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 atm. Calculate the Henry's law constant for ${\mathrm{N}}_2$ in units of mol/L \cdot atm for Henry's law in the form $C=kP,$ 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 nitrogen in water is approximately $1.04\times {10}^{-3}\mathrm{mol}/\mathrm{L}\cdot \mathrm{atm}$. The solubility of nitrogen in water at 0°C and a pressure of 1.10 atm is approximately $1.14\times {10}^{-3}\mathrm{mol}/\mathrm{L}$.

Step-by-step solution

Identify the given variables

We are given: - C = $8.21\times {10}^{-4}$ mol/L (solubility of N₂ in water at 0°C and 0.790 atm) - P₁ = 0.790 atm (initial pressure of N₂) - P₂ = 1.10 atm (new pressure of N₂)

Calculate the Henry's law constant (k)

Using the formula C = kP, we have: k = C/P₁ Plug in the given values: k = $\frac{8.21\times {10}^{-4}\mathrm{mol}/\mathrm{L}}{0.790\mathrm{atm}}$ Now, calculate the value of k: k ≈ $1.04\times {10}^{-3}\mathrm{mol}/\mathrm{L}\cdot \mathrm{atm}$

Calculate the solubility of nitrogen at the new pressure

Now, we need to find the solubility of N₂ (C₂) when the pressure above water is 1.10 atm. Using the formula C = kP, we have: C₂ = kP₂ Plug in the given values: C₂ = $1.04\times {10}^{-3}\mathrm{mol}/\mathrm{L}\cdot \mathrm{atm}$ × 1.10 atm Now, calculate the value of C₂: C₂ ≈ $1.14\times {10}^{-3}\mathrm{mol}/\mathrm{L}$

State the final results

1. The Henry's law constant for nitrogen in water is k ≈ $1.04\times {10}^{-3}\mathrm{mol}/\mathrm{L}\cdot \mathrm{atm}$. 2. The solubility of nitrogen in water at 0°C and a pressure of 1.10 atm is C₂ ≈ $1.14\times {10}^{-3}\mathrm{mol}/\mathrm{L}$.

Key concepts

Solubility of Gases

Understanding the solubility of gases in various solvents, such as water, is a key concept in chemistry and environmental sciences. It can be quite complex, as the solubility of a gas depends on multiple factors like temperature, the nature of the gas and solvent, and the pressure of the gas above the solution.

Solubility is defined as the maximum amount of gas that can dissolve in a solvent at a specific temperature and pressure. It is usually quantified as the concentration of the gas in the solvent, often expressed in moles per liter (mol/L). According to Henry's Law, the solubility of a gas in a liquid is directly proportional to the partial pressure of that gas above the liquid.

For students, it's important to grasp that when you increase the pressure of a gas above the solution, more gas molecules collide with the surface, increasing the likelihood of dissolution. Conversely, when the pressure is lower, fewer gas molecules dissolve into the solution. Temperature also plays a role; typically, gases are less soluble in warmer liquids.

Real-Life Application

Understanding gas solubility is crucial in fields like oceanography where the oxygen supply for marine life depends on the solubility of oxygen in seawater—which varies with temperature and pressure.

Gas Concentration in Solution

Although the solubility of gases might seem abstract, it is a concept that is constantly in play around us. The concentration of a gas in a solution is a direct measure of how much of that gas has dissolved. It is crucial to distinguish between solubility, which refers to a physical constant based on specific conditions, and concentration, which indicates how much gas is actually dissolved at any given time.

In the context of Henry's Law, the concentration ($C$) of a gas is sometimes described as a product of the Henry's Law constant ($k$) and the partial pressure of the gas ($P$). The formula used is $C=kP$. This relationship shows that for a constant temperature, the amount of gas dissolved in a liquid is directly proportional to the pressure of that gas above the liquid. This concept is vital when working with processes like carbonation of beverages, where carbon dioxide concentration is controlled by pressure.

Practical implications

Understanding the relationship between gas concentration and pressure aids in diverse applications, from designing systems for breathable air in submarines to estimating how pollutants disperse in bodies of water.

Partial Pressure

Partial pressure refers to the pressure that a single gas in a mixture would exert if it occupied the entire volume on its own. In mixtures of gases, such as the Earth's atmosphere, each gas exerts a pressure independently. This is an essential principle in the study of gases and their behaviors.

Henry's Law uses the concept of partial pressure to explain how gases dissolve in liquids. The law's formula, $C=kP$, connects the solubility of a gas directly to its partial pressure: as the partial pressure of the gas above a solution increases, so does its solubility, and vice versa. This principle is used in decompression calculations for divers to prevent decompression sickness, for instance.

Application in Breathing

Partial pressure is also fundamental to understanding how gases are exchanged in the lungs during breathing. Oxygen enters the blood because it has a higher partial pressure in the air than in the blood, highlighting the importance of pressure differences in biological processes.