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.

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.

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.