Calculate the concentration of \(\mathrm{CO}_{2}\) in a soft drink bottle after the bottle is opened and sits at \(25^{\circ} \mathrm{C}\) under a \(\mathrm{CO}_{2}\) partial pressure of \(3.0 \times 10^{-4} \mathrm{~atm}\). Henry's law constant for \(\mathrm{CO}_{2}\) in water is \(3.1 \times 10^{-2}\) mol/litre-atm at this temperature.

Gas solubility refers to the ability of a gas to dissolve in a liquid. This attribute is of significant importance not just in chemistry, but also in fields like environmental science and food technology. To understand how solubility works, picture a bottle of your favorite carbonated drink. When sealed, the gas (carbon dioxide, in this case) is under high pressure, staying dissolved in the liquid. However, as soon as you open the bottle, the pressure above the liquid drops, and the gas begins to escape, forming bubbles.

One key factor that determines the solubility of a gas in a liquid is temperature. Typically, gases become less soluble as the temperature increases. This is why your drink seems to lose its fizz faster on a hot day than on a cold one. Additionally, the nature of the gas and the liquid play roles in how well the gas will dissolve. Different gases have varying solubilities in various liquids based on their physical and chemical properties. For a refined understanding, we look at Henry's Law, which specifically quantifies the relationship between pressure and solubility for gases in liquids at a constant temperature.

Partial pressure is the pressure that a gas in a mixture of gases would exert if it alone occupied the entire volume. Imagine being at a concert with a crowd of people. The overall noise is loud, but if only one person were to sing, that voice would have its own distinct 'pressure' or sound level. In the same manner, each gas in a mixture contributes to the total pressure of the system, according to its proportion in the mixture.

When it comes to solving problems involving Henry's Law, knowing the partial pressure of the gas in question is crucial. It's the value you're going to use to determine how much of the gas will dissolve in a particular liquid at a given temperature. In our soft drink example, the carbon dioxide in the bottle has a partial pressure, and it's this pressure that determines how much CO2 remains in solution, giving the drink its characteristic fizziness.

Concentration calculation involves determining the amount of a substance within a particular volume. Just as bakers must measure out ingredients to ensure a recipe turns out correctly, chemists need to calculate concentration to predict and understand the behavior of solutions. Concentration is a measure of how 'crowded' a solution is with a particular solute. In the context of our fizzy beverage, after opening the bottle, the concentration of CO2 in the liquid will change as the gas escapes.

To calculate the concentration from Henry's Law, you multiply the gas's partial pressure by its solubility constant at the given temperature. Using the formula from the exercise solution, \( C = kP \), we determined the concentration of CO2 in the soft drink. The subtleties of this calculation lie in the precise use of units for pressure (atmospheres, in this case) and the concentration (moles per liter). It's these details that translate scientific laws into practical tools for understanding and designing everyday systems like carbonated beverages.