Problem 94 - Chemistry The Central Science In Si Units

Question

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.

Step-by-Step Solution

Answer

(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 1: 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.

Step 2: 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: \[ k_H = \frac{0.109 \text{ M}}{15.2 \text{ kPa}} \approx 0.00717 \text{ M/kPa} \].

Step 3: 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.)

Step 4: Calculate Molar Concentration of Radon in Water

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

Key Concepts

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°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’s 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.