Problem 25 - Chemistry The Central Science In Si Units

Question

You have a gas at \(25^{\circ} \mathrm{C}\) confined to a cylinder with a movable piston. Which of the following actions would double the gas pressure? (a) Lifting up on the piston to double the volume while keeping the temperature constant; (b) Heating the gas so that its temperature rises from \(25^{\circ} \mathrm{C}\) to \(50^{\circ} \mathrm{C}\), while keeping the volume constant; (c) Pushing down on the piston to halve the volume while keeping the temperature constant.

Step-by-Step Solution

Answer

Option (c) will double the gas pressure.

Step 1: Understanding the Gas Laws

We can analyze the problem using Boyle's Law and Gay-Lussac’s Law. Boyle’s Law states that \( P_1V_1 = P_2V_2 \) for a constant temperature, meaning pressure and volume are inversely related. Gay-Lussac's Law says \( \frac{P_1}{T_1} = \frac{P_2}{T_2} \) when volume is constant, meaning pressure is directly proportional to temperature.

Step 2: Analyze Action (a) – Doubling the Volume

If the volume of the gas is doubled by lifting up on the piston at constant temperature, Boyle's Law implies that the pressure will be halved, since \( P_2 = \frac{P_1 \times V_1}{2V_1} = \frac{P_1}{2} \). Therefore, option (a) would not double the pressure.

Step 3: Analyze Action (b) – Increasing Temperature

By increasing the temperature from \(25^{\circ} \mathrm{C}\) to \(50^{\circ} \mathrm{C}\), the Kelvin temperatures are \(298 \: K\) to \(323 \: K\) respectively. Using Gay-Lussac’s Law: \( \frac{P_2}{P_1} = \frac{323}{298} \approx 1.084 \). This results in a slight increase in pressure, not a doubling.

Step 4: Analyze Action (c) – Halving the Volume

If the volume is halved by pushing down on the piston, keeping temperature constant, Boyle's Law implies that the pressure will double: \( P_2 = \frac{P_1 \times V_1}{\frac{V_1}{2}} = 2P_1 \). Therefore, this action will double the gas pressure.

Key Concepts

Boyle's Law

In a world where gases love to dance to the tune of temperature and pressure, Boyle’s Law stands as a key principle. This law helps us understand the relationship between the pressure and volume of a gas at constant temperature. Simply put, Boyle’s Law states that the pressure of a gas is inversely proportional to its volume when the temperature is held constant.
This can be expressed with the equation: \( P_1V_1 = P_2V_2 \). Here, \( P \) represents pressure, and \( V \) represents volume.

• If you increase the volume of the container, the pressure of the gas decreases because the gas molecules have more space to move around.
• Conversely, if you decrease the volume, the pressure increases because the molecules get squeezed together, colliding more frequently with the walls.

This principle is beautifully displayed in action (c) from the exercise. By halving the volume, the gas molecules become more compact, causing the pressure to double. No heating required, just some good, old-fashioned compression!

Gay-Lussac's Law

Let's give a warm welcome to Gay-Lussac's Law, a fascinating gas law that tells us about the interplay between pressure and temperature. This law shows that when the volume is held constant, the pressure of a gas is directly proportional to its temperature in Kelvin. The formula is expressed as: \( \frac{P_1}{T_1} = \frac{P_2}{T_2} \), where \( P \) stands for pressure and \( T \) for temperature.

• As the temperature of a gas increases, its pressure also rises because the heat makes the gas molecules move faster, hitting the walls of the container more often and with greater force.
• Conversely, if the temperature decreases, the pressure drops as well, since the molecules slow down.

In the exercise, when the temperature goes up from \(25^{\circ}C\) to \(50^{\circ}C\), even though we see a rise in pressure, it isn't enough to double it. The subtle increase in pressure is calculated through the law's formula, highlighting how temperature needs significant changes to greatly influence pressure under constant volume.

Pressure and Volume Relationship

The relationship between pressure and volume is a captivating dance in the realm of gases, governed primarily by Boyle's Law. When temperature is constant, any change in a gas's volume will inversely affect its pressure. Think of it this way:

• A shrinking volume makes the pressure push upwards as the same number of particles now have less space to roam.
• Expanding the volume gives particles more room, so pressure reduces and they bounce off the walls less frequently.

Boyle's Law embodies this interaction perfectly with its equation \( P_1V_1 = P_2V_2 \). It's the fundamental way gases communicate through adjustments in pressure and volume. Recognizing and predicting these changes are crucial in many scientific and real-world applications, from understanding how syringes work to anticipating weather patterns. In essence, grasping the inverse relationship between pressure and volume sets the stage for mastering gas behaviors.