Problem 89 - Chemistry Matter and Change

Question

At a pressure of 1.5 atm, the solubility of a gas is 0.54 g/L. Calculate the solubility when the pressure is doubled.

Step-by-Step Solution

Answer

The solubility of the gas when the pressure is doubled is approximately \(1.08 \text{ g/L}\).

Step 1: Recall Henry's Law formula

Henry's Law states that the solubility of a gas (S) in a liquid is directly proportional to the partial pressure (P) of the gas above the liquid. Mathematically, it can be written as: \[S = k \cdot P\] where k is the Henry's Law constant (unique for each gas and liquid pair).

Step 2: Calculate the Henry's Law constant

First, we need to find the value of the Henry's Law constant (k) using the given solubility (S) and pressure (P). From the question, we have S = 0.54 g/L and P = 1.5 atm. Plug these values into the formula: \[k = \frac{S}{P}\] \[k = \frac{0.54 \text{ g/L}}{1.5 \text{ atm}}\] \[k \approx 0.36 \frac{\text{g/L}}{\text{atm}}\]

Step 3: Calculate the solubility at the new pressure

The problem states that the pressure is doubled, so the new pressure (P') is 2 * P = 3.0 atm. Now, using the Henry's Law formula, we will calculate the solubility (S') at this new pressure: \[S' = k \cdot P'\] \[S' = 0.36 \frac{\text{g/L}}{\text{atm}} \cdot 3.0 \text{ atm}\] \[S' \approx 1.08 \text{ g/L}\]

Step 4: State the final answer

The solubility of the gas when the pressure is doubled is approximately 1.08 g/L.

Key Concepts

Gas Solubility

Gas solubility is a measure of how well a gas can dissolve in a liquid. It depends on several factors, but pressure is a primary one. When a gas dissolves, its molecules spread out and interact with the liquid molecules. This interaction is affected by the type of liquid and gas, temperature, and crucially, pressure. Understanding gas solubility helps in predicting and calculating how gases will behave when mixed with liquids. For instance, knowing the solubility is important in various real-world applications like carbonated beverages, where carbon dioxide is dissolved under pressure in water. As pressure changes, the amount of gas that can dissolve in a liquid changes too, making it a dynamic characteristic. It is essential for students to grasp how solubility is not a constant value but varies with conditions, especially with changes in pressure and temperature.

Pressure Effect

The effect of pressure on gas solubility is quite significant. According to Henry's Law, the solubility of a gas increases as the pressure increases. This is because, under higher pressure, gas molecules are forced closer together, increasing their interaction with the liquid molecules. For instance, in the given exercise, when the pressure is doubled, the solubility of the gas also increases. This relationship is linear, meaning if the pressure doubles, so does the solubility. This principle is crucial to understand in fields like chemistry and environmental science. It has practical implications, such as in the processes of carbonation in beverages or in the medical field, where gas solubility can affect the behavior of gases in liquids under different pressures. Students often find it helpful to visualize this effect by imagining a balloon filled with gas that shrinks under high pressure, forcing the gas particles into the liquid more effectively.

Henry's Law Constant

The Henry’s Law constant, represented as k, is a key component in calculating gas solubility. It is unique to each gas-liquid pair and indicates how much gas will dissolve in a liquid at a specific pressure. Understanding this constant is crucial for accurately calculating the changes in solubility under varying conditions, as shown in the original exercise. In the exercise, the constant was determined using the formula \( k = \frac{S}{P} \), where S is the solubility and P is the pressure. Once k is known, it can predict solubility at different pressures using the equation \( S = k \cdot P \). This constant offers insight into the potential of a gas to dissolve in a specific liquid, guiding practical applications like industrial gas purification or even in scientific research.

Chemistry Calculations

Chemistry calculations involving gas solubility require an understanding of algebra and proportionality, particularly using equations like Henry's Law. These calculations allow us to predict and quantify changes in physical states of substances under different conditions. As demonstrated in the exercise, solving such problems starts with identifying the known values—like pressure and solubility—and using the given formulas to find unknowns, such as the Henry's Law constant or the new solubility. With correct substitution and manipulation of the formulas, the calculation steps become straightforward. These calculations teach consistency and accuracy, as they can be applied to broader chemistry problems beyond just gas solubility. They illustrate how fundamental Math skills intertwine with chemistry concepts to solve practical and theoretical challenges effectively in scientific fields.