Question

Calculate the final volume when $125\mathrm{mL}$ of argon gas undergoes a pressure change from $705\mathrm{mm}\mathrm{Hg}$ to $385\mathrm{mm}\mathrm{Hg}$. Assume that the temperature remains constant.

Short answer

The final volume is approximately 228.57 mL.

Step-by-step solution

Identify the Known Variables

We are given the initial volume $V_1=125\mathrm{mL}$ and the initial pressure $P_1=705\mathrm{mm}\mathrm{Hg}$. The final pressure $P_2$ after the change is $385\mathrm{mm}\mathrm{Hg}$.

Determine the Applicable Formula

Since the temperature remains constant, Boyle's Law applies here: $P_1{V}_1=P_2{V}_2$, where $V_2$ is the final volume we want to determine.

Substitute the Known Values

Using the equation from Boyle's Law, substitute the known values: $705\mathrm{mm}\mathrm{Hg}\times 125\mathrm{mL}=385\mathrm{mm}\mathrm{Hg}\times V_2$.

Solve for the Unknown Variable

Rearrange the equation to solve for $V_2$:$$V_2=\frac{705\mathrm{mm}\mathrm{Hg}\times 125\mathrm{mL}}{385\mathrm{mm}\mathrm{Hg}}$$

Calculate the Final Volume

Perform the calculation:$$V_2=\frac{705\times 125}{385}\approx 228.57\mathrm{mL}$$

Key concepts

Pressure-Volume Relationship

Boyle's Law is a fundamental concept that helps establish the relationship between pressure and volume for a gas, assuming temperature remains constant. This principle describes how pressure and volume are inversely related:

• As the volume of a gas increases, the pressure decreases.
• Conversely, as the volume decreases, the pressure increases.

This relationship is mathematically expressed as $P_1{V}_1=P_2{V}_2$, where:

• $P_1$ and $P_2$ are the initial and final pressures, respectively.
• $V_1$ and $V_2$ are the initial and final volumes, respectively.

In the exercise provided, we applied this relationship to find the final volume by knowing the initial conditions and the change in pressure.

Ideal Gas Law

While Boyle's Law focuses on the pressure-volume relationship at constant temperature, the ideal gas law expands this to include temperature, providing a more comprehensive view of gas behavior. The ideal gas law is given by the formula $PV=nRT$, where:

• $P$ is the pressure of the gas.
• $V$ is the volume.
• $n$ is the number of moles of gas.
• $R$ is the ideal gas constant.
• $T$ is the temperature in Kelvin.

In this context, Boyle's Law is essentially a special case of the ideal gas law, used when the number of moles and temperature are constant. Therefore, it's essential to understand both concepts when studying gas behavior as they provide frameworks for predicting how a gas might react under various conditions.

Gas Behavior

Understanding gas behavior is crucial in fields ranging from chemistry to physics and engineering. At a molecular level, gases are composed of particles in constant, random motion. This motion causes gases to expand and fill their Containers completely. Here's what influences and characterizes gas behavior:

• Pressure: Results from collisions of gas particles with the walls of their container.
• Volume: The space occupied by the gas. Gases tend to expand to fill their volume.
• Temperature: Reflects the kinetic energy of gas particles. Higher temperatures increase this energy, thereby increasing pressure if volume is constant.

By understanding these basic principles and how they integrate, we can predict and manipulate how gases will respond to changes in their environment, such as those showcased in the given exercise.