Question

A gas has an initial volume of \(638 \mathrm{~mL}\) and an initial pressure of 779 torr. What is its final volume in liters if its pressure is changed to 0.335 atm? Assume temperature and amount are held constant.

Short answer

The final volume is approximately 1.953 L.

Step-by-step solution

Identify Given Information

First, identify the given information in the problem. We have the initial volume \(V_1 = 638 \, \text{mL}\), initial pressure \(P_1 = 779 \, \text{torr}\), and final pressure \(P_2 = 0.335 \, \text{atm}\). We need to determine the final volume \(V_2\) and convert it to liters.

Convert Pressure to the Same Units

To use Boyle's Law, both pressures need to be in the same units. Convert 0.335 atm to torr using the conversion factor \(1 \, \text{atm} = 760 \, \text{torr}\). Calculate \(P_2 = 0.335 \, \text{atm} \times 760 \, \text{torr/atm} = 254.6 \, \text{torr}\).

Apply Boyle's Law

Boyle's Law states that the product of the initial pressure and volume is equal to the product of the final pressure and volume, assuming constant temperature and amount of gas: \(P_1V_1 = P_2V_2\). Substitute the known values: \(779 \, \text{torr} \times 638 \, \text{mL} = 254.6 \, \text{torr} \times V_2\).

Solve for Final Volume in Milliliters

Rearrange the equation to solve for the final volume \(V_2\): \(V_2 = \frac{779 \, \text{torr} \times 638 \, \text{mL}}{254.6 \, \text{torr}}\). Calculate \(V_2 \approx 1952.7 \, \text{mL}\).

Convert Final Volume to Liters

Convert the final volume from milliliters to liters since there are 1000 mL in one liter: \(V_2 \approx \frac{1952.7 \, \text{mL}}{1000} = 1.953 \, \text{L}\).

Conclusion

The final volume of the gas is approximately \(1.953 \, \text{L}\), given the changes in pressure with constant temperature and quantity.

Key concepts

Gas Laws

Boyle's Law is an essential concept in chemistry and physics that helps us understand the behavior of gases. Boyle's Law states that under constant temperature, the pressure and volume of a gas are inversely proportional. This means that if you increase the pressure on a gas, its volume decreases, and vice versa, as long as the temperature remains unchanged. This relationship can be expressed mathematically as:\[ P_1V_1 = P_2V_2 \]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

Understanding Boyle's Law is crucial when dealing with real-world gas scenarios, such as those in syringes, balloons, and even your car tires!

Pressure Conversion

Understanding how to convert pressure units is crucial when applying Boyle's Law, because pressures need to be in the same units to perform accurate calculations. In this exercise, we encountered pressures in both torr and atm units. Here's how we convert these units:We use the conversion factor:\[ 1 \, \text{atm} = 760 \, \text{torr} \]For the given exercise, the final pressure is converted from atm to torr:\[ 0.335 \, \text{atm} \times 760 \, \text{torr/atm} = 254.6 \, \text{torr} \]This step ensures that all pressures are uniformly expressed, making it easier to plug into the Boyle's Law equation and find the final volume.

Volume Calculations

Calculating the change in volume of a gas requires careful consideration of units and application of Boyle's Law. Initially, we have the volume:

• Initial Volume \(V_1 = 638 \, \text{mL}\)
• And after using conversions, solving the final pressure and volume equation gets us \(V_2\).

By solving the equation:\[ V_2 = \frac{P_1V_1}{P_2} = \frac{779 \, \text{torr} \times 638 \, \text{mL}}{254.6 \, \text{torr}} \approx 1952.7 \, \text{mL} \]Finally, for ease of use and clarity, converting the obtained volume from milliliters to liters gives us:\[ 1952.7 \, \text{mL} \div 1000 = 1.953 \, \text{L} \]This conversion helps us present the volume in the standard scientific notation.

Constant Temperature

A key assumption in the application of Boyle's Law is that the temperature remains constant. This is crucial for the law to hold true. In real-world applications, if the temperature were to change, we would need to account for additional gas laws, such as Charles's Law or the Combined Gas Law. However, when temperature is constant: - No extra energy is added to increase the kinetic energy of the gas molecules. - Their speed, and thus the internal pressure specifically related to temperature, remains the same. This assumption simplifies calculations significantly and allows us to focus solely on the relationship between pressure and volume.