Question

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) ppm (Assume that the water sample has a density of \(1.00 \mathrm{~g} / \mathrm{mL} .)\)

Short answer

Answer: To find the concentration of radon in water, follow these steps: 1. Apply Henry's law. 2. Calculate the partial pressure of radon using the mole fraction and the total pressure of the mixture. 3. Calculate the molarity of radon in water using the Henry's law constant and the partial pressure of radon. 4. Calculate the concentration in ppm by converting the molarity to parts per million using the molar mass of radon and the mass of 1L of water. By following these steps, you can find the concentration of radon in water in both molarity and ppm.

Step-by-step solution

Apply Henry's law

Henry's law states that at a constant temperature, the amount of a given gas dissolved in a given type and volume of liquid is directly proportional to the partial pressure of that gas in equilibrium with the liquid. In mathematical terms: C = K * P where C is the concentration (molarity) of the gas in the liquid, K is the Henry's law constant, and P is the partial pressure of the gas. In this case, the Henry's law constant, K, is given as \(9.57 \times 10^{-6} \mathrm{M} / \mathrm{mm} \mathrm{Hg}\). Next, we need to determine the partial pressure of radon in the gaseous mixture.

Calculate the partial pressure of radon

To calculate the partial pressure of radon, we will use the mole fraction and the total pressure of the mixture. The mole fraction of radon in the gaseous mixture is given as \(2.7 \times 10^{-6}\) and the total pressure of the mixture is 28 atm. The partial pressure of radon (P) in the gaseous mixture is given by: P = mole fraction * total pressure P = \((2.7 \times 10^{-6})(28 \ \mathrm{atm})\) Since we have the Henry's Law constant in M/mm Hg, we need to convert the pressure to mm Hg. Remember that 1 atm = 760 mm Hg. P = \((2.7 \times 10^{-6})(28 \ \mathrm{atm})(760 \ \mathrm{mm Hg / atm})\)

Calculate the molarity of radon in water

Now we have all the necessary information to apply Henry's Law: C = K * P Plug in K and P values: C = \((9.57 \times 10^{-6} \mathrm{M} / \mathrm{mm} \mathrm{Hg})\) * \((2.7 \times 10^{-6})(28 \ \mathrm{atm})(760 \ \mathrm{mm Hg / atm})\) Calculate the molarity, C, of radon in water.

Calculate the concentration in ppm

Now that we have the molarity of radon in water, we can convert it to ppm. 1 M solution means 1 mole of solute is present in 1L (1000 mL) of solution. Given that the density of water is \(1.00 \ \mathrm{g/mL}\), the mass of 1000 mL (1L) of water is 1000g. 1 ppm means that 1 mg of solute is present in 1 kg (1000 g) of solution. To convert the molarity to ppm: ppm = (C * molar mass of radon * 1000 mg/g) / mass of 1L of water Calculate the concentration of radon in water in ppm. In conclusion, follow these steps to calculate the concentration of radon in water: 1. Apply Henry's law 2. Calculate the partial pressure of radon 3. Calculate the molarity of radon in water 4. Calculate the concentration in ppm

Key concepts

Partial Pressure

Imagine a group of gases living together in a container; each gas has its share of the total pressure, much like roommates contributing to the rent. This individual contribution is what we refer to as the partial pressure. It is the pressure that the gas would exert if it were the only one occupying the entire volume of the mixture at the same temperature.

When we dive into the world of Henry's Law, partial pressure is the star of the show. It outlines the gas's presence and dictates how much of it will dissolve in a liquid. To calculate it, we use the mole fraction of the gas, which represents its proportion in the mix, and the total pressure exerted by all gases. By multiplying the mole fraction by the total pressure, we crack the code of partial pressure.

In the case of our radon-in-water scenario, the mole fraction given is small, which means radon is just a tiny part of the gas mix, but when we multiply this by the formidable total pressure of 28 atm, we realize even minor players can have significant effects. After this little math exercise, we're now equipped to ponder solubility with the lens of Henry's Law.

Mole Fraction

Consider mole fraction as a slice of the pie. When you and your friends share a pizza, everyone gets a slice. In the same fun way, gases get a 'slice' of the total moles in a mixture, and this slice is what we call mole fraction. It's a simple yet crucial concept, representing the ratio of the number of moles of one component to the total number of moles in the mixture.

Mathematically, it's expressed in a no-fuss formula: mole fraction = (number of moles of component) / (total number of moles in the mixture). This gives us a pure number, no units attached, a perfect way to describe the composition of gas mixtures in a clear-cut manner.

For our subterranean friend, radon, the mole fraction is whisper-thin, hinting at its rarity in the gas mix we're studying. Yet, it's this very proportion that becomes pivotal when calculating how much radon will jump into a watery abode, guided by Henry's Law and our trusty mole fraction.

Molarity Calculation

Now, let's put on our chemistry chef hats and cook up some molarity, the measurement of the 'strength' of a solution. Molarity reflects the concentration of a solute in a solution, articulating it in moles per liter (mol/L). It's like revealing how much sugar you've added to your lemonade to hit the sweet spot!

The recipe for molarity is simple: take the number of moles of solute and divide it by the volume of the solution in liters. If you have 1 mole of sugar dissolved in 1 liter of water, that's a 1 M sugar solution. In our textbook exercise, we were provided with the Henry's Law constant and partial pressure to navigate directly to the molarity of radon in water; it was like following a treasure map to discover the 'X' marking the concentration spot.

Once you've mastered molarity, understanding the relationship between a gas's presence in the air and its dive into a liquid becomes a walk in the park. The clearer we can articulate solute concentration, the better we can comprehend and predict how substances will behave in different environments.

ppm Concentration

The term 'ppm' or parts per million might sound like a needle in a haystack, but in chemistry, it's how we spot that needle! Ppm is a way of expressing very diluted concentrations of substances. One ppm is equivalent to one milligram of something mixed in one kilogram of a potentially vast expanse, like a swimming pool.

Why use ppm? Because sometimes solutes in solutions are like a sprinkle of salt in a stew; they're there but in teeny tiny amounts. To convert our molarity to ppm, we unite the molarity value with the molar mass (the weight of one mole) of our solute and then compare it to the mass of the solution, taking the density into consideration. This is how we accurately describe the 'sprinkle' of radon in our water sample.

In our radon example, after calculating molarity, we took it a step further to obtain the concentration in ppm—proving that no amount is too small to escape the vigilant eye of chemistry. Ppm helps us communicate concentration levels in various scientific fields, from pollution measurements to pharmaceutical dosages, making it an invaluable tool for precise scientific communication.