Question

The presence of the radioactive gas radon \((\mathrm{Rn})\) in well water obtained from aquifers that lie in rock deposits presents a possible health hazard in parts of the United States. (a) Assuming that the solubility of radon in water with 1 atm pressure of the gas over the water at \(30^{\circ} \mathrm{C}\) is \(7.27 \times 10^{-3} M\), what is the Henry's law constant for radon in water at this temperature? (b) A sample consisting of various gases contains \(3.5 \times 10^{-6}\) mole fraction of radon. This gas at a total pressure of 32 atm 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 \(30^{\circ}\mathrm{C}\) is \(7.27 \times 10^{-3} \frac{\text{M}}{\text{atm}}\). The molar concentration of radon in water when in equilibrium with the given gas mixture is \(8.14 \times 10^{-7} \text{M}\).

Step-by-step solution

(a) Determine the Henry's law constant for radon in water

To find the Henry's law constant for radon in water, we can use the formula for Henry's law: \[K_{H} = \frac{C}{P}\] where \(K_{H}\) is the Henry's law constant, \(C\) is the concentration of radon in water, and \(P\) is the pressure of radon over the water. Since we are given the solubility of radon in water with 1 atm pressure of the gas, \(C = 7.27 \times 10^{-3}M\) and \(P = 1\text{ atm}\). Plugging these values into the formula gives: \[K_{H} = \frac{7.27 \times 10^{-3}\text{M}}{1\text{ atm}}\]

(a) Calculate the Henry's law constant

Now we can calculate the Henry's law constant for radon in water at \(30^{\circ}\mathrm{C}\): \[K_{H} = 7.27 \times 10^{-3} \frac{\text{M}}{\text{atm}}\]

(b) Determine the pressure of radon in the gas mixture

In part (b), we are given the mole fraction of radon in the gas mixture and the total pressure of the mixture. To find the partial pressure of radon in the mixture, we can use the formula: \[P_{\mathrm{Rn}} = x_{\mathrm{Rn}} \times P_{\mathrm{total}}\] where \(P_{\mathrm{Rn}}\) is the partial pressure of radon, \(x_{\mathrm{Rn}}\) is the mole fraction of radon, and \(P_{\mathrm{total}}\) is the total pressure of the mixture. We are given \(x_{\mathrm{Rn}} = 3.5 \times 10^{-6}\) and \(P_{\mathrm{total}} = 32 \text{ atm}\). Plugging these values into the formula gives: \[P_{\mathrm{Rn}} = (3.5 \times 10^{-6})(32 \text{ atm})\]

(b) Calculate the pressure of radon in the gas mixture

Now we can calculate the partial pressure of radon in the gas mixture: \[P_{\mathrm{Rn}} = 1.12 \times 10^{-4} \text{ atm}\]

(b) Determine the molar concentration of radon in water

Using the Henry's law constant from part (a), we can now find the concentration of radon in water when it is in equilibrium with the gas mixture: \[C_{\mathrm{Rn}} = K_{H} \times P_{\mathrm{Rn}}\] where \(C_{\mathrm{Rn}}\) is the concentration of radon in water, \(K_{H} = 7.27 \times 10^{-3} \frac{\text{M}}{\text{atm}}\) is the Henry's law constant, and \(P_{\mathrm{Rn}} = 1.12 \times 10^{-4} \text{ atm}\) is the partial pressure of radon in the gas mixture. Plugging these values into the formula gives: \[C_{\mathrm{Rn}} = (7.27 \times 10^{-3} \frac{\text{M}}{\text{atm}})(1.12 \times 10^{-4} \text{ atm})\]

(b) Calculate the molar concentration of radon in water

Now we can calculate the molar concentration of radon in water at equilibrium with the gas mixture: \[C_{\mathrm{Rn}} = 8.14 \times 10^{-7} \text{M}\]

Key concepts

Solubility of Gases

Understanding the solubility of gases in liquids is crucial for a variety of scientific and industrial processes. Essentially, solubility refers to how much of a particular gas can dissolve in a liquid under certain conditions. This process is driven by intermolecular forces between gas molecules and the liquid, as well as external factors such as temperature and pressure.

A key point to remember is that at higher pressures, more gas molecules will

Partial Pressure

Partial pressure plays a pivotal role in determining the solubility of gases in liquids. It is defined as the pressure that a gas would exert if it alone occupied the volume of the mixture at the same temperature. According to Dalton's Law of Partial Pressures, the total pressure exerted by a mixture of non-reacting gases is equal to the sum of the partial pressures of individual gases.

Molar Concentration

Molar concentration, denoted as Molarity and represented by the symbol M, is a way to quantify the amount of a substance in a solution. It is defined as the number of moles of solute per liter of solution. This measure is intrinsic to the study of solutions and is particularly relevant in understanding how the solute interacts with the solvent, in terms of both chemical reactions and physical processes like dissolving.

Gas Solubility in Water

Gas solubility in water is a specialized aspect of solubility that has significant implications for environmental and biological systems. The solubility of a gas in water is determined by the inherent nature of the gas, the temperature of the water, and the partial pressure of the gas above the water. Henry's Law provides a relationship to predict and calculate the solubility of gases in liquids, stating that at a constant temperature, the amount of gas that dissolves in a type and volume of liquid is directly proportional to the partial pressure of that gas in equilibrium with the liquid.