Question

The concentration of Ar in the ocean at \(25^{\circ} \mathrm{C}\) is \(11.5 \mu \mathrm{M} .\) The Henry's law constant for \(\mathrm{Ar}\) is \(1.5 \times 10^{-3}\) \(\mathrm{mol} \mathrm{L}^{-1} \mathrm{atm}^{-1} .\) Calculate the mass of \(\mathrm{Ar}\) in a liter of ocean water. Calculate the partial pressure of \(\mathrm{Ar}\) in the atmosphere.

Short answer

The mass of Ar in a liter of ocean water is approximately 0.46 μg. The partial pressure of Ar in the atmosphere is approximately 0.0077 atm.

Step-by-step solution

Convert concentration into moles

Given that the concentration of Ar in the ocean is 11.5 μM, this means there are \( 11.5 \times 10^{-6} \) moles of Ar in one litre of ocean water.

Determine mass of Ar per litre

Using the molar mass of Ar, which is approximately 40 g/mol, we calculate the mass of Ar in one litre of water. This is done by multiplying the value obtained in the first step by the molar mass: \( mass = n \times molar \ mass = 11.5 \times 10^{-6} \times 40 \) g.

Calculate the partial pressure of Ar in the atmosphere

We utilize Henry's law, which states that the dissolved concentration of a gas is directly proportional to the partial pressure of that gas above the solution. Rearranging the formula, we can solve for the partial pressure of Ar: \( P = [A] / K_H \), where \(P\) is the partial pressure, \([A]\) is the molar concentration of Ar, and \(K_H\) is Henry's law constant. Substituting the known values, \( P = 11.5 \times 10^{-6} / 1.5 \times 10^{-3} \) atm.

Key concepts

Partial Pressure Calculation

Understanding partial pressure is crucial in several scientific applications, including diving, altitude physiology, and the solubility of gases in liquids. 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.

In the context of Henry's law, we look specifically at the pressure exerted by a particular gas that is in equilibrium with its dissolved form in a solution. To calculate the partial pressure (\(P\textsubscript{gas}\)) of a gas, we can rearrange Henry's law equation:
\[ P\textsubscript{gas} = \frac{[A]}{K\textsubscript{H}} \]
where \( [A] \) is the concentration of the gas in solution and \( K\textsubscript{H} \) is the Henry's law constant for the gas.

This relationship implies that as the concentration of a gas in a solution increases, so does its partial pressure in the atmosphere above the solution. Conversely, a decrease in the concentration of the gas in solution will result in a decrease in its partial pressure. This concept is vital in several fields, including environmental science and chemical engineering, where it's used to predict how gases will behave under different pressures.

Gas Solubility in Water

The solubility of gas in water or any other solvent is a measure of how much gas can be dissolved in the solvent at a given temperature and pressure. Henry's law provides us with a way to understand the relationship between gas solubility and partial pressures for a range of applications including the beverage industry, where carbon dioxide solubility determines the fizziness of soft drinks, and in oceanography, where the solubility of gases like oxygen affects marine life.

According to Henry's law, at a constant temperature, the solubility of a gas in a liquid is directly proportional to the partial pressure of the gas above the liquid. The law states as an equation: \[ S\textsubscript{gas} = K\textsubscript{H} \times P\textsubscript{gas} \]
where \( S\textsubscript{gas} \) represents the solubility of the gas, \( K\textsubscript{H} \) is the Henry's law constant, and \( P\textsubscript{gas} \) is the partial pressure of the gas. Changes in temperature and pressure can significantly influence the solubility, with higher pressures increasing solubility and higher temperatures generally decreasing it. This concept helps scientists and engineers to design and optimize processes involving the dissolution of gases in liquids.

Molar Mass Determination

The molar mass of a substance is a fundamental property used in chemistry to calculate the mass of a given substance that contains exactly one mole of particles, be they atoms, molecules, or other units. In the provided exercise, knowing the molar mass of argon (\(Ar\textsubscript{2}\)) is essential for the determination of the mass of argon in a liter of ocean water.

The molar mass is typically expressed in grams per mole (g/mol). For argon, the average molar mass is approximately 40 g/mol. By using the number of moles calculated and the molar mass, the mass of argon in ocean water is given by the equation: \[ mass = n \times \text{molar mass} \]
where \( n \) is the number of moles of the gas. The determination of a substance's molar mass often begins with the periodic table where the atomic masses of the elements are listed. Understanding how to find and make use of molar mass is pivotal in stoichiometry, which assists in making sense of chemical reactions and the quantities of reactants and products involved.