The partial pressure of ethane over a saturated solution containing \(6.56 \times 10^{-2} \mathrm{~g}\) of ethane is 1 bar. If the solution contains \(5.00 \times 10^{-2} \mathrm{~g}\) of ethane, then what shall be the partial pressure of the gas.

Partial pressure is a key concept when it comes to understanding how gases behave in mixtures. When we have a mixture of gases, like the air we breathe, each gas exerts a certain pressure as if it were alone in the container. This pressure is what we call the partial pressure. It depends on factors such as the amount of the gas present and the temperature of the system.

In the given exercise, ethane gas in contact with its solution is creating pressure that can be referred to as the partial pressure of ethane. By knowing the partial pressure exerted by a gas above a solution, we can get an idea about its concentration in the solution. This brings us to Henry's Law, which has a fundamental role in explaining the relationship between the amount of gas dissolved in the solution and the partial pressure of that gas.

Gas solubility, specifically in chemistry and environmental science, is an important aspect of how gases dissolve in solvents, such as water. The amount of gas that dissolves in a certain volume of liquid is dictated by the solubility of the gas under specific conditions, usually under constant temperature. Henry's Law provides 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.

Considering our case with ethane gas, its solubility in the solution can be predicted and calculated using the law. The implication is that if the partial pressure of ethane above the solution increases, more ethane will dissolve in the solution until a new equilibrium is established. Conclusively, understanding gas solubility allows us to predict how changes in conditions like pressure will affect the concentration of a dissolved gas.

Proportionality plays a critical role in many areas of chemistry. It serves as the basis for understanding relationships between different properties, like the solubility of a gas in a liquid and the partial pressure of the gas. This concept is evidenced in the expression of Henry's Law, which states that the solubility of a gas is directly proportional to its pressure.

In the original exercise, a direct proportion helps us relate the partial pressures to the amounts of dissolved ethane. When the initial amount of ethane changes, we can use this relationship to calculate the new partial pressure. Simplicity is key in conveying such concepts: think of proportionality as a seesaw in a playground; as one side goes up, the other goes down by the same amount. This direct ratio allows chemists to easily predict how one variable will change when another is altered, which is exactly what we did when finding the new partial pressure for a different amount of ethane.