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For the absorption of a gas (like carbon dioxide) into a liquid (like water) Henry's law states that partial pressure of the gas is proportional to the mole fraction of the gas in the liquid-gas solution with the constant of proportionality being Henry's constant. A bottle of soda pop \(\left(\mathrm{CO}_{2}-\mathrm{H}_{2} \mathrm{O}\right)\) at room temperature has a Henry's constant of \(17,100 \mathrm{kPa}\). If the pressure in this bottle is \(120 \mathrm{kPa}\) and the partial pressure of the water vapor in the gas volume at the top of the bottle is neglected, the concentration of the \(\mathrm{CO}_{2}\) in the liquid \(\mathrm{H}_{2} \mathrm{O}\) is (a) \(0.003 \mathrm{~mol}-\mathrm{CO}_{2} / \mathrm{mol}\) (b) \(0.007 \mathrm{~mol}-\mathrm{CO}_{2} / \mathrm{mol}\) (c) \(0.013 \mathrm{~mol}-\mathrm{CO}_{2} / \mathrm{mol}\) (d) \(0.022 \mathrm{~mol}-\mathrm{CO}_{2} / \mathrm{mol}\) (e) \(0.047 \mathrm{~mol}-\mathrm{CO}_{2} / \mathrm{mol}\)

Step by step solution

01

Understand Henry's Law

Henry's Law states that the partial pressure of a gas (\(P_{\text{gas}}\)) is proportional to the mole fraction of the gas (X) in the liquid-gas solution, with the proportionality constant being Henry's constant (K_H). Mathematically, it can be represented as: \(P_{\text{gas}} = K_H \times X\)

02

Identify the given information

We are given the following information: - Henry's constant (K_H) for CO2-H2O solution at room temperature = 17,100 kPa - The pressure in the bottle (P_bottle) = 120 kPa. Since the partial pressure of water vapor is neglected, we can consider this the partial pressure of CO2.

03

Apply Henry's Law to find the mole fraction of CO2 in the liquid

Using the given information and Henry's Law, we can find the mole fraction of CO2 in the liquid solution: \(P_{\text{CO}_{2}} = K_H \times X_{\text{CO}_{2}}\) Plug in the values: \(120 \,\text{kPa} = 17100\, \text{kPa} \times X_{\text{CO}_{2}}\) Solve for \(X_{\text{CO}_{2}}\): \(X_{\text{CO}_{2}} = \frac{120}{17100}\) \(X_{\text{CO}_{2}} = 0.007 \, \text{mol CO}_{2} \, / \, \text{mol}\)

04

Determine the concentration of CO2 in the liquid

We have now found the mole fraction of CO2 in the liquid, which is also equal to the concentration of CO2 in the liquid: Concentration of CO2 = \(0.007 \, \text{mol CO}_{2} \,/\, \text{mol}\) Thus, the correct answer is (b) \(0.007 \, \text{mol CO}_{2} \,/\, \text{mol}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Partial Pressure

Understanding partial pressure is crucial when studying the behavior of gases. It refers to the pressure that a single gas in a mixture of gases would exert if it occupied the entire volume on its own. In simpler terms, it's the contribution of each gas in a mixture to the total pressure.

When considering Henry's Law, the partial pressure of the gas above a liquid solution is directly related to how much of that gas gets dissolved in the liquid. For instance, in a bottle of soda, the partial pressure of carbon dioxide (CO2) is an indicator of how much CO2 is present in the water as a dissolved gas. In our exercise, by calculating the partial pressure, we can use Henry's Law to determine the concentration of CO2 dissolved in water.

Mole Fraction

The mole fraction is a way of expressing the concentration of a component in a mixture. It is defined as the ratio of the number of moles of one component to the total number of moles of all components in the mixture.

To calculate the mole fraction, you would use the formula:\[\begin{equation}X_{component} = \frac{n_{component}}{n_{total}}\end{equation}\]
where \(n_{component}\) is the number of moles of the component of interest, and \(n_{total}\) is the total number of moles of all components in the mixture. This dimensionless quantity helps us understand the composition of the solution without worrying about the actual amounts. In the context of gas absorption, the mole fraction can be used to describe the proportion of gas molecules dissolved in the liquid.

Gas Absorption in Liquid

Gas absorption in liquid involves the transfer of a gas into a liquid medium. The extent to which a gas is absorbed depends on factors like temperature, pressure, and the nature of the gas and liquid. According to Henry's Law, at a constant temperature, the amount of gas that dissolves in a liquid is directly proportional to the partial pressure of that gas in contact with the liquid.

This principle is particularly evident in the creation of carbonated beverages, where CO2 is dissolved under high pressure to create the fizz. When you open a bottle of soda, you decrease the partial pressure of CO2, leading to the release of gas from the liquid and the formation of bubbles.

Concentration Calculation

The concentration calculation is essential in chemistry to determine the amount of substance within a specific volume of a solution. It's often expressed as molarity, which is the number of moles of a solute per liter of solution. In relation to Henry's Law, the concentration of a gas in a liquid can be calculated once the mole fraction is known.

To transition from a mole fraction to concentration, under the assumption that the liquid's composition remains relatively unchanged, the mole fraction can be treated as an actual concentration. This simplification is useful for understanding the solubility of gases in liquids and is fundamental in various fields including environmental engineering, physiology, and beverage production.