Question

Challenge The solubility of a gas at 7 atm of pressure is 0.52 \(\mathrm{g} / \mathrm{L}\) . How many grams of the gas would be dissolved per 1 \(\mathrm{L}\) if the pressure was raised to 10 \(\mathrm{atm}\) ?

Short answer

The solubility of the gas at 10 atm is 0.364 g/L. Thus, 0.364 grams of the gas would be dissolved per 1 L if the pressure was raised to 10 atm.

Step-by-step solution

List down the given values

We are given the following values: - The solubility of the gas (\(C_1\)) at 7 atm (\(P_1\)) is 0.52 g/L - The pressure (\(P_2\)) raised to 10 atm We need to determine the solubility of the gas (\(C_2\)) at 10 atm.

Write down the Henry's Law formula

According to Henry's Law, the relationship between the solubility and pressure can be written as: \(C_1P_1 = C_2P_2\)

Plug in the given values and solve for \(C_2\)

Substitute the given values in the formula: \(C_1P_1 = C_2P_2\) \((0.52\, \mathrm{g/L})(7\, \mathrm{atm}) = C_2(10\, \mathrm{atm})\) Now solve for \(C_2\): \(C_2 = \frac{(0.52\, \mathrm{g/L})(7\, \mathrm{atm})}{10\, \mathrm{atm}}\)

Calculate the solubility at 10 atm

Now, just perform the calculations: \(C_2 = \frac{(0.52)(7)}{10}\) \(C_2 = \frac{3.64}{10}\) \(C_2 = 0.364\, \mathrm{g/L}\) The solubility of the gas at 10 atm is 0.364 g/L. Thus, 0.364 grams of the gas would be dissolved per 1 L if the pressure was raised to 10 atm.

Key concepts

Gas Solubility

Gas solubility refers to how much of a gas can dissolve in a liquid under certain conditions. It is measured as the amount of gas (in grams) dissolved in a specific volume of liquid (usually 1 liter). The higher the solubility, the more gas can be dissolved in the liquid. For gases, solubility is directly related to the temperature and pressure of the system.

When considering solubility, one must know that different gases have varying solubility levels depending on their nature. For instance, carbon dioxide is quite soluble in water, whereas oxygen is less so. Solubility can change when the temperature or pressure changes, which is critical in many industrial and natural processes.

• Temperature Dependency: Unlike solids, gases become less soluble as the temperature increases because gases tend to expand and escape from liquids.
• Pressure Dependency: According to Henry's Law, the solubility of a gas in a liquid is directly proportional to the pressure of the gas above the liquid.

Pressure Changes

Pressure changes significantly affect the solubility of gases in liquids. According to Henry's Law, an increase in the pressure of a gas above a liquid will increase the solubility of the gas in that liquid. This principle is crucial in understanding the behavior of gases in various environments.

Henry's Law is formulated as:
\[ C = kP \]Where:

• \( C \) is the solubility of the gas
• \( k \) is Henry's Law constant
• \( P \) is the partial pressure of the gas

In practice, this means that if you double the pressure of the gas, the solubility also doubles, assuming the temperature stays constant. Understanding pressure's role helps in processes like carbonating beverages or in diving, where pressure changes affect how gases dissolve in blood.

Solubility Calculation

Solubility calculation often involves using formulas derived from Henry's Law. Calculating solubility helps us predict how much of a gas can dissolve in a liquid at specific conditions. The exercise provided is a classic example of calculating solubility changes due to pressure adjustments.

To perform such calculations, we set up an equation using the initial and final conditions of the gas pressure and its solubility:
\[ C_1P_1 = C_2P_2 \]Given:

• \( C_1 = 0.52 \, \text{g/L} \) at \( P_1 = 7 \, \text{atm} \)
• \( P_2 = 10 \, \text{atm} \)

We solve for \( C_2 \):
\[ C_2 = \frac{(0.52 \, \text{g/L})(7 \, \text{atm})}{10 \, \text{atm}} = 0.364 \, \text{g/L} \]This straightforward calculation shows how increased pressure changes the solubility of a gas, giving a practical application of Henry's Law.

Chemistry Problems

Chemistry problems involving gas solubility often test your understanding of core concepts like Henry's Law, pressure impacts, and solubility relationships. Solving these problems requires a systematic approach, starting with identifying what is known and what needs to be found.

Typical steps include:

• Identify Variables: Note the initial conditions, such as initial pressure and solubility, and the target conditions.
• Use Formulas Appropriately: Apply the correct formulas, being careful to follow units and conversion factors.
• Check for Consistency: After calculations, review results to ensure they make physical sense.

Through practice, solving these chemistry problems becomes easier as one develops an intuitive understanding of how gases behave under different conditions. Mastery of these concepts is not only important academically but also in fields like environmental science, engineering, and beyond.