Problem 19 - Chemistry The Molecular Science

Question

Some butane, the fuel used in backyard grills, is placed in a sealed 3.50 -L container at \(25^{\circ} \mathrm{C} ;\) its pressure is \(735 \mathrm{mmHg}\). You transfer the gas to a sealed 15.0 - \(\mathrm{L}\) container, also at \(25^{\circ} \mathrm{C}\). Calculate the pressure of the gas in the larger container.

Step-by-Step Solution

Answer

The pressure in the larger container is 171.5 mmHg.

Step 1: Understand the Problem

We have some butane gas that is transferred from one container to another without any change in temperature. We need to find out what happens to the pressure when the volume changes from 3.50 L to 15.0 L, while the temperature remains constant at 25°C.

Step 2: Use Boyle's Law

Since the temperature is constant, we can apply Boyle's Law. Boyle's Law states that the product of the pressure and volume of a gas is constant when the temperature and the amount of gas do not change. In mathematical terms, \( P_1 \times V_1 = P_2 \times V_2 \), where \(P_1\) and \(V_1\) are the initial pressure and volume, and \(P_2\) and \(V_2\) are the final pressure and volume.

Step 3: Plug in the Known Values

We are given that the initial pressure \( P_1 = 735 \text{ mmHg} \) and the initial volume \( V_1 = 3.50 \text{ L} \). The final volume \( V_2 = 15.0 \text{ L} \) and we need to find \( P_2 \). Use the formula: \[ P_2 = \frac{P_1 \times V_1}{V_2} \]

Step 4: Calculate the Final Pressure

Substitute the known values into the equation: \[ P_2 = \frac{735 \text{ mmHg} \times 3.50 \text{ L}}{15.0 \text{ L}} \] Calculate the result: \[ P_2 = \frac{2572.5}{15.0} \] \[ P_2 = 171.5 \text{ mmHg} \]

Key Concepts

Gas Pressure Calculations

Calculating gas pressure involves understanding the relationship between pressure, volume, and temperature in a gas system. In Boyle's Law problems, we focus on the pressure and volume, assuming temperature remains constant. To find the pressure of a gas in a new container:

• Use the initial conditions of the gas, such as its starting pressure and volume.
• Apply Boyle's Law, which states that the product of pressure and volume is constant if temperature is stable.
• Substitute the known values into the equation to find the new pressure after a change in volume.

This step-by-step substitution helps students understand how gas pressure changes in different situations. In our example, transferring butane between containers highlights how volume expansion reduces pressure.

Gas Law Problems

Solving gas law problems requires identifying which gas law applies to the given situation. For Boyle's Law problems, this means recognizing cases where temperature doesn't change. The general steps for these types of problems are:

• Identify if temperature is constant, as this directs which law applies.
• Choose Boyle's Law because it interrelates pressure and volume at constant temperature.
• Use the formula ₁V₁ = P₂V₂ , rearrange it to solve for the unknown.

The structured use of formulas simplifies the process and improves accuracy. Understanding how to map problem conditions to the correct law is essential for effective problem-solving.

Volume and Pressure Relationship

The relationship between volume and pressure in gases is best described by Boyle's Law. It states that pressure and volume have an inverse relationship when temperature is constant. As the volume of a gas increases:

• Pressure decreases because the same number of gas molecules has more space to move.
• The force exerted on the container walls drops due to increased space.

Conversely, as volume decreases, pressure increases. This fundamental concept helps explain phenomena such as how balloons work or why air pressures vary with altitude. By keeping the temperature constant, Boyle's Law simplifies the study of how volume affects pressure.