At \(0^{\circ} \mathrm{C}\) and \(1.00 \mathrm{atm},\) as much as \(0.70 \mathrm{g}\) of \(\mathrm{O}_{2}\) can dissolve in \(1 \mathrm{L}\) of water. At \(0^{\circ} \mathrm{C}\) and \(4.00 \mathrm{atm},\) how many grams of \(\mathrm{O}_{2}\) dissolve in 1 L of water?

Gas solubility is the measure of how much of a particular gas can dissolve in a liquid at a specific temperature and pressure. This concept is critical to understand in various fields, including environmental science, engineering, and physical chemistry.

When a gas comes into contact with a liquid, molecules from the gas phase can enter the liquid phase. The solubility of the gas in the liquid will depend on several factors, including the nature of the gas, the type of liquid, the temperature, and most notably, the pressure of the gas above the liquid. Higher solubility means that more gas can be dissolved in the liquid.
In the given exercise, we examine the solubility of oxygen ($O\_2$) in water. The provided data allows us to understand how the solubility of oxygen changes with pressure—a concept extensively described by Henry's Law.

Partial pressure is an essential concept when discussing gas solubility. It refers to the pressure that a single gas component in a mixture of gases would exert if it alone occupied the entire volume. In simpler terms, it's the contribution each gas in a mixture makes to the total pressure.

In a mixture of gases, each gas's partial pressure is proportional to its mole fraction. This concept helps us understand how gases behave under different conditions and is especially useful in calculations involving gas solubility in liquids.
When analyzing the solubility of a gas in a liquid, it is the partial pressure of that specific gas above the liquid that is considered, not the total pressure of all gases present. This is why Henry's Law relates the solubility of a gas in a liquid directly to the partial pressure of the gas—not the overall atmospheric pressure.

The interplay between solubility and pressure in a liquid-gas system is governed by Henry's Law, which states that at a constant temperature, the solubility of a gas in a liquid is directly proportional to the pressure of that gas above the liquid.

Mathematically, this correlation can be expressed using the equation $S\; =\; kP$, where $S$ is the solubility of the gas, $k$ is Henry's law constant (which can vary depending on the gas and the liquid), and $P$ represents the partial pressure of the gas.
As pressure increases, more gas molecules are 'pushed' into the liquid, and solubility increases. Conversely, if the pressure decreases, gas molecules are 'released' from the liquid, and solubility decreases. Understanding this relationship is crucial for predicting how changes in pressure can affect the concentration of a gas in a liquid, as illustrated by our textbook exercise, wherein the oxygen solubility increases when the pressure above the water rises.