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Chapter 13: Problem 94

The presence of the radioactive gas radon (Rn) in well water presents a possible health hazard in parts of the United States. (a) Assuming that the solubility of radon in water with \(15.2 \mathrm{kPa}\) pressure of the gas over the water at \(30^{\circ} \mathrm{C}\) is \(0.109 \mathrm{M}\), what is the Henry's law constant for radon in water at this temperature? (b) A sample consisting of various gases contains 4.5 -ppm radon (mole fraction). This gas at a total pressure of \(5.07 \mathrm{MPa}\) is shaken with water at \(30^{\circ} \mathrm{C} .\) Calculate the molar concentration of radon in the water.

Short Answer

Expert verified
(a) The Henry's law constant is approximately 0.00717 M/kPa. (b) The molar concentration of radon in the water is approximately 1.635 x 10^-4 M.

Step by step solution

01

Understanding Henry's Law

Henry's law states that the concentration of a gas dissolved in a liquid is directly proportional to the pressure of the gas above the liquid. The law is expressed as \( C = k_H \times P \), where \( C \) is the concentration of the gas in the liquid, \( k_H \) is the Henry's law constant, and \( P \) is the partial pressure of the gas.
02

Calculate Henry's Law Constant

Given that the solubility of radon at a pressure of \( 15.2 \text{ kPa} \) is \( 0.109 \text{ M} \), use Henry's law to find \( k_H \). Rearrange the equation to \( k_H = \frac{C}{P} \).Substitute the known values: \[ k_H = \frac{0.109 \text{ M}}{15.2 \text{ kPa}} \approx 0.00717 \text{ M/kPa} \].
03

Convert PPM to Partial Pressure

4.5 ppm means 4.5 parts of radon per million parts of the gas mixture by mole ratio. Convert this to a mole fraction by dividing by 1,000,000:\[ x_{Rn} = \frac{4.5}{1,000,000} = 4.5 \times 10^{-6} \].Find the partial pressure of radon: \[ P_{Rn} = x_{Rn} \times P_{total} = 4.5 \times 10^{-6} \times 5.07 \text{ MPa} = 2.2815 \times 10^{-2} \text{ kPa} \].(Note: Convert MPa to kPa, remembering 1 MPa = 1000 kPa.)
04

Calculate Molar Concentration of Radon in Water

Use Henry's law with the calculated Henry's constant to find the molar concentration of radon:\[ C = k_H \times P_{Rn} = 0.00717 \text{ M/kPa} \times 2.2815 \times 10^{-2} \text{ kPa} \approx 1.635 \times 10^{-4} \text{ M} \].

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Key Concepts

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

Gas Solubility
Gas solubility refers to how much of a gas can dissolve in a liquid. The more soluble a gas is, the more of it can be found in the liquid at equilibrium for a given set of conditions. Solubility can depend on various factors such as pressure, temperature, and the nature of the gas and liquid themselves.

For radon in water, solubility is particularly important because radon is a radioactive gas which can pose health risks if present in significant amounts. In our context, the solubility is given as 0.109 M (molarity), which means 0.109 moles of radon can dissolve in one liter of water at a specific pressure and temperature (in this case, 15.2 kPa and 30掳C).

This relationship is governed by Henry's Law, which provides a direct connection between gas solubility and pressure.
Partial Pressure
Partial pressure is the pressure contributed by a single gas in a mixture of gases. It's important because it determines the solubility of that gas when mixed with other gases and interacting with liquids.

When a gas mixture is in contact with a liquid, each gas has its own partial pressure based on its concentration (or mole fraction) in the mixture and the mixture's total pressure.

In this exercise, radon has a partial pressure derived from its mole fraction in a mixture (4.5 ppm) and the total pressure of the gas mix (5.07 MPa). By converting 5.07 MPa to kPa, and calculating based on Henry's law, we use partial pressure to determine how much radon will dissolve in water.
Concentration Calculation
Concentration calculation involves determining how much of a substance is present in a given volume of liquid. It's often expressed in moles per liter (Molarity).

Henry鈥檚 Law provides the formula for calculating the concentration of a dissolved gas: \[ C = k_H \times P \] where \( C \) is the concentration, \( k_H \) is the Henry's law constant, and \( P \) is the partial pressure of the gas.

In this scenario, we first calculate the Henry's law constant using the initial conditions given (15.2 kPa pressure and 0.109 M concentration), finding it to be approximately 0.00717 M/kPa.

This constant is then used with the computed partial pressure of radon to find its concentration in water under the new conditions. This approach shows the practical application of theoretical concepts, allowing us to understand the environmental concentrations of gases like radon.

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Most popular questions from this chapter

Which of the following in each pair is likely to be more soluble in hexane, \(\mathrm{C}_{6} \mathrm{H}_{14}:\) (a) \(\mathrm{CCl}_{4}\) or \(\mathrm{CaCl}_{2},(\mathbf{b})\) benzene \(\left(\mathrm{C}_{6} \mathrm{H}_{6}\right)\) or glycerol, \(\mathrm{CH}_{2}(\mathrm{OH}) \mathrm{CH}(\mathrm{OH}) \mathrm{CH}_{2} \mathrm{OH},\) (c) Octanoic acid, \(\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{CH}_{2} \mathrm{CH}_{2} \mathrm{CH}_{2} \mathrm{CH}_{2} \mathrm{CH}_{2} \mathrm{COOH},\) or acetic acid, \(\mathrm{CH}_{3} \mathrm{COOH}\) ? Explain your answer in each case.

You make a solution of a nonvolatile solute with a liquid solvent. Indicate if each of the following statements is true or false. (a) The freezing point of the solution is unchanged by addition of the solvent. (b) The solid that forms as the solution freezes is nearly pure solute. (c) The freezing point of the solution is independent of the concentration of the solute. (d) The boiling point of the solution increases in proportion to the concentration of the solute. \((\mathbf{e})\) At any temperature, the vapor pressure of the solvent over the solution is lower than what it would be for the pure solvent.

You make two solutions of a nonvolatile solute with a liquid solvent, \(0.01 M\) and \(1.00 M .\) Indicate whether each of the following statements is true or false. (a) The vapor pressure of the concentrated solution is higher than that of the diluted solution. (b) The osmotic pressure of the concentrated solution is higher than that of the diluted solution. (c) The boiling point of the concentrated solution is higher than that of the diluted solution. (d) The freezing point of the concentrated solution is higher than that of the diluted solution.

Breathing air that contains \(4.0 \%\) by volume \(\mathrm{CO}_{2}\) over time causes rapid breathing, throbbing headache, and nausea, among other symptoms. What is the concentration of \(\mathrm{CO}_{2}\) in such air in terms of \((\mathbf{a})\) mol percentage, \((\mathbf{b})\) molarity, assuming 101.3 kPa pressure and a body temperature of \(37^{\circ} \mathrm{C} ?\)

\(\mathrm{KBr}\) is relatively soluble in water, yet its enthalpy of solution is \(+19.8 \mathrm{~kJ} / \mathrm{mol}\). Which of the following statements provides the best explanation for this behavior? (a) Potassium salts are always soluble in water. (b) The entropy of mixing must be unfavorable. (c) The enthalpy of mixing must be small compared to the enthalpies for breaking up water-water interactions and K-Br ionic interactions. (d) \(\mathrm{KBr}\) has a high molar mass compared to other salts like \(\mathrm{NaCl}\)
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