Question

At \(20^{\circ} \mathrm{C}(293 \mathrm{~K})\) and a partial pressure of \(10^{5} \mathrm{~Pa}\), the solubility of \(\mathrm{CO}_{2}\) in water is \(0.169 \mathrm{~g} / 100 \mathrm{~mL}\). What is the solubility of \(\mathrm{CO}_{2}\) at \(33.3 \times 10^{5} \mathrm{~Pa}\) ?

Short answer

The solubility of \( \text{CO}_2 \) at \( 33.3 \times 10^5 \ Pa \) is approximately \( 5.63 \ g/100 \ mL \).

Step-by-step solution

Understanding Henry's Law

Henry's Law states that the solubility of a gas in a liquid is directly proportional to its partial pressure above the liquid. Mathematically, Henry's law can be expressed as \( S = k_H \times P \), where \( S \) is the solubility of the gas, \( k_H \) is Henry's law constant, and \( P \) is the partial pressure.

Determine Initial Conditions

We are given an initial solubility of \( 0.169 \ g/100 \ mL \) at a partial pressure of \( 10^5 \ Pa \). Let this initial solubility be \( S_1 = 0.169 \ g/100 \ mL \), and partial pressure \( P_1 = 10^5 \ Pa \).

Calculate Henry's Law Constant

According to Henry's Law, \( S = k_H \times P \). Rearranging to solve for \( k_H \), we have \( k_H = \frac{S_1}{P_1} = \frac{0.169}{10^5} \ g \cdot m^{-3} \cdot Pa^{-1}\).

Determine New Condition

The new partial pressure given is \( P_2 = 33.3 \times 10^5 \ Pa \). We need to find the new solubility \( S_2 \) using Henry's Law: \( S_2 = k_H \times P_2 \).

Calculate New Solubility

Substitute the values into the equation: \( S_2 = \frac{0.169}{10^5} \times 33.3 \times 10^5 = 0.169 \times 33.3 \ g/100 \ mL \). Calculate this product to find \( S_2 \).

Final Calculation

\( S_2 = 0.169 \times 33.3 = 5.6277 \ g/100 \ mL \). Thus, the solubility of \( \text{CO}_2 \) at the new pressure is approximately \( 5.63 \ g/100 \ mL \).

Key concepts

Gas Solubility

Gas solubility refers to the ability of a gas to dissolve in a liquid. This concept is important especially in various natural and industrial processes. For example, carbon dioxide dissolves in water to create carbonated beverages, while oxygen needs to dissolve in blood to support life. The solubility of a gas depends on several factors, including:

• Temperature: Increasing the temperature generally decreases gas solubility because gases tend to expand and escape from liquids as they are heated.
• Nature of Gas and Liquid: Some gases dissolve better in water (like carbon dioxide and oxygen), while others might dissolve well in oils or other solvents.
• Partial Pressure: More about this in the next section, since this directly influences the amount of a gas that can dissolve in a liquid.

Understanding these factors helps in predicting how much gas can be dissolved under certain conditions.

Partial Pressure

Partial pressure is a concept related to gases in mixtures, like those in the atmosphere. When we discuss gas solubility, partial pressure plays an essential role. It is defined as the pressure that a single gas in a mixture would exert if it alone occupied the entire volume of the original mixture at the same temperature. Therefore, it affects how much of the gas can dissolve in a liquid.According to Henry's Law, which is central to understanding partial pressure and solubility, the solubility of a gas is directly proportional to its partial pressure above the liquid. This means that increasing the partial pressure will increase the solubility of the gas. In mathematical terms, we have:\[ S = k_H \times P \]Where:

• \( S \) is the solubility of the gas.
• \( k_H \) is Henry’s law constant, which varies depending on the gas-liquid pair and the temperature.
• \( P \) is the partial pressure of the gas.

Thus, knowing the partial pressures helps in determining how gases can behave under different conditions.

Solubility Calculation

Solubility calculation involves determining how much of a gas will dissolve in a specific amount of liquid under set conditions. The exercise provided an example of this by calculating the solubility of carbon dioxide in water under different pressures. Here's a simplified approach to calculate it using Henry's Law:1. **Identify Initial Conditions:** - Initial solubility \( S_1 = 0.169 \ g/100 \ mL \). - Initial pressure \( P_1 = 10^5 \ Pa \).2. **Calculate Henry’s Law Constant:** To calculate \( k_H \), use the formula: \[ k_H = \frac{S_1}{P_1} \]3. **Determine New Conditions:** - New pressure \( P_2 = 33.3 \times 10^5 \ Pa \).4. **Calculate New Solubility:** Substituting into the Henry’s Law equation, determine the new solubility \( S_2 \): \[ S_2 = k_H \times P_2 \ = \frac{0.169}{10^5} \times 33.3 \times 10^5 \ = 5.63 \ g/100 \ mL \]Through these steps, you can predict how the solubility of a gas changes with varying pressures, aiding in practical applications like carbonated drink manufacturing or chemical reactors.