Question

Calculate the value of ${\mathit{c}}_{\mathbf{p}}$ at$\mathbf{298}\mathbf{}\mathit{K}$ and $\mathbf{1}\mathbf{}\mathit{a}\mathit{t}\mathit{m}$pressure predicted for ${\mathbf{I}}_{\mathbf{2}}\mathbf{\left(}\mathbf{g}\mathbf{\right)}$and $\mathbf{H}\mathbf{I}\mathbf{\left(}\mathbf{g}\mathbf{\right)}$by the classical equipartition theorem. Compare the predicted results with the experimental results (see Appendix D) and calculate the percent of the measured value that arises from vibration motions.

Short answer

The heat capacity at constant pressure,$C_p$ value for ${\text{I}}_2$and $\text{HI}$ is calculated as, $C_p\left({\text{I}}_2\right)=C_p\left(\text{HI}\right)=29.1\text{J}/\text{molK}$.

Step-by-step solution

Given data

The value of $C_p\left({\text{I}}_2\right)=36.90\text{J}/\text{molK}$ and $C_p\left(\text{HI}\right)=29.16\text{J}/\text{molK}$.

Concept of the heat capacity

The Joseph Black was the first to investigate the idea of specific heat; he observed that equal mass of different substances require varying quantities of heat to create the same temperature rise.

Expression of specific heat at constant pressure and volume

Two specific heats are defined for gases, one for constant volume $\left(C_v\right)$and one for constant pressure $\left(C_p\right)$.

The relation between the two is obtained from the first law of thermodynamics $C_p=C_v+R$.

The general rule for determine$C_v$ , $C_v=\frac{f}{2}R$.

Where f is the number of degrees of freedom.

Expression of the di-atomic molecule

For diatomic molecules, two rotational degrees of freedom are added to the three translational degrees of freedom, corresponding to the rotation about two perpendicular axes through the center of the molecule.

That gives us 5 total degrees of freedom.

This means that:

$$\begin{gathered} C_v=\frac{5}{2}R \\ {C}_p=\frac{5}{2}R+R \\ {C}_p=\frac{7}{2}R \end{gathered}$$

Calculation of Cp (heat capacity at constant pressure) for I2  and HI

Calculation of Both${\text{I}}_2$and HI are diatomic gases is shown below.

$$\begin{gathered} C_p\left({\text{I}}_2\right)=C_p\left(\text{HI}\right) \\ =\frac{7}{2}R \\ {C}_p\left({\text{I}}_2\right)=C_p\left(\text{HI}\right) \\ =\frac{7}{2}\times 8.314 \end{gathered}$$

Calculate further as given below.

$$\begin{gathered} C_p\left({\text{I}}_2\right)=C_p\left(\text{HI}\right) \\ =29.1\text{J}/\text{molK} \end{gathered}$$

Calculation of experimental value

Compare to the experimental values.

$$\begin{gathered} C_p\left({\text{I}}_2\right)=36.90\text{J}/\text{molK} \\ {C}_p\left(\text{HI}\right)=29.16\text{J}/\text{molK} \end{gathered}$$

As we can see that the data differs from the theoretical value. This is due to the vibrational motion.

Calculation of fraction of vibrational motion for  (iodine) molecule

Calculation of fraction of vibration motion for$I_2$is shown below.

$$\begin{gathered} C_p{\left({\text{I}}_2\right)}_{\text{exp}}-C_p{\left({\text{I}}_2\right)}_{\text{calculated}}=36.90-29.1 \\ =7.8\text{J}/\text{molK} \\ \frac{\text{vibrational motion}}{C_p}=\frac{7.8}{36.90}\times 100\% \\ \frac{\text{vibrational motion}}{C_p}=21.14\% \end{gathered}$$

Calculation of fraction of vibrational motion for  HI

Here, the calculation of fraction of vibrational motion for HI is shown below.

$$\begin{gathered} C_p(\text{HI}{)}_{exp}-C_p(\text{HI}{)}_{\text{calculated}}=29.16-29.1 \\ =0.06\text{J}/\text{molK} \\ \frac{\text{vibrational motion}}{C_p}=\frac{0.06}{29.16}\times 100\% \\ \frac{\text{vibrational motion}}{C_p}=0.21\% \end{gathered}$$

Therefore, the heat capacity at constant pressure,$C_p$ value for ${\text{I}}_2$and is written as,$C_p\left({\text{I}}_2\right)=C_p\left(\text{HI}\right)=29.1\text{J}/\text{molK}$ .