text{J}}{\left(100\text{K}\right)\left(1.38\times {10}^{-23}\text{J}/\text{K}\right)}\right)\end{array}\right]\\ & =& 2\left[\exp \left(0\right)+\exp \left(-0.725\right)+\exp \left(-1.812\right)+\exp \left(-2.899\right)+\exp \left(-4.348\right)\right]\\ & \approx & 3.431\end{array}$

Therefore, the value of $q$ is $3.431$ at $100\text{K}$.

Substitute the value of energies, degeneracies, $k$ and $T=200\text{K}$ in the equation (2).

$\begin{array}{rll}q& =& 2\left[\begin{array}{c}\exp \left(-\frac{0\text{J}}{\left(200\text{K}\right)\left(1.38\times {10}^{-23}\text{J}/\text{K}\right)}\right)+\exp \left(-\frac{1\times {10}^{-21}\text{J}}{\left(200\text{K}\right)\left(1.38\times {10}^{-23}\text{J}/\text{K}\right)}\right)\\ +\exp \left(-\frac{2.5\times {10}^{-21}\text{J}}{\left(200\text{K}\right)\left(1.38\times {10}^{-23}\text{J}/\text{K}\right)}\right)+\exp \left(-\frac{4\times {10}^{-21}\text{J}}{\left(200\text{K}\right)\left(1.38\times {10}^{-23}\text{J}/\text{K}\right)}\right)\\ +\exp \left(-\frac{6\times {10}^{-21}\text{J}}{\left(200\text{K}\right)\left(1.38\times {10}^{-23}\text{J}/\text{K}\right)}\right)\end{array}\right]\\ & =& 2\left[\exp \left(0\right)+\exp \left(-0.362\right)+\exp \left(-0.906\right)+\exp \left(-1.449\right)+\exp \left(-2.174\right)\right]\\ & \approx & 4.898\end{array}$

Therefore, the value of $q$ is $4.898$ at $200\text{K}$.

Substitute the value of energies, degeneracies, $k$ and $T=300\text{K}$ in the equation (2).

$\begin{array}{rll}q& =& 2\left[\begin{array}{c}\exp \left(-\frac{0\text{J}}{\left(300\text{K}\right)\left(1.38\times {10}^{-23}\text{J}/\text{K}\right)}\right)+\exp \left(-\frac{1\times {10}^{-21}\text{J}}{\left(300\text{K}\right)\left(1.38\times {10}^{-23}\text{J}/\text{K}\right)}\right)\\ +\exp \left(-\frac{2.5\times {10}^{-21}\text{J}}{\left(300\text{K}\right)\left(1.38\times {10}^{-23}\text{J}/\text{K}\right)}\right)+\exp \left(-\frac{4\times {10}^{-21}\text{J}}{\left(300\text{K}\right)\left(1.38\times {10}^{-23}\text{J}/\text{K}\right)}\right)\\ +\exp \left(-\frac{6\times {10}^{-21}\text{J}}{\left(300\text{K}\right)\left(1.38\times {10}^{-23}\text{J}/\text{K}\right)}\right)\end{array}\right]\\ & =& 2\left[\exp \left(0\right)+\exp \left(-0.242\right)+\exp \left(-0.604\right)+\exp \left(-0.966\right)+\exp \left(-1.449\right)\right]\\ & \approx & 5.894\end{array}$

Therefore, the value of $q$ is $5.894$ at $300\text{K}$.

Substitute the value of energies, degeneracies, $k$ and $T=500\text{K}$ in the equation (2).

$\begin{array}{rll}q& =& 2\left[\begin{array}{c}\exp \left(-\frac{0\text{J}}{\left(500\text{K}\right)\left(1.38\times {10}^{-23}\text{J}/\text{K}\right)}\right)+\exp \left(-\frac{1\times {10}^{-21}\text{J}}{\left(500\text{K}\right)\left(1.38\times {10}^{-23}\text{J}/\text{K}\right)}\right)\\ +\exp \left(-\frac{2.5\times {10}^{-21}\text{J}}{\left(500\text{K}\right)\left(1.38\times {10}^{-23}\text{J}/\text{K}\right)}\right)+\exp \left(-\frac{4\times {10}^{-21}\text{J}}{\left(500\text{K}\right)\left(1.38\times {10}^{-23}\text{J}/\text{K}\right)}\right)\\ +\exp \left(-\frac{6\times {10}^{-21}\text{J}}{\left(500\text{K}\right)\left(1.38\times {10}^{-23}\text{J}/\text{K}\right)}\right)\end{array}\right]\\ & =& 2\left[\exp \left(0\right)+\exp \left(-0.145\right)+\exp \left(-0.362\right)+\exp \left(-0.580\right)+\exp \left(-0.870\right)\right]\\ & \approx & 7.080\end{array}$

Therefore, the value of $q$ is $7.080$ at $500\text{K}$.

Substitute the value of energies, degeneracies, $k$ and $T=1000\text{K}$ in the equation (2).

$\begin{array}{rll}q& =& 2\left[\begin{array}{c}\exp \left(-\frac{0\text{J}}{\left(1000\text{K}\right)\left(1.38\times {10}^{-23}\text{J}/\text{K}\right)}\right)+\exp \left(-\frac{1\times {10}^{-21}\text{J}}{\left(1000\text{K}\right)\left(1.38\times {10}^{-23}\text{J}/\text{K}\right)}\right)\\ +\exp \left(-\frac{2.5\times {10}^{-21}\text{J}}{\left(1000\text{K}\right)\left(1.38\times {10}^{-23}\text{J}/\text{K}\right)}\right)+\exp \left(-\frac{4\times {10}^{-21}\text{J}}{\left(1000\text{K}\right)\left(1.38\times {10}^{-23}\text{J}/\text{K}\right)}\right)\\ +\exp \left(-\frac{6\times {10}^{-21}\text{J}}{\left(1000\text{K}\right)\left(1.38\times {10}^{-23}\text{J}/\text{K}\right)}\right)\end{array}\right]\\ & =& 2\left[\exp \left(0\right)+\exp \left(-0.0725\right)+\exp \left(-0.1812\right)+\exp \left(-0.2898\right)+\exp \left(-0.4348\right)\right]\\ & \approx & 8.320\end{array}$

Therefore, the value of $q$ is $8.320$ at $1000\text{K}$.

On comparing the partition function values it is observed that, the value of partition function increases with the increase in temperature. At low temperatures the increase of $50\text{K}$ results in the increase in partition function of about $1.00$ unit, while at higher temperatures with the increase of $500\text{K}$ in temperature, the increase in partition function is of about $1.00$ unit. Thus, this shows as the temperature increases the increment in the values of partition function decreases.

Conclusion

The values of partition function at given temperatures $50\text{K}$, $100\text{K}$, $200\text{K}$, $300\text{K}$, $500\text{K}$ and $1000\text{K}$ is $2.530$, $3.431$, $4.898$, $5.894$, $7.080$ and $8.320$ respectively. The fact that leveling off of the value of $q$ as the temperature increases has been validated.