Problem 149 - Chemistry The Science in Context

Question

The volume of a sample of propane gas at 12.5 atm is \(10.6 \mathrm{L}\) What volume does the gas occupy if the pressure is reduced to 1.05 atm and the temperature remains constant?

Step-by-Step Solution

Answer

Answer: Approximately 125 L.

Step 1: Identify the given parameters

We have been given the initial pressure (P1) as 12.5 atm, the initial volume (V1) as 10.6 L, and the final pressure (P2) as 1.05 atm. We need to determine the final volume (V2).

Step 2: Write down the Boyle's Law equation

According to Boyle's Law, when the temperature is constant, the product of the initial pressure and volume is equal to the product of the final pressure and volume. So, the formula to use is: P1 * V1 = P2 * V2

Step 3: Substitute the given values

Next, we will substitute the given values for P1, V1, and P2 into the equation: (12.5 atm) * (10.6 L) = (1.05 atm) * V2

Step 4: Solve for V2

Now, we need to solve for the final volume (V2). First, multiply the numbers on the left side of the equation: 131.25 atm L = (1.05 atm) * V2 Then, divide both sides of the equation by the final pressure (1.05 atm) to isolate V2: V2 = 131.25 atm L / 1.05 atm

Step 5: Calculate the final volume

Finally, perform the division to find the final volume: V2 ≈ 125 L So, the volume of the propane gas when the pressure is reduced to 1.05 atm and the temperature remains constant is approximately 125 liters.

Key Concepts

Gas Volume Calculations

Calculating the volume of a gas involves understanding how pressure changes affect it, especially under constant temperature conditions. In gas calculations, Boyle's Law becomes very useful. Boyle's Law tells us that pressure and volume have an inverse relationship. This means if one increases, the other decreases, all while maintaining a constant temperature. When calculating gas volumes:

• Measure or provide the initial pressure and volume of the gas.
• Determine if the temperature is constant; this validation is essential for Boyle's Law application.
• Apply Boyle's Law, using the formula: \( P_1 \times V_1 = P_2 \times V_2 \) to find the unknown volume.

In our example, we started with a gas volume of 10.6 liters at a pressure of 12.5 atm. By reducing the pressure to 1.05 atm, the new volume was calculated to expand to approximately 125 liters. Such calculations emphasize how accurately we can predict gas behavior in variable pressure scenarios.

Pressure-Volume Relationship

The pressure-volume relationship is a core aspect of understanding gas behavior. Boyle’s Law is particularly helpful in showcasing this relationship when temperature remains constant. The law states that for a given amount of gas, pressure and volume are inversely proportional. This means if the pressure on a gas increases, the volume decreases, and vice versa. To further comprehend this concept:

• Recognize that "inversely proportional" means that their product remains constant: \( P \times V = ext{constant} \)
• Understand that this principle only holds true for conditions where the temperature and the amount of gas do not change.
• In real-world applications, this concept helps in predicting how gases will occupy space under different pressures.

In our problem, reducing the pressure significantly from 12.5 atm to 1.05 atm while maintaining the temperature led to an increase in the volume of propane gas. This is precisely what Boyle's Law predicts when applied correctly.

Propane Gas Behavior

When dealing with propane gas, understanding its behavior under different pressures is crucial, especially in constant temperatures. Propane behaves according to the ideal gas laws under most conditions and fits well within the predictions of Boyle's Law when temperature doesn't change. Key points on propane gas behavior include:

• Propane is a common hydrocarbon used in heating and cooking.
• In gas calculations, considering propane as an ideal gas is generally accurate, provided the conditions are not extreme (high pressure or very low temperature).
• Variations in pressure will impact propane's volume, as depicted through Boyle's Law.

In our specific calculation, the propane gas's volume increased to approximately 125 liters when the pressure dropped to 1.05 atm from 12.5 atm. Such calculations illustrate the predictability and reliability of using Boyle's Law for propane in standard conditions.