Question

The solubility of a gas at 37.0 \(\mathrm{kPa}\) is 1.80 \(\mathrm{g} / \mathrm{L}\) . At what pressure will the solubility reach 9.00 \(\mathrm{g} / \mathrm{L} ?\)

Short answer

The new pressure required when the solubility of the gas increases to 9.00 g/L is \(P_2 = 185.0 \thinspace \mathrm{kPa}\).

Step-by-step solution

Write down the given values and Henry's Law

In this exercise, we are given the initial solubility (S1) as 1.80 g/L and the initial pressure (P1) as 37.0 kPa. We want to find the new pressure (P2) when the solubility (S2) is increased to 9.00 g/L. According to Henry's Law, the solubility of a gas is directly proportional to the pressure, which can be written as: \[S_1/P_1 = S_2/P_2\]

Solve for the new pressure P2

We want to find P2, so we can rewrite the equation from Step 1 as: \[P_2 = P_1 \times (\frac{S_2}{S_1})\] Now, substitute the given values into the equation: \[P_2 = 37.0 \thinspace \mathrm{kPa} \times (\frac{9.00 \thinspace \mathrm{g/L}}{1.80 \thinspace \mathrm{g/L}})\]

Calculate the new pressure P2

By performing the calculation, we get: \[P_2 = 37.0 \thinspace \mathrm{kPa} \times 5\] \[P_2 = 185.0 \thinspace \mathrm{kPa}\] Thus, when the solubility of the gas is increased to 9.00 g/L, the pressure required is 185.0 kPa.

Key concepts

Gas Solubility

Gas solubility refers to how much of a gas can dissolve in a liquid under specific conditions. It's an important concept in chemistry, helping us understand how gases behave in different environments. The gas solubility in a liquid depends on factors like temperature and pressure. Usually, more gas can dissolve in a liquid when the temperature is lower and the pressure is higher.

When we refer to the solubility of a gas, it's often in terms of grams per liter (g/L). This tells us the amount of gas that has dissolved in a certain volume of liquid. In the given exercise, the initial solubility is 1.80 g/L. This number changes based on pressure or temperature alterations.

Understanding gas solubility is crucial for various applications, including environmental sciences, engineering, and medicine. For instance, it helps explain why fish can absorb oxygen in water and why divers must carefully manage their ascent rate to avoid decompression sickness.

Pressure-Solubility Relationship

The pressure-solubility relationship is a fundamental aspect of Henry's Law. This relationship describes how gas solubility in a liquid increases with increasing pressure. When pressure above a liquid rises, more gas molecules are "pushed" into the liquid, enhancing solubility. This is why you get more carbon dioxide in a soda bottle when it's sealed tight compared to when it's opened.

Henry's Law states that the solubility of a gas is directly proportional to its partial pressure over the solution. Mathematically, it means if you double the pressure, the solubility should also double, provided the temperature remains constant.

In the exercise, this relationship is used to calculate the new pressure required for a fivefold increase in solubility of the gas from 1.80 g/L to 9.00 g/L. The solving steps clearly illustrate how knowing one set of solubility and pressure allows you to predict another set using this direct relationship.

Direct Proportionality in Chemistry

Direct proportionality is a key concept in chemistry and other sciences. It explains situations where an increase in one quantity leads to a proportional increase in another. This is expressed by the formula:

• When two variables, say X and Y, are directly proportional, you can write it as \( X = k \times Y \) where \( k \) is a constant.
• A graph of one variable against the other will be a straight line through the origin.

In the context of Henry's Law, the solubility of a gas (S) is directly proportional to its pressure (P) above the solution, making \( S \propto P \). This direct relationship simplifies calculations in chemistry because knowing how much one variable changes allows you to directly calculate the change in the other.

In the provided exercise, you see this principle in action. The solubility is initially given as 1.80 g/L at 37 kPa, showing a clear direct proportional relationship. Increasing the solubility to 9.00 g/L (five times greater) requires increasing the pressure to 185 kPa, also five times greater, demonstrating the direct proportionality between solubility and pressure.