Chapter 18, Problem 18.59E

Interpretation Introduction

Interpretation:

The set of equations (or a small program) to evaluate the constant-volume heat capacity for a moleculeis to be stated. The graph of the result is to be plotted. The trend for the same is to be stated. The heat capacity versus temperature (say from $298\text{K}$ to $1000\text{K}$) for ${\text{H}}_{\text{2}}\text{O}$ and ${\text{CH}}_4$ is to be determined using these algorithm.

Concept introduction:

The heat capacity at constant volume for nonlinear polyatomic molecule is given by the formula,

$C_{\text{V}}=Nk\left[\frac{3}{2}+\frac{3}{2}+{\displaystyle \sum _{j=1}^{3N^{*}-6}{\left(\frac{{\theta }_j}{T}\right)}^2\cdot \frac{e^{-{\theta }_j/T}}{{\left(1-e^{-{\theta }_j/T}\right)}^2}}\right]$

Where,

• $N$ is the Avogadro number.

• $T$ is the temperature.

• $k$ is the Boltzmann constant.

• ${\theta }_j$ is the rotational temperature.

Answer to Problem 18.59E

The set of equations (or a small program) to evaluate the constant-volume heat capacity for a molecule are,

• $Nk$

• ${\left(\frac{{\theta }_j}{T}\right)}^2$

• $e^{-{\theta }_j/T}$

• $1-e^{-{\theta }_j/T}$

• ${\left(1-e^{-{\theta }_j/T}\right)}^2$

• $\frac{e^{-{\theta }_j/T}}{{\left(1-e^{-{\theta }_j/T}\right)}^2}$

• ${\left(\frac{{\theta }_j}{T}\right)}^2\cdot \frac{e^{-{\theta }_j/T}}{{\left(1-e^{-{\theta }_j/T}\right)}^2}$

The plot between $C_{\text{V}}$ and $T$ for ${\text{CH}}_4$ is,

The plot between $C_{\text{V}}$ and $T$ for ${\text{H}}_{\text{2}}\text{O}$ is,

Explanation of Solution

The heat capacity at constant volume for nonlinear polyatomic molecule is given by the formula,

$C_{\text{V}}=Nk\left[\frac{3}{2}+\frac{3}{2}+{\displaystyle \sum _{j=1}^{3N^{*}-6}{\left(\frac{{\theta }_j}{T}\right)}^2\cdot \frac{e^{-{\theta }_j/T}}{{\left(1-e^{-{\theta }_j/T}\right)}^2}}\right]$…(1)

The set of equations(or a small program) to evaluate the constant-volume heat capacity for a molecule are shown below.

• $Nk$

• ${\left(\frac{{\theta }_j}{T}\right)}^2$

• $e^{-{\theta }_j/T}$

• $1-e^{-{\theta }_j/T}$

• ${\left(1-e^{-{\theta }_j/T}\right)}^2$

• $\frac{e^{-{\theta }_j/T}}{{\left(1-e^{-{\theta }_j/T}\right)}^2}$

• ${\left(\frac{{\theta }_j}{T}\right)}^2\cdot \frac{e^{-{\theta }_j/T}}{{\left(1-e^{-{\theta }_j/T}\right)}^2}$

The vibrational temperatures for ${\text{CH}}_4$ are $\text{7}\text{.54}\text{K}$.

Substitute the value of vibrational temperatures for ${\text{CH}}_4$ and $\text{298}\text{K}$ in equation (1).

$C_{\text{V}}=\left(6.02\times {10}^{23}{\text{mol}}^{-1}\right)\left(1.381\times {10}^{-23}{\text{JK}}^{-1}\right)\left\{\frac{3}{2}+\frac{3}{2}+{\left(\frac{7.54\text{K}}{298\text{K}}\right)}^2\cdot \frac{\exp \left(-\frac{7.54\text{K}}{298\text{K}}\right)}{{\left[1-\exp \left(-\frac{7.54\text{K}}{298\text{K}}\right)\right]}^2}\right\}$

The value of $C_{\text{V}}$ at different temperature is shown below.

$T$${\left(\frac{{\theta }_j}{T}\right)}^2$$\frac{e^{-{\theta }_j/T}}{{\left(1-e^{-{\theta }_j/T}\right)}^2}$$C_{\text{V}}$

$\begin{array}{rrrr}\hfill 298& \hfill 88804& \hfill 2439943& \hfill 20275925\\ \hfill 348& \hfill 121104& \hfill 4537656& \hfill 37707923\\ \hfill 398& \hfill 158404& \hfill 7763309& \hfill 64513102\\ \hfill 448& \hfill 200704& \hfill 12463117& \hfill 1.04E+08\end{array}$

$\begin{array}{rrrr}\hfill 498& \hfill 248004& \hfill 19029702& \hfill 1.58E+08\\ \hfill 548& \hfill 300304& \hfill 27902097& \hfill 2.32E+08\\ \hfill 598& \hfill 357604& \hfill 39565745& \hfill 3.29E+08\\ \hfill 648& \hfill 419904& \hfill 54552498& \hfill 4.53E+08\end{array}$

$\begin{array}{rrrr}\hfill 698& \hfill 487204& \hfill 73440615& \hfill 6.1E+08\\ \hfill 748& \hfill 559504& \hfill 96854769& \hfill 8.05E+08\\ \hfill 798& \hfill 636804& \hfill 1.25E+08& \hfill 1.04E+09\\ \hfill 848& \hfill 719104& \hfill 1.6E+08& \hfill 1.33E+09\end{array}$

$\begin{array}{rrrr}\hfill 898& \hfill 806404& \hfill 2.01E+08& \hfill 1.67E+09\\ \hfill 948& \hfill 898704& \hfill 2.5E+08& \hfill 2.08E+09\\ \hfill 998& \hfill 996004& \hfill 3.07E+08& \hfill 2.55E+09\end{array}$

The plot between $C_{\text{V}}$ and $T$ is shown below.

Figure 1

The three vibrational temperatures for ${\text{H}}_{\text{2}}\text{O}$ are $\text{13}\text{.4}\text{K}$, $\text{20}\text{.9}\text{K}$ and $\text{40}\text{.1}\text{K}$.

Substitute the value of vibrational temperatures for ${\text{H}}_{\text{2}}\text{O}$ and $\text{298}\text{K}$ in equation (1).

$C_{\text{V}}=\left(6.02\times {10}^{23}{\text{mol}}^{-1}\right)\left(1.381\times {10}^{-23}{\text{JK}}^{-1}\right)\left\{\begin{array}{l}\frac{3}{2}+\frac{3}{2}+{\left(\frac{13.4\text{K}}{298\text{K}}\right)}^2\cdot \frac{\exp \left(-\frac{13.4\text{K}}{298\text{K}}\right)}{{\left[1-\exp \left(-\frac{13.4\text{K}}{298\text{K}}\right)\right]}^2}\\ +{\left(\frac{20.9\text{K}}{298\text{K}}\right)}^2\cdot \frac{\exp \left(-\frac{20.9\text{K}}{298\text{K}}\right)}{{\left[1-\exp \left(-\frac{20.9\text{K}}{298\text{K}}\right)\right]}^2}\\ +{\left(\frac{40.1\text{K}}{298\text{K}}\right)}^2\cdot \frac{\exp \left(-\frac{40.1\text{K}}{298\text{K}}\right)}{{\left[1-\exp \left(-\frac{40.1\text{K}}{298\text{K}}\right)\right]}^2}\end{array}\right\}$

The value of $C_{\text{V}}$ at different temperature is shown below.

$T$${\left(\frac{{\theta }_j}{T}\right)}^2\cdot \frac{e^{-{\theta }_j/T}}{{\left(1-e^{-{\theta }_j/T}\right)}^2}$$C_{\text{V}}$

$\begin{array}{rrr}\hfill 298& \hfill 2.997914& \hfill 24.91267\\ \hfill 448& \hfill 2.999077& \hfill 24.92233\\ \hfill 598& \hfill 2.999482& \hfill 24.92569\\ \hfill 748& \hfill 2.999669& \hfill 24.92725\end{array}$

$8982.9997724.92809$

The plot between $C_{\text{V}}$ and $T$ for ${\text{H}}_{\text{2}}\text{O}$ is shown below.

Figure 2

Conclusion

The set of equations (or a small program) to evaluate the constant-volume heat capacity for a molecule are,

• $Nk$

• ${\left(\frac{{\theta }_j}{T}\right)}^2$

• $e^{-{\theta }_j/T}$

• $1-e^{-{\theta }_j/T}$

• ${\left(1-e^{-{\theta }_j/T}\right)}^2$

• $\frac{e^{-{\theta }_j/T}}{{\left(1-e^{-{\theta }_j/T}\right)}^2}$

• ${\left(\frac{{\theta }_j}{T}\right)}^2\cdot \frac{e^{-{\theta }_j/T}}{{\left(1-e^{-{\theta }_j/T}\right)}^2}$

The plot between $C_{\text{V}}$ and $T$ for ${\text{CH}}_4$ and ${\text{H}}_{\text{2}}\text{O}$ is shown in Figure 1 and Figure 2.