Question

The data from Problem P9.20 can be expressed in terms of the molality rather than the mole fraction of \(\mathrm{Br}_{2}\). Use the data from the following table and a graphical method to determine the Henry's law constant for \(\mathrm{Br}_{2}\) in \(\mathrm{CCl}_{4}\) at \(25^{\circ} \mathrm{C}\) in terms of molality. $$\begin{array}{lllc} m_{B r_{2}} & P(\text { Torr }) & m_{B r_{2}} & P(\text { Torr }) \\ \hline 0.026 & 1.52 & 0.086 & 5.43 \\ 0.028 & 1.60 & 0.157 & 9.57 \\ 0.039 & 2.39 & 0.158 & 9.83 \\ 0.067 & 4.27 & 0.167 & 10.27 \end{array}$$

Short answer

To determine the Henry's law constant for \(\mathrm{Br}_{2}\) in \(\mathrm{CCl}_{4}\) at \(25^{\circ} \mathrm{C}\) in terms of molality, plot the given data points with molality (m) on the x-axis and partial pressure (P) on the y-axis. Create a best fit straight line through the data points and calculate its slope using the formula \(\mathrm{slope} = \frac{\Delta P}{\Delta m}\). The slope of the line will be equal to the Henry's law constant (H).

Step-by-step solution

Plot the data points

For plotting the data points, create a graph with molality (m) on the x-axis and partial pressure (P) on the y-axis. Plot individual data points given in the table.

Create a best fit line

Use a straight line that best fits the data points. This line should minimize the cumulative distance between the line and the data points.

Find the slope of the line

Calculate the slope of the best fit line. The slope can be found using two points on the line and the following formula: $$\mathrm{slope} = \frac{\Delta P}{\Delta m} = \frac{P_2 - P_1}{m_2 - m_1}$$ Where P1 and P2 are the partial pressures, m1 and m2 are the molalities of two points on the line.

Determine the Henry's law constant

The slope of the line found in step 3 is equal to the Henry's law constant, H. So the value of H can be directly taken from the slope. Thus, by following the above steps, the Henry's law constant for \(\mathrm{Br}_{2}\) in \(\mathrm{CCl}_{4}\) at \(25^{\circ} \mathrm{C}\) in terms of molality can be determined.

Key concepts

Molality

Molality is a measure of the concentration of a solute in a solution. In chemistry, it's defined as the number of moles of solute per kilogram of solvent. Unlike molarity, which depends on the volume of the solution, molality is temperature-independent as it's based on the mass of the solvent, making it particularly useful when dealing with scenarios involving temperature changes.

To calculate the molality (\textbf{m}), you would use the following formula:
\[m = \frac{\text{moles of solute}}{\text{kilograms of solvent}}\]
This calculation comes into play when examining Henry's law, which involves the solubility of gases in liquids under various conditions of pressure and temperature.

Partial Pressure

Partial pressure refers to the pressure that a specific gas in a mixture of gases would exert if it alone occupied the entire volume of the mixture at the same temperature. It's a key concept in understanding gas-liquid interactions, such as those described by Henry's law.

The partial pressure (\textbf{P}), in the context of this exercise, is directly proportional to the molality of \(\mathrm{Br}_{2}\) in \(\mathrm{CCl}_{4}\) at a constant temperature. The given data points (\textbf{m} and \textbf{P}) can be plotted to visualize this relationship and to further determine the Henry's law constant.

Graphical Method

The graphical method involves visual data analysis where information is plotted on a graph to identify trends, correlations, and patterns. It is useful in scenarios where a theoretical or empirical relationship between variables is being investigated.

In the exercise, the molality of \(\mathrm{Br}_{2}\) is plotted on the x-axis, while the partial pressure is on the y-axis. Each point represents a different concentration of \(\mathrm{Br}_{2}\) and its corresponding partial pressure in \(\mathrm{CCl}_{4}\). The graphical method here not only provides a visual representation of the data but also sets the stage for the next concept: the best fit line.

Best Fit Line

A best fit line is a straight line that runs through the data points on a graph in a way that minimizes the overall distance between the line and each of the points. This is often referred to as the least squares method. In our exercise, finding a best fit line among the data points of partial pressure and molality helps to visually and mathematically discern the relationship described by Henry's law.

By determining the slope of the best fit line using the formula mentioned earlier, we calculate the rate of change of pressure with respect to molality. This slope is equivalent to the Henry's law constant (\textbf{H}) for \(\mathrm{Br}_{2}\) in \(\mathrm{CCl}_{4}\), effectively linking the theoretical constant with a graphical representation of experimental data.