Question

A sample of air at 15.0 psi compresses from \(555 \mathrm{~mL}\) to \(275 \mathrm{~mL}\). If the temperature remains constant, what is the final pressure in psi?

Short answer

The final pressure is approximately 30.27 psi.

Step-by-step solution

Identify Given Information

We are given the initial pressure \( P_1 = 15.0 \, \text{psi} \), the initial volume \( V_1 = 555 \, \text{mL} \), and the final volume \( V_2 = 275 \, \text{mL} \). We need to find the final pressure \( P_2 \).

Apply Boyle's Law

Since the temperature remains constant, we can apply Boyle's Law, which states that \( P_1 V_1 = P_2 V_2 \). This equation implies that the product of the pressure and volume of a gas sample is constant when the temperature is constant.

Solve for Final Pressure

Rearrange the Boyle's Law equation to solve for \( P_2 \): \( P_2 = \frac{P_1 V_1}{V_2} \). Substitute the values: \( P_2 = \frac{15.0 \, \text{psi} \times 555 \, \text{mL}}{275 \, \text{mL}} \).

Calculate the Final Pressure

Perform the calculation: \( P_2 = \frac{15.0 \, \text{psi} \times 555}{275} = 30.27 \, \text{psi} \). Therefore, the final pressure \( P_2 \) is approximately 30.27 psi.

Key concepts

Gas Laws

Gas laws are fundamental concepts in chemistry that describe how gases behave under different conditions of pressure, volume, and temperature. One of the key gas laws is Boyle's Law, which specifically explores the relationship between pressure and volume at a constant temperature. These laws are essential for understanding and predicting the behavior of gases in various scientific and real-world applications. Learning about these laws helps to form the foundation for advanced studies in physical chemistry and thermodynamics.

The gas laws are based on the ideal gas behavior where the parameters of a gas, such as pressure, volume, and temperature, are related. Understanding these relationships is crucial for solving problems involving gases in confined spaces. Boyle's Law, Charles's Law, and Avogadro's Law together build the ideal gas equation, but individually, each gives insight into how a single variable impacts the gas’s condition.

• Boyle's Law: Describes how pressure and volume are inversely proportional when temperature is held constant.
• Charles's Law: Looks at the direct proportionality between volume and temperature with constant pressure.
• Avogadro's Law: Examines how volume is directly proportional to the number of gas molecules when pressure and temperature are constant.

Pressure and Volume Relationship

The relationship between pressure and volume is a direct application of Boyle's Law, which states that for a given mass of gas at a constant temperature, the product of the initial pressure and volume is equal to the product of the final pressure and volume. In simple words, if you compress a gas by decreasing its volume, the pressure increases, and vice versa.

This relationship is expressed mathematically as \[ P_1 \times V_1 = P_2 \times V_2 \]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.

To predict one of these variables when the others are known, you can rearrange the equation as needed. The inverse relationship between pressure and volume means that doubling the pressure of a gas will halve its volume, as long as the temperature is consistent. This is what was applied in the original exercise to find the final pressure when the initial pressure and volumes were known.

Introductory Chemistry

In the realm of introductory chemistry, understanding how gases react to changes in their environment is foundational. Students are first introduced to this concept through simple exercises that utilize the basic gas laws to build a strong comprehension of these ideas.

At this level, it's crucial to get comfortable with identifying known values in a problem and seeing how they interact under an assumed state of constant temperature or pressure. This is the building block of more complex chemistry concepts. For instance, in the exercise presented, students learned about Boyle's Law and how to apply it through a step-by-step approach. This teaches them to be methodical, ensuring they check each variable and perform calculations to reach the correct conclusions. Without recognizing these fundamental principles, advancing in chemistry would be challenging. The foundational skills learned during these exercises, such as rearranging equations and performing basic algebra, are invaluable. They help to develop critical thinking and problem-solving abilities which are necessary for tackling more advanced topics in chemistry and other scientific disciplines.