Question

A mass of gas is under a pressure \(760 \mathrm{~mm} \mathrm{Hg}\) and occupies a volume of \(525 \mathrm{ml}\). If the pressure were doubled, what volume would the gas now occupy? Assume the temperature is constant.

Short answer

The gas would now occupy a volume of \(262.5\,\text{ml}\) when the pressure is doubled and the temperature remains constant.

Step-by-step solution

Identify the given variables

In this problem, we are given the following information: - Initial pressure, \(P_1 = 760\,\text{mm}\, \text{Hg}\) - Initial volume, \(V_1 = 525\,\text{ml}\) - The pressure is doubled, which means the final pressure, \(P_2 = 2P_1 = 2 \times 760\,\text{mm}\, \text{Hg}\) We want to find the final volume, \(V_2\).

Apply Boyle's Law

Since the temperature is constant, we can use Boyle's Law, which is given by the formula: \(P_1V_1 = P_2V_2\). We'll first plug in the known values for \(P_1\), \(V_1\), and \(P_2\) and then solve for \(V_2\).

Solve for the final volume

We now have the following equation: \[760\,\text{mm}\, \text{Hg} \times 525\,\text{ml} = (2 \times 760\,\text{mm}\, \text{Hg}) \times V_2\] Divide both sides of the equation by \(2 \times 760\,\text{mm}\, \text{Hg}\) to solve for \(V_2\): \[V_2 = \frac{760\,\text{mm}\, \text{Hg} \times 525\,\text{ml}}{2 \times 760\,\text{mm}\, \text{Hg}}\]

Calculate the final volume

Now, we can simplify the equation and calculate the value of \(V_2\): \[V_2 = \frac{760 \times 525}{2 \times 760}\,\text{ml} = \frac{525}{2}\,\text{ml} = 262.5\,\text{ml}\]

State the answer

The gas would now occupy a volume of \(262.5\,\text{ml}\) when the pressure is doubled and the temperature remains constant.

Key concepts

Understanding Boyle's Law

Boyle's Law is an essential principle in the field of chemistry and physics, particularly when studying the behavior of gases. It describes the inverse relationship between pressure and volume for a given mass of an ideal gas that is maintained at a constant temperature. Simply put, if the volume increases, the pressure decreases, and vice versa, as long as the temperature and amount of gas do not change.

This relationship is mathematically expressed as: \[P_1V_1 = P_2V_2\]Where:

• \(P_1\) and \(V_1\) are the initial pressure and volume,
• \(P_2\) and \(V_2\) are the final pressure and volume.

In the context of the provided problem, the initial pressure is \(760 \text{mm Hg}\), and the volume is \(525 \text{ml}\). When the pressure is doubled, the volume that the gas would occupy can be calculated using Boyle's Law, leading to the result of \(262.5 \text{ml}\).

Examining the Pressure-Volume Relationship

The pressure-volume relationship explored by Boyle's Law is a practical tool for predicting how gas will behave under different pressures. When we say that pressure and volume are inversely related, we are indicating that as one value goes up, the other must come down in order to keep the product of pressure and volume constant. This relationship only holds when temperature is constant - a condition known as isothermal conditions.

In practical situations, this means that divers must deal with increased pressure and decreased lung volume as they descend underwater, and packaging sealed at high altitudes can expand or even burst when brought to lower altitudes.

Visualizing the Pressure-Volume Graph

When plotted on a graph, pressure and volume have a hyperbolic relationship, with pressure on the y-axis and volume on the x-axis. This hyperbolic curve demonstrates that no matter the initial pressure or volume, as one increases, the other must decrease, illustrating the inverse relationship described by Boyle's Law.

Applying the Ideal Gas Law to Boyle's Principle

While Boyle's Law focuses on the pressure-volume relationship at constant temperature, the Ideal Gas Law offers a broader view, encompassing temperature and amount of gas in addition to pressure and volume. The Ideal Gas Law is expressed as: \[PV = nRT\]Where:

• \(P\) is the pressure,
• \(V\) is the volume,
• \(n\) is the number of moles of the gas,
• \(R\) is the ideal gas constant,
• \(T\) is the temperature in kelvins.

This formula is vital in situations where the temperature and the amount of gas also change. When we simplify this formula by holding the temperature and the number of moles constant, we essentially derive Boyle's Law. This underscores how Boyle's Law is a special case within the broader context of the Ideal Gas Law. It's important for students to understand this connection because it allows for a more comprehensive understanding of gas behavior in different scenarios.