Question

The presence of the radioactive gas radon \((\mathrm{Rn})\) 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 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 MPa is shaken with water at \(30^{\circ} \mathrm{C} .\) Calculate the molar concentration of radon in the water.

Short answer

The Henry's law constant for radon in water at 30°C is 7.17 M/MPa. The molar concentration of radon in the water when in contact with a gas sample containing a radon mole fraction of 4.5 ppm and a total pressure of 5.07 MPa is approximately \(1.636 \times 10^{-4}\) M.

Step-by-step solution

(a) Finding the Henry's law constant

Using the given solubility for radon in water (0.109 M) and the pressure of the gas over the water (15.2 kPa), we can now find the Henry's law constant for radon in water at 30°C using the Henry's law equation: \(C = k_HP\) Rearrange the equation to solve for \(k_H\): \(k_H = \frac{C}{P}\) Plug in the given values and calculate \(k_H\): \(k_H = \frac{0.109\text{ M}}{15.2\text{ kPa}}\) Make sure that the pressure is given in the same units: \(k_H = \frac{0.109\text{ M}}{0.0152\text{ MPa}} = 7.17 \text{ M/MPa}\) So, the Henry's law constant for radon in water at 30°C is 7.17 M/MPa.

(b) Finding the molar concentration of radon in the water

We are given the mole fraction of radon in the gas mixture (4.5 ppm) and the total pressure of the gas mixture (5.07 MPa). First, we need to determine the partial pressure of radon in the gas mixture: Mole fraction, \(x = \frac{moles\, of\, Rn}{total\, moles}\) So, partial pressure of radon, \(P_{Rn} = x_{Rn} \times Total\, pressure\) Plug in the given values and calculate the radon partial pressure: \(P_{Rn} = (4.5 \times 10^{-6}) \times 5.07\text{ MPa} = 0.000022815\text{ MPa}\) Now, we can use the Henry's law constant calculated in part (a) and the radon partial pressure to find the molar concentration of radon in the water: \(C = k_HP_{Rn}\) Plug in the values and calculate the molar concentration: \(C = (7.17\text{ M/MPa}) \times 0.000022815\text{ MPa} = 1.636 \times 10^{-4}\text{ M}\) Thus, the molar concentration of radon in the water is approximately \(1.636 \times 10^{-4}\) M.

Key concepts

Solubility

Solubility refers to the ability of a substance, known as the solute, to dissolve in a solvent, creating a solution. It's a key concept in chemistry, denoting how much of the solute can dissolve in a given amount of solvent at a specified temperature and pressure.
Solubility is often expressed in terms of molarity, which measures the concentration of the solute in the solution. In simple terms, it tells us how many moles of the solute are present in one liter of the solution.
There are several factors that influence solubility, including:

• Temperature: Generally, the solubility of solid solutes in liquid solvents increases with temperature.
• Pressure: For gases, increased pressure usually increases solubility.
• Nature of the solute and solvent: Polar solutes typically dissolve better in polar solvents.

Understanding the solubility principles is crucial when working with gases in solutions, as it enables us to predict how different conditions might affect the outcome.

Radon in water

Radon is a radioactive gas that can dissolve in water, posing certain health risks, especially when it's present in drinking water. Radon in well water can release into the air during activities like showers or dishwashing. Therefore, it is important to assess and manage its solubility in water to mitigate these risks.
The solubility of radon in water is determined by its concentration in the water under specific conditions. In this context, it's measured using Henry's Law, a fundamental principle that relates the concentration of a gas in a liquid to the partial pressure of that gas above the liquid.
For example, if the solubility of radon at 15.2 kPa pressure is 0.109 M, it implies that this is the maximum molarity of radon that can dissolve in the water at this pressure, serving as a significant indicator for environmental and health safety assessments.

Partial pressure

Partial pressure is a component of a gas mixture's total pressure, representing the pressure that one individual gas exerts if it occupied the entire volume by itself. It's a crucial concept in determining how gases behave when dissolved in liquids, such as water.
The partial pressure of a gas is calculated using the mole fraction of the gas in the mixture and the total pressure of the mixture. For instance, if radon makes up a small proportion of a gas mixture, its partial pressure is a fraction of the total pressure proportional to its mole fraction.

• Formula: \[ P_{Rn} = x_{Rn} \times \text{Total pressure} \]

Calculating the partial pressure is an essential step when applying Henry's Law to determine the solubility of gases in liquids. Knowing the partial pressures helps in predicting how much of a gas like radon will dissolve in water at a given temperature, which is vital for assessing the associated risks and remedies.