Question

A sample of \(\mathrm{CO}_{2}\) gas has a pressure of \(56.5 \mathrm{mm}\) Hg in a 125-mL. flask. The sample is transferred to a new flask, where it has a pressure of \(62.3 \mathrm{mm}\) Hg at the same temperature. What is the volume of the new flask?

Short answer

The volume of the new flask is approximately 113.3 mL.

Step-by-step solution

Identify Known Values

First, identify the known values from the problem.Initial pressure \( P_1 = 56.5 \, \text{mm Hg} \)Initial volume \( V_1 = 125 \, \text{mL} \)Final pressure \( P_2 = 62.3 \, \text{mm Hg} \)The temperature is constant.

Apply 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. This can be expressed as:\[ P_1 \times V_1 = P_2 \times V_2 \]Where \( P_1 \) and \( V_1 \) are the initial pressure and volume, and \( P_2 \) and \( V_2 \) are the final pressure and volume.

Solve for the New Volume

Rearrange the equation from Boyle's Law to solve for the new volume \( V_2 \):\[ V_2 = \frac{P_1 \times V_1}{P_2} \]Substitute the known values into the equation:\[ V_2 = \frac{56.5 \, \text{mm Hg} \times 125 \, \text{mL}}{62.3 \, \text{mm Hg}} \]

Calculate the New Volume

Perform the calculation:\[ V_2 = \frac{7062.5}{62.3} \approx 113.3 \, \text{mL} \]

Conclusion

The volume of the new flask is approximately \( 113.3 \, \text{mL} \). This is consistent with Boyle's Law, as the volume decreases when pressure increases at constant temperature.

Key concepts

Gas Pressure

Gas pressure is the force exerted by gas molecules as they collide with the walls of their container. Imagine tiny invisible balls bouncing around inside a box; each time they hit the walls of the box, they apply a tiny force. When you add up all these tiny forces from countless collisions, you get the gas pressure. The more often and the more forcefully these molecules collide, the higher the pressure.
Gas pressure is measured using different units, with common ones being millimeters of mercury (mm Hg), Pascals (Pa), or atmospheres (atm). In scientific problems, it’s crucial to know the units being used, as they help to determine how many collisions are happening inside the container at a given time.
In our example with the \(\mathrm{CO}_2\) sample, pressure was given in mm Hg. Keeping track of pressure changes is essential to calculating other properties, like volume, under the ideal gas law.

Volume Calculations

Calculating the volume of a gas involves understanding how it expands or contracts when conditions such as temperature or pressure change. Boyle's Law is particularly useful for these calculations, as it demonstrates the relationship between pressure and volume in an ideal gas. According to Boyle's Law, when the temperature is kept constant, the pressure of a gas is inversely proportional to its volume.
This means if pressure increases, volume decreases and vice versa, maintaining a constant product of pressure and volume. The formula for Boyle's Law is: \[ P_1 \times V_1 = P_2 \times V_2 \] In our exercise, the initial and final pressures (\(P_1\) and \(P_2\)) and the initial volume (\(V_1\)) are provided. Using the rearranged formula, the final volume \(V_2\) is determined by: \[ V_2 = \frac{P_1 \times V_1}{P_2} \] By substituting the given values, you can solve for the new volume, ensuring you keep the units consistent throughout your calculations for accurate results. This approach is foundational to understanding gas behavior in chemistry.

Ideal Gas Behavior

Ideal gas behavior is a theoretical model that assumes gas particles have no volume and experience no intermolecular forces, behaving similarly in simple conditions (low pressure and high temperature). While no real gas perfectly fits this model, it provides useful approximations for understanding and calculating gas properties under many conditions.
Boyle's Law fits into this model, addressing how gases respond to pressure changes. Real gases tend to deviate from ideal behavior, especially at high pressures or low temperatures, where intermolecular forces and the volume of gas particles become significant.
For most calculations involving gases, assuming ideal behavior simplifies the problem enough to yield practical insights without requiring complex adjustments. The given exercise assumes a constant temperature and applies Boyle's Law to estimate gas volume changes, embodying a typical scenario where ideal gas principles sufficiently describe the situation.
Understanding when and how to apply these assumptions helps in interpreting laboratory results and predicting the behavior of gaseous substances in various scientific applications.