Question

The Henry's law constant for \(\mathrm{N}_{2}\) in water at \(25^{\circ} \mathrm{C}\) is \(8.4 \times 10^{-7} \mathrm{~mol} \mathrm{~L}^{-1} \mathrm{mmHg}^{-1}\). Calculate the solubility of \(\mathrm{N}_{2}\) in \(\mathrm{mol} / \mathrm{L}\) if its partial pressure is \(1520 \mathrm{mmHg}\). Calculate the solubility when the \(\mathrm{N}_{2}\) partial pressure is 20. mmHg.

Short answer

The solubilities are \(1.28 \times 10^{-3} \text{ mol/L}\) and \(1.68 \times 10^{-5} \text{ mol/L}\) for pressures of 1520 mmHg and 20 mmHg, respectively.

Step-by-step solution

Understand Henry's Law

Henry's law states that the solubility (s) of a gas in a liquid is directly proportional to the partial pressure of the gas (P) above the liquid. The formula is expressed as: \[ s = kH \times P \]where \(kH\) is the Henry's law constant and \(P\) is the partial pressure.

Substitute First Pressure Value

For the first scenario, we need to find the solubility of \( \mathrm{N}_2\) when the partial pressure \( P \) is 1520 mmHg. Substitute the known values into Henry's law formula:\[ s = (8.4 \times 10^{-7} \text{ mol} \text{ L}^{-1} \text{ mmHg}^{-1}) \times 1520 \text{ mmHg} \].

Calculate Solubility for First Pressure

Perform the multiplication to find the solubility:\[ s = 8.4 \times 10^{-7} \times 1520 = 1.2768 \times 10^{-3} \text{ mol} \text{ L}^{-1} \].

Substitute Second Pressure Value

For the second scenario, find the solubility of \( \mathrm{N}_2\) when the partial pressure \( P \) is 20 mmHg.Substitute the values:\[ s = (8.4 \times 10^{-7} \text{ mol} \text{ L}^{-1} \text{ mmHg}^{-1}) \times 20 \text{ mmHg} \].

Calculate Solubility for Second Pressure

Perform the multiplication to find the solubility:\[ s = 8.4 \times 10^{-7} \times 20 = 1.68 \times 10^{-5} \text{ mol} \text{ L}^{-1} \].

Key concepts

Gas Solubility

Gas solubility is a crucial concept in understanding how gases interact with liquids. Essentially, gas solubility refers to the amount of a gas that can be dissolved in a liquid at a given temperature and pressure. The greater the solubility, the more gas can be held within the liquid. Solubility is influenced by several factors:

• Temperature: Generally, gas solubility decreases as the temperature increases, because gases tend to escape from liquids more readily at higher temperatures.
• Pressure: According to Henry's Law, solubility is directly proportional to the partial pressure of the gas above the liquid.
• Nature of the Gas and Liquid: Some gases dissolve better in certain liquids due to chemical affinity or polarity.

When applying Henry’s Law to calculate solubility, we use the formula \[ s = k_H \times P \]where

• \(s\) is the solubility,
• \(k_H\) is the Henry's law constant,
• \(P\) is the partial pressure of the gas.

Understanding these basics will help in moving to more specific calculations related to solutions and gases.

Partial Pressure

In a gas mixture, such as air, each gas exerts its own pressure termed as partial pressure. The concept of partial pressure is fundamental in Dalton's Law of Partial Pressures, which states that the total pressure of a gaseous mixture is the sum of the partial pressures of each individual gas present. The formula to express this is:\[P_{ ext{total}} = P_1 + P_2 + P_3 + ext{...} + P_n\]where each \(P_i\) represents the partial pressure of a different gas in the mixture. Partial pressure is also critical when applying Henry’s Law, as it determines the amount of gas that will dissolve in a liquid. The higher the partial pressure, the more gas will dissolve until equilibrium is reached. This idea is widely used in chemical engineering, environmental science, and medicine, especially for gases dissolved in blood or environmental gases.
In the case of nitrogen (\(\mathrm{N}_2\)) discussed earlier, by knowing its partial pressure, you can easily calculate its solubility in water using Henry’s Law. Understanding how to manipulate and apply these pressures can effectively predict and control gas behaviors in different settings.

Chemical Calculations

Chemical calculations form the backbone of quantitative chemistry. They involve the application of mathematical operations to solve problems related to chemical quantities. Some common types of chemical calculations include:

• Stoichiometry: Calculations of reactants and products in chemical reactions, often involving mole ratios.
• Concentration: Determining the amount of a substance within a given volume, such as molarity (mol/L).
• Gas Laws: Employing laws like Boyle's or Charles's to solve problems involving gases.

In the example of Henry's Law calculations, it's important to identify known values carefully, such as the Henry's law constant and the gas's partial pressure, before substituting them into the law's formula. Being thorough with conversions and unit consistency is key to avoiding errors. Using these mathematical approaches helps in finding exact solutions to how chemicals behave in varied circumstances.