Question

Suppose you had a 4.10 - \(\mathrm{L}\) sample of neon gas at \(21^{\circ} \mathrm{C}\) and a pressure of \(0.959 \mathrm{~atm}\). What would be the volume of this gas if the pressure were increased to 1.210 atm while the temperature remained constant?

Short answer

The final volume is 3.25 L.

Step-by-step solution

Understand the Given Data

You are provided with an initial volume of the gas, \( V_1 = 4.10 \, L \), an initial pressure, \( P_1 = 0.959 \, atm \), and a final pressure, \( P_2 = 1.210 \, atm \). The temperature remains constant, which allows us to use Boyle's Law for the solution.

Boyle's Law Formula

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 \( P_1 V_1 = P_2 V_2 \). We use this formula to find the final volume, \( V_2 \).

Rearrange the Equation

Rearrange the equation \( P_1 V_1 = P_2 V_2 \) to solve for \( V_2 \). This gives us \( V_2 = \frac{P_1 V_1}{P_2} \).

Substitute Known Values

Substitute the known values into the rearranged equation: \( V_2 = \frac{0.959 \, atm \times 4.10 \, L}{1.210 \, atm} \).

Calculate the Final Volume

Perform the calculation to find \( V_2 \). Therefore, \( V_2 = \frac{0.959 \times 4.10}{1.210} = 3.25 \, L \). This is the volume of the gas after the change in pressure.

Key concepts

Understanding Gas Laws

Gas laws describe the behavior of gases under various conditions. They are crucial for understanding how pressure, volume, and temperature relate to each other. There are several well-known gas laws, such as Boyle's Law, Charles's Law, and Avogadro's Law, but in this context, we focus on Boyle's Law. It's essential because it explains the relationship between pressure and volume when temperature is constant. Gas laws are essential in many fields, including chemistry and physics, because they help predict how a gas will behave in different scenarios. Knowing these principles allows scientists and engineers to design everything from engines to weather balloons.

Pressure and Volume Relationship

The relationship between pressure and volume is at the heart of Boyle's Law. When you increase the pressure on a gas, its volume decreases, and vice versa, provided the temperature remains constant. This relationship is inversely proportional, which means as one goes up, the other goes down. In mathematical terms, this relationship is expressed as: \[ P_1 V_1 = P_2 V_2 \] Here, \( P_1 \) and \( V_1 \) are the initial pressure and volume, whereas \( P_2 \) and \( V_2 \) are the final pressure and volume after the change. This formula helps predict the new volume or pressure when one changes.To put it simply:

• Increase in pressure = Decrease in volume
• Decrease in pressure = Increase in volume

A practical example would be compressing air in a syringe. If you press the plunger, you're increasing the pressure, causing the air inside to occupy a smaller volume.

Role of Constant Temperature

Constant temperature is a crucial condition in applying Boyle's Law. Temperature can have a significant effect on gas behavior. If the temperature changes, the relationship between pressure and volume won't follow Boyle's Law. At a constant temperature, the average kinetic energy of gas molecules doesn't change. This steadiness allows the inverse relationship of pressure and volume to remain predictable. In the exercise example, we increase the pressure of the neon gas from 0.959 atm to 1.210 atm. By keeping the temperature constant, we can directly apply Boyle's Law to determine the resulting volume. Remember: Boyle's Law only applies accurately if temperature remains unchanged. When temperature varies, other gas laws, like Charles's Law, might come into play.