Question

A sample of gas at \(30 .{ }^{\circ} \mathrm{C}\) has a pressure of \(2.0 \mathrm{~atm}\) in a sealed 1.0 - \(\mathrm{L}\) container. Calculate the pressure it will exert in a 4.0 -L container. The temperature does not change.

Short answer

The final pressure is 0.5 atm.

Step-by-step solution

Understand the Problem

We are given a gas contained in a 1.0 L container that initially exerts a pressure of 2.0 atm. We need to find the pressure when this gas is transferred to a 4.0 L container, with the temperature remaining constant.

Apply Boyle's Law

Boyle's Law states that for a given mass of gas at constant temperature, the pressure is inversely proportional to the volume. This can be expressed as \( P_1 V_1 = P_2 V_2 \), where \( P_1 \) and \( V_1 \) are the initial pressure and volume, and \( P_2 \) and \( V_2 \) are the final pressure and volume.

Substitute Given Values

We start with a pressure \( P_1 = 2.0 \) atm and volume \( V_1 = 1.0 \) L. The final volume \( V_2 = 4.0 \) L. Substitute these values into the equation: \( 2.0 \times 1.0 = P_2 \times 4.0 \).

Solve for Final Pressure \( P_2 \)

Rearrange the equation to solve for \( P_2 \): \( P_2 = \frac{2.0 \times 1.0}{4.0} \). Calculate the value to find \( P_2 = 0.5 \) atm.

Key concepts

Gas Pressure

Gas pressure is an essential concept in understanding how gases behave. It refers to the force that the gas exerts on the walls of its container. This force is caused by gas particles colliding with those walls.
Usually, pressure is measured in units like atm, psi, or Pascal, with atmospheres (atm) being common.
Gas pressure is influenced by a few key factors:

• Temperature: Increasing temperature typically increases pressure because particles move faster and collide more often.
• Volume: Changing the volume of a container can directly affect the pressure, which Boyle's Law helps to explain.
• Amount of Gas: Adding more gas increases the number of particles and thus the pressure.

Understanding gas pressure is crucial for various scientific disciplines, including chemistry and physics, because it helps us predict how gases will react under different conditions.

Volume and Pressure Relationship

In the realm of gases, there's an important relationship between volume and pressure illustrated by Boyle's Law. This law states that for a fixed amount of gas at a constant temperature, the pressure of a gas is inversely proportional to its volume. Simply put, if you decrease the volume, the gas pressure increases, and vice versa.
Mathematically, Boyle's Law can be expressed with the formula: \[ P_1 V_1 = P_2 V_2 \]Here, \( P_1 \) and \( V_1 \) are the original pressure and volume, whereas \( P_2 \) and \( V_2 \) are the final values. This equation helps us calculate how pressure changes when the volume of a gas is changed, assuming temperature stays the same.
To apply this to a real situation:

• If you have a balloon and you squeeze it to make it smaller, the pressure inside will increase.
• If you let the balloon expand, the pressure decreases as the volume increases.

This fundamental relationship is widely used in many practical applications, such as breathing mechanisms, aerosol sprays, and more.

Ideal Gas Law

The Ideal Gas Law offers a more comprehensive framework for the behaviors of gases by combining several fundamental gas laws into one equation. While Boyle's Law focuses solely on pressure and volume at constant temperature, the Ideal Gas Law includes temperature and the amount of gas as variables.
The Ideal Gas Law is represented by: \[ PV = nRT \]In this equation:

• \( P \) stands for pressure
• \( V \) stands for volume
• \( n \) represents the moles of gas
• \( R \) is the universal gas constant
• \( T \) represents temperature in Kelvin

This equation helps predict how gases behave under different conditions by accounting for changes in volume, pressure, temperature, or amount of gas.
For example, if you know any three of these variables, you can determine the fourth. The Ideal Gas Law is instrumental in scenarios like calculating fuel efficiency and even understanding atmospheric science.