The presence of the radioactive gas radon (Rn) in well water presents a possible health hazard in parts of the United States. (b) A sample consisting of various gases contains 3.5 × 10^-6 mole fraction of radon. This gas at a total pressure of 32 atm is shaken with water at 30 °C. Calculate the molar concentration of radon in the water.

Verified step by step guidance

1. First, understand that the problem involves calculating the molar concentration of radon in water using Henry's Law, which relates the solubility of a gas in a liquid to the partial pressure of the gas above the liquid.
2. Identify the given values: the mole fraction of radon in the gas mixture is 3.5 × 10^-6, and the total pressure of the gas mixture is 32 atm.
3. Calculate the partial pressure of radon using the formula: <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msub><mi>P</mi><mi>Rn</mi></msub><mo>=</mo><msub><mi>X</mi><mi>Rn</mi></msub><mo>&#xD7;</mo><mi>P</mi></mrow></math>, where <math xmlns='http://www.w3.org/1998/Math/MathML'><msub><mi>X</mi><mi>Rn</mi></msub></math> is the mole fraction of radon and <math xmlns='http://www.w3.org/1998/Math/MathML'><mi>P</mi></math> is the total pressure.
4. Use Henry's Law to find the molar concentration of radon in water: <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msub><mi>C</mi><mi>Rn</mi></msub><mo>=</mo><msub><mi>k</mi><mi>H</mi></msub><mo>&#xD7;</mo><msub><mi>P</mi><mi>Rn</mi></msub></mrow></math>, where <math xmlns='http://www.w3.org/1998/Math/MathML'><msub><mi>k</mi><mi>H</mi></msub></math> is the Henry's Law constant for radon at 30 °C.
5. Look up the value of the Henry's Law constant for radon at 30 °C in a reliable source, then substitute the values into the equation to find the molar concentration of radon in the water.

Key Concepts

Mole Fraction

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 a specific component to the total number of moles of all components in the mixture. In this case, the mole fraction of radon indicates its proportion relative to other gases present, which is crucial for calculating its concentration in water.

Henry's Law

Henry's Law states that the amount of gas that dissolves in a liquid at a given temperature is directly proportional to the partial pressure of that gas above the liquid. This principle is essential for determining how much radon will dissolve in water when it is in contact with the gas at a specific pressure, allowing us to calculate its molar concentration in the water.

Molar Concentration

Molar concentration, or molarity, is defined as the number of moles of solute per liter of solution. It provides a measure of how concentrated a solution is and is critical for understanding the amount of radon that can be expected to dissolve in water under the given conditions. This concept is key to solving the problem by converting the amount of radon from its mole fraction and pressure into a concentration value.