Question

The carbon dioxide gas dissolved in a sample of water in a partly filled, sealed container has reached equilibrium with irs partial pressure in the air above the solution. Explain what happens to the solubility of the \(\mathrm{CO}_{2}\) if (a) the partial pressure of the \(\mathrm{CO}_{2}\) gas is doubled by the addition of more \(\mathrm{CO}_{2}\); (b) the total pressure of the gas above the liquid is doubled by the addirion of nitrogen.

Short answer

For (a) the solubility of \( \mathrm{CO}_{2} \) will double, while for (b) the solubility of \( \mathrm{CO}_{2} \) will remain unchanged.

Step-by-step solution

Understanding Henry's Law

Henry's Law states that the solubility of a gas in a liquid at a given temperature is directly proportional to the partial pressure of that gas above the liquid. Mathematically, it can be represented as \( S_g = k_H P_g \), where \( S_g \) is the solubility of the gas, \( k_H \) is Henry’s law constant, and \( P_g \) is the partial pressure of the gas.

Scenario (a): Increasing \( \mathrm{CO}_{2} \) Partial Pressure

When the partial pressure of \( \mathrm{CO}_{2} \) is doubled, according to Henry's Law, the solubility of \( \mathrm{CO}_{2} \) in water will also double. This is because the solubility is directly proportional to the partial pressure of the dissolved gas.

Scenario (b): Increasing Total Pressure with Nitrogen

Adding nitrogen to the container increases the total pressure but does not affect the partial pressure of \( \mathrm{CO}_{2} \) gas. Since Henry's Law depends only on the partial pressure of the gas in question, the solubility of \( \mathrm{CO}_{2} \) in the water will remain unchanged.

Key concepts

Partial Pressure

Understanding partial pressure is crucial when discussing the solubility of gases in liquids. Partial pressure refers to the pressure that a single component of a gas mixture would exert if it alone occupied the entire volume of the mixture.
The importance of partial pressure emerges from Dalton's Law, which states that 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, when dealing with a gas dissolved in a liquid, it's the partial pressure of the gas above the solution that's taken into account, rather than the total pressure of all gases present.
Scenario (a) of the exercise highlights the direct relationship between the solubility of a gas and its partial pressure. When the partial pressure of CO₂ is increased, the amount of CO₂ that the water can dissolve increases proportionally, assuming temperature remains constant.

Solubility of Gases

The solubility of gases in liquids is a measure of how much of a particular gas will dissolve in a given amount of liquid at a specific temperature and pressure. Various factors affect solubility, including the nature of the gas and the solvent, temperature, and pressure.
As described by Henry's Law, at a constant temperature, the solubility of a gas in a liquid is proportional to the partial pressure of that gas above the liquid. This principle is widely used in industries such as beverage carbonation, where CO₂ is dissolved into drinks under high pressure to create fizziness.
In our exercise, we've established that doubling the partial pressure of CO₂ leads directly to the doubling of its solubility in water. This principle is foundational in understanding how gases become more or less soluble under varying pressure conditions.

Chemical Equilibrium

Chemical equilibrium is a state in a chemical process when the concentrations of reactants and products remain constant over time because the forward and reverse reactions occur at the same rate. This balance doesn't imply that the reactants and products are equal in concentration, but rather that their ratio doesn't change with time.
In the context of gas-liquid equilibrium, when a gas is dissolved in a liquid, equilibrium is reached when the rate of gas molecules entering the liquid phase is equal to the rate of molecules exiting to the gas phase. It's important to note that while Henry's Law provides insights into the initial solubility of gases in liquids, chemical equilibrium describes the dynamic but stable state achieved after the dissolving process has occurred over time.
Changes in conditions, such as pressure or temperature, can 'shift' the equilibrium, as predicted by Le Chatelier's Principle, which states that a system at equilibrium will adjust to counteract a change in conditions.

Gas-Liquid Equilibrium

Gas-liquid equilibrium pertains to the balance between a gas phase and a liquid phase when a gas is dissolved in a liquid. This equilibrium is dynamic, meaning that molecules of gas are continually dissolving into the liquid and escaping back into the gas phase.
Scenario (b) from our exercise tackles a non-reacting gas: nitrogen. Adding nitrogen increases the total pressure but does not disrupt the partial pressure of CO₂. Thus, it does not affect the CO₂ solubility in water, demonstrating that for a gas-liquid equilibrium, the solubility is linked to its own partial pressure, irrespective of other gases present.
This principle is essential when analyzing systems containing multiple gases, and it allows us to predict that changes in the composition of the gas phase won't affect the solubility of a gas unless they alter its partial pressure.