Question

A sample of a gas has a pressure of \(100 . \mathrm{mmHg}\) in a sealed \(125-\mathrm{mL}\). flask. This gas sample is transferred to another flask with a volume of \(200 . \mathrm{mL}\). Calculate the new pressure. Assume that the temperature remains constant.

Short answer

The new pressure is 62.5 mmHg.

Step-by-step solution

Identify Given Values

Note the initial and final volumes and the initial pressure:- Initial Pressure \( P_1 = 100 \, \text{mmHg} \)- Initial Volume \( V_1 = 125 \, \text{mL} \)- Final Volume \( V_2 = 200 \, \text{mL} \)We need to find the final pressure \( P_2 \).

State Boyle's Law

Boyle's Law explains the relationship between the pressure and volume of a gas at constant temperature. It states that the product of the initial pressure and volume is equal to the product of the final pressure and volume:\[ P_1 \times V_1 = P_2 \times V_2 \]

Rearrange Boyle's Law Formula

To find the final pressure \( P_2 \), rearrange the formula:\[ P_2 = \frac{P_1 \times V_1}{V_2} \] This formula allows us to calculate the new pressure given the other known values.

Insert Known Values

Substitute the given values into the rearranged Boyle's Law formula:\[ P_2 = \frac{100 \, \text{mmHg} \times 125 \, \text{mL}}{200 \, \text{mL}} \]

Calculate the Final Pressure

Perform the calculation to find the final pressure:\[ P_2 = \frac{12500 \, \text{mmHg} \, \text{mL}}{200 \, \text{mL}} = 62.5 \, \text{mmHg} \] So, the new pressure in the 200 mL flask is 62.5 mmHg.

Key concepts

Boyle's Law

Boyle's Law is a fundamental principle in chemistry and physics that describes how the pressure of a gas tends to decrease as the volume increases, provided the temperature remains constant. This law, discovered by Robert Boyle in the 17th century, is crucial in understanding gas behavior.According to Boyle's Law, the product of the initial pressure and the volume of a gas remains constant during any transformation. This can be mathematically described by the equation:\[ P_1 \times V_1 = P_2 \times V_2 \]where:- \( P_1 \) is the initial pressure of the gas- \( V_1 \) is the initial volume- \( P_2 \) is the final pressure after a change in volume- \( V_2 \) is the final volumeBoyle's Law helps predict how a change in volume will affect the pressure of a gas in a sealed container, assuming no other variables such as temperature change.

Pressure and Volume Relationship

The relationship between pressure and volume in gases, as noted by Boyle's Law, is inversely proportional. This means that as the volume of a gas increases, the pressure decreases, and vice versa. When the gas in a sealed container is compressed, the molecules are forced into a smaller space, increasing the pressure. Conversely, when the container's volume increases, the molecules have more room to move, resulting in reduced pressure. Understanding this pressure-volume relationship is key to solving problems involving gases. For example, in the given exercise, a gas stored initially in a 125 mL flask at 100 mmHg is transferred to a 200 mL flask. Since the volume increases, the pressure decreases, as shown by the calculation of the new pressure (62.5 mmHg). This relationship is pivotal in many real-world applications, such as designing medical syringes and engines where control over gas pressure is essential.

Sealed Gas Samples

Sealed gas samples refer to gases that are contained within a closed system, preventing any exchange of matter with their surroundings. In such a system, factors like pressure, volume, and temperature become critical in determining the behavior of the gas. In the context of Boyle's Law, working with sealed gas samples implies that we can assume the number of gas molecules remains constant. Thus, any changes in pressure or volume must result from internal adjustments within the contained system, rather than external influences. Sealed gas environments are often used in scientific experiments to maintain controlled conditions and ensure that the results are consistent and reliable. By knowing the initial conditions and applying Boyle's Law, we can easily predict how a gas will behave when transferred from one sealed container to another, as demonstrated in the exercise by calculating the new pressure as the gas was moved to a larger volume.