Question

The solubility of nitrogen in water is \(8.21 \times 10^{-4} \mathrm{mol} / \mathrm{L}\) at \(0^{\circ} \mathrm{C}\) when the \(\mathrm{N}_{2}\) pressure above water is 0.790 \(\mathrm{atm}\) . Calculate the Henry's law constant for \(\mathrm{N}_{2}\) in units of \(\mathrm{mol} / \mathrm{L} \cdot\) atm for Henry's law in the form \(C=k P,\) where \(C\) is the gas concentration in mol/L. Calculate the solubility of \(\mathrm{N}_{2}\) in water when the partial pressure of nitrogen above water is 1.10 atm at \(0^{\circ} \mathrm{C} .\)

Short answer

The Henry's law constant for N₂ is approximately \(1.04 \times 10^{-3} \frac{\mathrm{mol}}{\mathrm{L}\cdot\mathrm{atm}}\), and the solubility of N₂ in water when the partial pressure of nitrogen is 1.10 atm at \(0^{\circ} \mathrm{C}\) is approximately \(1.14 \times 10^{-3} \mathrm{mol}/\mathrm{L}\).

Step-by-step solution

(Step 1: Extract the given data)

(We are given the solubility of nitrogen (\(\mathrm{mol}/\mathrm{L}\)) and the partial pressure above water (\(\mathrm{atm}\)). Let's extract: Solubility of N₂ in water: \(8.21 \times 10^{-4} \mathrm{mol} / \mathrm{L}\) Partial pressure of N₂ above water: \(0.790 \mathrm{atm}\))

(Step 2: Calculate the Henry's law constant)

(Using the equation \(C=kP\), we can solve for the Henry's law constant \(k\) by rearranging the equation to \(k=C/P\). Plugging in the given values, we have: \(k = \frac{8.21 \times 10^{-4} \mathrm{mol} / \mathrm{L}}{0.790 \mathrm{atm}}\)) Next, perform the calculation to find the value of \(k\): \(k = \frac{8.21 \times 10^{-4} \mathrm{mol} / \mathrm{L}}{0.790 \mathrm{atm}} \approx 1.04 \times 10^{-3} \frac{\mathrm{mol}}{\mathrm{L}\cdot\mathrm{atm}}\)

(Step 3: Calculate the new solubility of N₂)

(We are now asked to find the solubility of N₂ when the partial pressure is 1.10 atm. Using the same equation as the previous step, rearrange to find the concentration: \(C = kP\)). We have \(k = 1.04 \times 10^{-3} \frac{\mathrm{mol}}{\mathrm{L}\cdot\mathrm{atm}}\) and \(P = 1.10 \mathrm{atm}\). Multiply these values to find the new solubility of N₂ in water: \(C = \left(1.04 \times 10^{-3} \frac{\mathrm{mol}}{\mathrm{L}\cdot\mathrm{atm}}\right) (1.10 \mathrm {atm}) \approx 1.14 \times 10^{-3} \mathrm{mol}/\mathrm{L}\) The solubility of N₂ in water when the partial pressure of nitrogen is 1.10 atm at \(0^{\circ} \mathrm{C}\) is approximately \(1.14 \times 10^{-3} \mathrm{mol}/\mathrm{L}\).

Key concepts

Solubility

Solubility refers to the ability of a substance, specifically a gas in this context, to dissolve in a solvent, such as water. The solubility of a gas depends on various factors, including the partial pressure of the gas over the solvent and the temperature. According to the exercise, nitrogen has a solubility of \(8.21 \times 10^{-4} \ \mathrm{mol}/\mathrm{L}\) at a temperature of \(0^{\circ} \mathrm{C}\) with its vapor pressure over the water measured at 0.790 atm.

It is important to note that solubility helps predict how much of a gas can be absorbed by a liquid under certain conditions. In practical terms, this provides essential information for processes like oxygenation in aquatic environments, where gas solubility impacts aquatic life sustainability. People studying chemical engineering or atmospheric science frequently consider solubility to understand and design efficient systems for gas-liquid interactions.

Gas Concentration

Gas concentration is a measure of the amount of a solute gas present in a given volume of solvent. In the context of the exercise, the gas concentration is given in the units of \(\mathrm{mol}/\mathrm{L}\). The concentration essentially indicates how many molecules of gas are present in a specific volume of water at equilibrium.

Henry's Law, represented by the equation \(C = kP\), highlights the direct relationship between gas concentration \(C\) and the partial pressure \(P\) of the gas. The term \(k\) in the equation is the Henry's Law constant, which varies with the nature of the solvent and gas. To determine \(k\), one would divide the initial gas concentration by its initial partial pressure. For instance, using the given data, the constant is calculated using the concentration \(8.21 \times 10^{-4} \ \mathrm{mol}/\mathrm{L}\) and the pressure 0.790 atm, which yields an important parameter used to estimate concentrations under different pressures.

Partial Pressure

Partial pressure plays a critical role in determining the solubility of a gas in a solvent according to Henry's Law. It refers to the pressure exerted by an individual gas in a mixture, such as nitrogen in the atmosphere above a solvent like water. The exercise outlines that nitrogen's partial pressure is 0.790 atm initially, and this affects its concentration and solubility in the water.

The core idea here is that gas molecules exert pressure against the liquid surface, which drives their solubility. An increased partial pressure means more gas molecules are available to dissolve into the solvent, thus increasing the solubility. This concept is essential in various scientific fields, including chemistry and environmental science, where understanding how gases interact and dissolve in liquids under different conditions is crucial.

Temperature Effect

Although the exercise focuses on solubility at \(0^{\circ} \mathrm{C}\), understanding the temperature's impact provides a deeper insight. Generally, gases are more soluble in liquids at lower temperatures. This inverse relationship means that as the temperature increases, gas solubility typically decreases.

The rationale behind this behavior lies in kinetic theory. As temperature rises, gas molecules gain kinetic energy, making them more likely to escape from the liquid phase back into the gas phase, thereby reducing solubility. Conversely, at lower temperatures, molecules have less kinetic energy, resulting in higher solubility. Temperature effects are crucial for applications ranging from beverage carbonation to understanding aquatic life ecology and the rates of gas absorption and release in natural water bodies.