Question

A sample of nitrogen gas has a pressure of \(67.5 \mathrm{mm} \mathrm{Hg}\) in a \(500 .\) -mL flask. What is the pressure of this gas sample when it is transferred to a \(125-\mathrm{mL}\) flask at the same temperature?

Short answer

The final pressure is 270 mm Hg.

Step-by-step solution

Understand Boyle's Law

Boyle's Law states that for a given mass of gas at constant temperature, the pressure of the gas is inversely proportional to its volume. Mathematically, this is 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.

Identify known values

From the problem, we know the initial pressure \( P_1 = 67.5 \, \text{mm Hg} \) and the initial volume \( V_1 = 500 \, \text{mL} \). The final volume \( V_2 \) is \( 125 \, \text{mL} \). We need to find the final pressure \( P_2 \).

Rearrange Boyle's Law Equation

To find the final pressure \( P_2 \), rearrange the equation \( P_1 V_1 = P_2 V_2 \) to solve for \( P_2 \). This gives us \( P_2 = \frac{P_1 V_1}{V_2} \).

Substitute known values

Substitute the known values into the rearranged equation: \( P_2 = \frac{67.5 \, \text{mm Hg} \times 500 \, \text{mL}}{125 \, \text{mL}} \).

Calculate the final pressure

Perform the calculation: \( P_2 = \frac{67.5 \times 500}{125} \). This gives \( P_2 = 270 \, \text{mm Hg} \).

Key concepts

Pressure-Volume Relationship

The pressure-volume relationship is a fundamental concept in the study of gases. It refers to how the pressure of a gas changes with its volume, while keeping the temperature constant. This principle is at the heart of Boyle's Law. Simply put, Boyle's Law tells us that if we decrease the volume of a gas, its pressure will increase, provided the temperature remains unchanged.

Here's why: As we compress the gas into a smaller space, the gas molecules have less room to move around. This forces them to collide more frequently with the walls of their container, thereby increasing the pressure.

To put it mathematically, the equation for Boyle's Law is:

• \( P_1 \times V_1 = P_2 \times V_2 \)

Where:

• \( P_1 \) is the initial pressure
• \( V_1 \) is the initial volume
• \( P_2 \) is the final pressure
• \( V_2 \) is the final volume

By understanding this relationship, we can solve problems involving changes in gas pressure and volume with ease.

Gas Laws

Gas laws, such as Boyle's Law, encompass a set of rules that describe how gases behave under different conditions. They are crucial for scientists and engineers to predict and manipulate the behavior of gases in a variety of settings—from chemical reactions to mechanical systems.

Boyle's Law is just one of several important gas laws. Others include Charles's Law, which relates volume and temperature, and Avogadro's Law, which relates volume and the amount of gas present.

These laws are important because they help us understand how gases will react to changes in pressure, volume, temperature, and quantity. This can be very useful, for example, when calculating how much air is needed to fill a balloon or how a gas will behave in a closed container.

In chemistry, understanding these relationships allows us to predict the results of experiments and ensure processes are carried out safely and efficiently.

Chemistry Problem-Solving

Chemistry problem-solving often involves applying fundamental principles, like Boyle’s Law, to determine unknown values in a system. By identifying the known quantities and using the right formula, such as \( P_1 V_1 = P_2 V_2 \), we can find unknown variables, like pressure or volume.

Here's a simple strategy:

• Start by identifying what is known and unknown in the problem.
• Select the suitable formula based on the known and unknown variables.
• Rearrange the equation to solve for the unknown variable.
• Substitute the known values into the equation.
• Perform the calculation to find the unknown value.

In the nitrogen gas problem, we knew the initial pressure and volume as well as the final volume. By using Boyle's Law, we were able to rearrange the equation to solve for the final pressure, leading us to the answer. This approach provides a logical and structured way to tackle chemistry problems, making complex systems simpler to understand.