Chapter 6, Problem 6.70E

Rearrange the Clausius-Clapeyron equation, equation 6.14 in terms of the pressure $p_2$ of a material. Plot the vapor pressures of ${\text{H}}_{\text{2}}\text{O}$ (the boiling point is $\text{100}°\text{C}$, ${\Delta }_{\text{vap}}H=40.71\text{kJ/mol}$), neon (the boiling point is $-\text{246}°\text{C}$, ${\Delta }_{\text{vap}}H=1.758\text{kJ/mol}$), and $\text{Li}$ (the boiling point is $\text{1342}°\text{C}$, ${\Delta }_{\text{vap}}H=134.7\text{kJ/mol}$). Although these three materials are very different, are there any similarities in the behavior of the vapor pressures as the temperature increases?

Interpretation Introduction

Interpretation:

The vapor pressures of ${\text{H}}_{\text{2}}\text{O}$, neon and $\text{Li}$ are to be plotted and any similarities in the behavior of the vapor pressures with the increase in temperature is to be stated.

Concept introduction:

The Clausius-Clapeyron equation can be obtained from the rearrangement and integration of Clapeyron equation. The Clausius-Clapeyron equation is generally used for gas-phase equilibria, to predict the equilibrium temperatures and pressures and also for the determination of enthalpy for phase transition.

Answer to Problem 6.70E

The plot between vapor pressure and temperature for ${\text{H}}_{\text{2}}\text{O}$, neon and $\text{Li}$ is shown below.

Figure 1

From the plot, the common observation is that for the given substances the increase in vapor pressure is slow until the normal boiling point.

Explanation of Solution

The Clausius-Clapeyron equation 6.14 is,

$\ln \frac{p_2}{p_1}=-\frac{\Delta H}{R}\left(\frac{1}{T_2}-\frac{1}{T_1}\right)$

Rearrange the given equation for the partial pressure $p_2$ as shown below.

$\begin{array}{rll}\ln \frac{p_2}{p_1}& =& -\frac{\Delta H}{R}\left(\frac{1}{T_2}-\frac{1}{T_1}\right)\\ \frac{p_2}{p_1}& =& e^{-\frac{\Delta H}{R}\left(\frac{1}{T_2}-\frac{1}{T_1}\right)}\\ p_2& =& p_1{e}^{-\frac{\Delta H}{R}\left(\frac{1}{T_2}-\frac{1}{T_1}\right)}\end{array}$

Given boiling point of ${\text{H}}_{\text{2}}\text{O}$, neon and $\text{Li}$ is $\text{100}°\text{C}$, $-\text{246}°\text{C}$ and $\text{1342}°\text{C}$ respectively and the enthalpy of vaporization is $40.71\text{kJ/mol}$, $1.758\text{kJ/mol}$ and $134.7\text{kJ/mol}$ respectively.

Calculation of partial pressure $p_2$ for ${\text{H}}_{\text{2}}\text{O}$.

$\Delta H$	$T_1$	$p_1$	$T_2$	$p_2$
$40.71\text{kJ/mol}$	$373\text{K}$	$1\text{atm}$	$300\text{K}$	$1.04\times {10}^{-5}\text{atm}$
$40.71\text{kJ/mol}$	$373\text{K}$	$1\text{atm}$	$500\text{K}$	$28.05\text{atm}$
$40.71\text{kJ/mol}$	$373\text{K}$	$1\text{atm}$	$750\text{K}$	$734.14\text{atm}$

Calculation of partial pressure $p_2$ for neon.

$\Delta H$	$T_1$	$p_1$	$T_2$	$p_2$
$1.758\text{kJ/mol}$	$27\text{K}$	$1\text{atm}$	$20\text{K}$	$0.064\text{atm}$
$1.758\text{kJ/mol}$	$27\text{K}$	$1\text{atm}$	$100\text{K}$	$5.72\text{atm}$
$1.758\text{kJ/mol}$	$27\text{K}$	$1\text{atm}$	$200\text{K}$	$817.31\text{atm}$

Calculation of partial pressure $p_2$ for $\text{Li}$.

$\Delta H$	$T_1$	$p_1$	$T_2$	$p_2$
$134.7\text{kJ/mol}$	$1615\text{K}$	$1\text{atm}$	$1500\text{K}$	$0.46\text{atm}$
$134.7\text{kJ/mol}$	$1615\text{K}$	$1\text{atm}$	$1600\text{K}$	$0.91\text{atm}$
$134.7\text{kJ/mol}$	$1615\text{K}$	$1\text{atm}$	$1800\text{K}$	$2.801\text{atm}$

The plot between vapor pressure and temperature for ${\text{H}}_{\text{2}}\text{O}$, neon and $\text{Li}$ is shown below.

Figure 1

To observe the change in the plot of lithium, the values of vapor pressure is considered till $10\text{atm}$ as shown below.

Figure 2

From the plot, the common observation is that for the given substances the increase in vapor pressure is slow until the normal boiling point. After the normal boiling point, the increase in vapor pressure is exponential. As the temperature increases from lower normal boiling point to higher values, the exponent value changes from negative to positive.

Conclusion

From the plot, the common observation is that for the given substances, the increase in vapor pressure is slow until the normal boiling point.

Conclusion

From the plot, the common observation is that for the given substances, the increase in vapor pressure is slow until the normal boiling point.