The carbon dioxide gas dissolved in a sample of water in a partially filled, sealed containcr has reached equilibrium with its 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 addition of nitrogen.

The concept of partial pressure is vital when discussing the behavior of gases, especially within mixtures. Partial pressure is the pressure that a single gas component in a mixture of gases would exert if it occupied the entire volume alone at the same temperature.

Imagine a container filled with different types of gases; each gas contributes to the total pressure inside the container. This individual contribution is what we call the 'partial pressure' of that gas. The total pressure, therefore, is the sum of all the partial pressures of the gases present. This principle is important because it dictates how gases will behave when they come into contact with liquids, influence their solubility, and even affect chemical reactions.

Regarding solubility, as demonstrated in the exercise, if the partial pressure of a specific gas (like CO₂) increases, the amount of the gas that dissolves in a liquid will also increase, assuming all other factors remain constant. This relationship is elegantly captured by Henry's Law.

Solubility is a measure of how much of a substance (solute) can dissolve in a given quantity of solvent to form a homogeneous solution at a specific temperature and pressure. In the context of gases dissolving in liquids, factors including the nature of the gas, the type of liquid, temperature, and pressure play significant roles.

Henry's Law provides a quantitative understanding of how gas solubility in liquids is affected by pressure. The law states that at a constant temperature, the amount of a given gas that dissolves in a given type and volume of liquid is directly proportional to the partial pressure of that gas in equilibrium with the liquid. Mathematically, this is expressed as \( S = kP \), where \( S \) is the solubility, \( k \) is the Henry's Law constant specific to the gas-liquid pair, and \( P \) is the partial pressure of the gas.

When we apply this law to a real-world scenario, such as a sealed container of carbonated water, we understand that increasing the partial pressure of CO₂ gas by adding more CO₂ would subsequently increase the amount of CO₂ that can be dissolved in the water.

Chemical equilibrium is a state in which the rates of the forward and reverse reactions are equal, resulting in no net change in the amounts of reactants and products. This dynamic process is essential in various chemical and biological systems.

When a gas such as CO₂ is dissolved in water, it is involved in equilibrium processes, including forming carbonic acid (H₂CO₃). The solution in the container from our exercise reaches a point where the rate at which CO₂ molecules enter the solution is equal to the rate at which they leave. This is chemical equilibrium.

Now, if we were to increase the partial pressure of CO₂ by injecting more into the container, the equilibrium would shift to accommodate more CO₂ molecules dissolving into the water until a new equilibrium is established (per Le Chatelier's Principle). Notably, the solubility of CO₂ increases, not just because of higher partial pressure, but also because the system adapts to reestablish equilibrium conditions.