The Henry's law constant for the solubility of radon in water at \(30^{\circ} \mathrm{C}\) is \(9.57 \times 10^{-6} \mathrm{M} / \mathrm{mm} \mathrm{Hg} .\) Radon is present with other gases in a sample taken from an aquifer at \(30^{\circ} \mathrm{C}\). Radon has a mole fraction of \(2.7 \times 10^{-6}\) in the gaseous mixture. The gaseous mixture is shaken with water at a total pressure of 28 atm. Calculate the concentration of radon in the water. Express your answers using the following concentration units. (a) molarity (b) \(\mathrm{ppm}\) (Assume that the water sample has a density of \(1.00 \mathrm{~g} / \mathrm{mL} .\) )

Molarity, often represented by the symbol 'M', is a measure of the concentration of a solute in a solution. It's defined as the number of moles of the solute divided by the volume of the solution in liters.

When calculating molarity in exercises involving Henry's Law, such as the solubility of radon in water, we first use the law to determine the molar concentration of the dissolved gas in solution. Henry's Law states that at a constant temperature, the concentration of a dissolved gas is directly proportional to the partial pressure of that gas above the solution.

Using Henry's Law with the given constant, we can convert the partial pressure of radon to its concentration in water — essentially, finding how many moles of radon will dissolve in one liter of water under the given conditions. For example, as in the provided solution, with a Henry's Law constant of 9.57 x 10^-6 M/mmHg and a calculated partial pressure in mmHg, we arrive at the molarity of radon in water.

Gas solubility in water can seem perplexing, but it is an important concept when dealing with solutions. It’s defined as the maximum amount of gas that can dissolve in water at a specific temperature and pressure.

Henry's Law provides a simple way to understand and calculate this solubility. According to this law, the solubility of a gas (how much of it can be dissolved) in water is proportional to the partial pressure of the gas above the liquid. A higher partial pressure means more gas will dissolve until equilibrium is reached.

For students to grasp this better, it is essential to highlight that temperature plays a role, too: solubility changes with temperature. At higher temperatures, gases generally have lower solubility in water. This is why soda goes 'flat' faster if it’s left out in the sun. In contrast, as shown in the exercise, solubility also increases as the partial pressure of the gas increases, which is quantitatively described by the constant provided in Henry's Law.

Partial pressure is a concept from gas laws which states that the pressure exerted by a single type of gas in a mixture of gases is the same as it would exert if it alone occupied the entire volume. In simple terms, it's like imagining that one specific gas in a mixture is alone and determining how much pressure it puts on the walls of the container.

To calculate partial pressure, you'll need to know the mole fraction of the gas in the mixture and the total pressure. In our example with radon, we multiply the given mole fraction by the total pressure to find the partial pressure of radon alone.

Understanding partial pressure is critical because it directly affects gas solubility. As per Henry's Law, the concentration of a dissolved gas is directly proportional to its partial pressure, which we applied in our initial steps to determine the molarity. Managing these calculations requires careful attention to units and proper multiplication of the mole fraction by the total pressure, which, in practical applications like scuba diving, is crucial for preventing conditions like the bends.