Question

The partial pressures of \(\mathrm{Br}_{2}\) above a solution containing \(\mathrm{CCl}_{4}\) as the solvent at \(25^{\circ} \mathrm{C}\) are found to have the values listed in the following table as a function of the mole fraction of \(\mathrm{Br}_{2}\) in the solution [Lewis G. N., and Storch, H. J. American Chemical Society \(39(1917): 2544]\). Use these data and a graphical method to determine the Henry's law constant for \(\mathrm{Br}_{2}\) in \(\mathrm{CCl}_{4}\) at \(25^{\circ} \mathrm{C}\). $$\begin{array}{cccc} x_{B r_{2}} & P(\text { Torr }) & x_{B r_{2}} & P(\text { Torr }) \\ \hline 0.00394 & 1.52 & 0.0130 & 5.43 \\ 0.00420 & 1.60 & 0.0236 & 9.57 \\ 0.00599 & 2.39 & 0.0238 & 9.83 \\ 0.0102 & 4.27 & 0.0250 & 10.27 \end{array}$$

Short answer

To determine the Henry's law constant for Br2 in CCl4 at 25°C, create a scatter plot using the given data with mole fraction of Br2 on the x-axis and partial pressure of Br2 on the y-axis. Find the best-fit line through the data points and calculate its slope. The slope of the line represents the Henry's law constant for Br2 in CCl4 at 25°C.

Step-by-step solution

Understand Henry's Law

Henry's law states that the partial pressure of a gas above a solution (P) is directly proportional to the mole fraction of the gas (x) in the solution. Mathematically, it can be written as: \[P = kH \cdot x\] where \(kH\) is the Henry's law constant.

Create a scatter plot using the given data

Using the data provided in the table, create a scatter plot with the mole fraction of Br2 (\(x_{Br_2}\)) on the x-axis and the partial pressure of Br2 (P) on the y-axis. You can use graphing software or draw it manually.

Find the best-fit line through the data points

Choose the line that best fits the data points in the scatter plot. This can be done manually or by using the software to calculate the line of best fit (also known as linear regression).

Calculate the slope of the best-fit line

The slope of the line represents the Henry's law constant, \(kH\). To calculate the slope, you can either use the graphing software if one was used, or calculate it manually by choosing two points on the line and using the formula for the slope of a line: \[kH = \frac{P_2 - P_1}{x_{Br_{2_2}} - x_{Br_{2_1}}}\]

Determine the Henry's law constant for Br2 in CCl4 at 25°C

After calculating the slope of the best-fit line, the Henry's law constant is determined as: \[kH = \text{(value of the slope)}\] This value represents the Henry's law constant for Br2 in CCl4 at 25°C.

Key concepts

Partial Pressures

Partial pressure is a fundamental concept in chemistry, particularly when dealing with gases. It refers to the pressure exerted by a single type of gas in a mixture of gases. Essentially, it's the pressure that gas would exert if it occupied the entire volume on its own. Understanding partial pressures is crucial when applying Henry's Law. According to this law, the partial pressure of the gas above the solution is directly proportional to the mole fraction of the gas in the solution. This relationship makes partial pressures a key factor in determining the solubility of gases in liquids.In a practical sense, when measuring the partial pressure of a gas over a solution, we are assessing the concentration of the gas within the solution — a vital step for calculating the Henry's law constant. To ensure clarity, imagine a container with a solution at the bottom and a space filled with gas above it. The pressure that the gas in the space exerts on the solution is its partial pressure, which we can measure and relate to its concentration within the solution, based on its mole fraction.

Mole Fraction

Mole fraction is another significant concept in chemical solutions, providing a dimensionless way of expressing the concentration of a component in a mixture. It is defined as the ratio of the number of moles of the component to the total number of moles of all components in the mixture. Represented by the symbol 'x', the mole fraction of a gas in a solution is crucial for Henry's Law. It helps to quantify the amount of gas dissolved at a given partial pressure. The relationship is a simple yet powerful tool that can predict how a gas behaves when mixed with a solvent, without the need for complex equipment.For any given temperature, as the mole fraction increases, the partial pressure also increases, provided the concentration of the gas in the solution doesn't exceed its solubility limit. This leads to a direct proportionality that Henry's Law describes, making the determination of the mole fraction elemental in finding the Henry's law constant.

Graphical Analysis

Graphical analysis is a method of interpreting data by representing it on a graph. This visual representation can reveal relationships and trends that might not be immediately evident from the raw data alone. In the context of Henry's Law, a graph can be used to illustrate the relationship between partial pressure and mole fraction. By plotting the provided data points — mole fraction on the x-axis and partial pressure on the y-axis — we can visually inspect how one variable affects the other. A linear trend suggests a direct proportionality, which is exactly what Henry's Law predicts.In our case, each dot on the graph represents the partial pressure and mole fraction of \(\mathrm{Br}_{2}\) at various concentrations in the solution. By analyzing the scatter plot, we can proceed to perform a linear regression, which helps us find the best line that fits through these data points, further used to determine the Henry's law constant for \(\mathrm{Br}_{2}\) in \(\mathrm{CCl}_{4}\).

Linear Regression

Linear regression is a statistical tool that’s used to create a straight line (best-fit line) through a set of data points. This line is the best representation of the relationship between the dependent and independent variables. In our context, the independent variable is the mole fraction and the dependent variable is the partial pressure.After plotting the data points on a graph, linear regression allows us to calculate the equation of the line that minimizes the distance between the data points and the line itself. This line’s slope, in the context of Henry's Law, is the Henry's law constant, \(kH\). By using software or manual calculation to perform linear regression, we end up with a precise value for the slope of the line. This value is crucial since it directly relates to how much of the gas \(\mathrm{Br}_{2}\) can be dissolved in \(\mathrm{CCl}_{4}\) at a given temperature. Through this method, we obtain an accurate and reliable Henry's Law constant.

Gas Solubility

Gas solubility is a measure of how much of a particular gas can be dissolved in a solvent at a given temperature and pressure. It's a critical parameter in various chemical processes and environmental studies. Henry's Law provides insight into gas solubility by correlating the concentration of a gas in a liquid with the partial pressure of that gas above the liquid.In practical terms, knowing the solubility of a gas helps us to predict how it will behave under different conditions. It can also influence the design of industrial processes, such as in fermentation or carbonated beverage production.In the context of the exercise, the solubility of \(\mathrm{Br}_{2}\) in \(\mathrm{CCl}_{4}\) can be understood in terms of Henry's Law. By determining the constant, we can predict how much \(\mathrm{Br}_{2}\) will dissolve in \(\mathrm{CCl}_{4}\) at 25°C for any given partial pressure. This understanding is essential for chemists and engineers who work with solutions and need to predict outcomes based on solubility data.