Calculate the concentration of \(\mathrm{CO}_{2}\) in a soft drink that is bottled with a partial pressure of \(\mathrm{CO}_{2}\) of 4 atm over the liquid at \(25^{\circ} \mathrm{C}\). The Henry's law constant for \(\mathrm{CO}_{2}\) in water at \(25^{\circ} \mathrm{C}\) is \(3.1 \times 10^{-2} \mathrm{~mol} /\) litre-atm .

Understanding how gases dissolve in liquids is essential in various scientific fields, including environmental engineering and the beverage industry, like in the case of carbonated soft drinks.

At the heart of this process is a basic principle known as Henry's Law. It tells us that at a constant temperature, the amount of gas that dissolves in a liquid is directly proportional to the pressure of the gas present above the liquid. This is a linear relationship, meaning if we double the pressure of the gas, the amount of gas dissolved in the liquid also doubles, keeping the temperature constant.

To put it more simply, more gas will dissolve in a liquid if we increase the pressure of that gas above the liquid. If you've ever opened a bottle of soda and watched it fizz, you've seen Henry's Law in action; when the bottle is sealed, the high pressure inside keeps more carbon dioxide dissolved, but releasing the pressure by opening the bottle allows the gas to escape, forming bubbles.

In the case of the soft drink example, the concentration of \(\mathrm{CO}_2\) in the beverage can be predicted by knowing the partial pressure of \(\mathrm{CO}_2\) above the liquid and the Henry's Law constant for \(\mathrm{CO}_2\) at the beverage's temperature.

The partial pressure of a gas is a measurement of the thermodynamic activity of the gas's molecules. It's essentially a way of expressing the pressure that the gas would exert if it alone occupied the volume of the mixture at the same temperature.

This concept is particularly important when we are dealing with mixtures of gases, such as the air we breathe, which is a mixture mainly of nitrogen, oxygen, and small amounts of other gases. The total pressure exerted by the air is the sum of the partial pressures of each of these individual gases.

In the context of calculating the concentration of gas in a liquid, as in the supplied textbook exercise, the partial pressure tells us how much pressure the gas contributes to the system. When we talk about carbonating beverages, the carbon dioxide occupies some space above the liquid and exerts a certain pressure—which is the partial pressure we use in our calculations for Henry's Law. The higher the partial pressure of \(\mathrm{CO}_2\), the more carbon dioxide gets dissolved in the soft drink, until it reaches equilibrium.

Aspiring students aiming to crack competitive exams like IIT-JEE must have a strong grasp of concepts across all branches of chemistry, including physical chemistry. This branch of chemistry deals with the understanding of the behavior of molecules, the kinetics of reactions, and the properties of gases, among other topics.

Physical chemistry is known for its mathematical approach to explaining the physical properties of molecules. For IIT-JEE, topics such as thermodynamics, chemical equilibrium, and solutions are often accompanied by rigorous problem-solving exercises that test a student's conceptual knowledge and application skills.

Henry's Law is an essential topic in the study of solutions, which is a part of physical chemistry, and understanding it can give IIT-JEE aspirants an edge in solving practical problems related to the concentration of gases in liquids. Problems like the one in the textbook exercise not only enhance comprehension of theoretical concepts but also develop skills for solving real-world challenges using scientific principles.