Question

A sample of air at 5.00 atm expands from \(1.75 \mathrm{~L}\) to \(2.50 \mathrm{~L} .\) If the temperature remains constant, what is the final pressure in atm?

Short answer

The final pressure is 3.50 atm.

Step-by-step solution

Identify the Relationship

Since temperature remains constant and the volume and pressure of a gas sample are changing, we can use Boyle's Law for this problem. Boyle's Law states that for a given mass of gas at constant temperature, the product of pressure and volume is constant: \( P_1 V_1 = P_2 V_2 \).

Assign Known Variables

Recognize the known initial pressure \( P_1 = 5.00 \) atm, initial volume \( V_1 = 1.75 \) L, and final volume \( V_2 = 2.50 \) L. We need to find the final pressure \( P_2 \).

Set Up the Equation

Using Boyle's Law, set up the equation: \( P_1 V_1 = P_2 V_2 \). Plug in the known values: \( 5.00 \times 1.75 = P_2 \times 2.50 \).

Solve for Final Pressure

Rearrange the equation to solve for \( P_2 \): \( P_2 = \frac{5.00 \times 1.75}{2.50} \).

Calculate the Final Pressure

Calculate \( P_2 \): \( P_2 = \frac{8.75}{2.50} = 3.50 \) atm.

Key concepts

Pressure-Volume Relationship

The pressure-volume relationship for gases is a fascinating topic often referred to as Boyle's Law. Imagine you have a balloon filled with air. If you squeeze it, making the balloon smaller, the air inside becomes more compressed, and thus the air pressure increases. This is much like when we apply Boyle’s Law, which expresses that pressure and volume of a gas share an inverse relationship at constant temperature. Easy to remember, right? When one goes up, the other goes down.

Boyle’s Law is mathematically expressed as: \[ P_1 V_1 = P_2 V_2 \] 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.

When you increase the volume of the gas, the gas spreads out thinner, reducing the pressure. Conversely, reducing the volume increases the pressure as the gas molecules are packed closer together. Understanding this relationship is critical for managing systems involving gases, from simple balloons to complex engineering systems like car engines.

Gas Laws

Gas laws encompass a set of rules that describe the behavior of gases under different conditions of pressure, volume, and temperature. They are essential in chemistry, explaining how gases act when pressure, volume, or temperature changes. Boyle’s Law is just one of these laws, with others including Charles's Law and Avogadro's Law. Together, they form the combined gas law, which integrates all these individual laws.

Boyle’s Law applies only when temperature is held constant, focusing specifically on the pressure-volume relationship. Imagine having a piston inside a cylinder. As you press the piston, reducing the space, gas molecules inside have less room to move, so pressure increases.

These laws not only help us in the laboratory setting but are also crucial for understanding natural phenomena and industrial processes like predicting weather patterns and designing chemical reactors.

Introductory Chemistry

Diving into introductory chemistry, students begin to unravel the mysteries of the world around them through the basic principles of matter and energy, where understanding gas behavior is fundamental. Learning about Boyle’s Law gives insight into how gases react under pressure, forming a foundational concept in chemistry and physics.

In early chemistry courses, students familiarize themselves with these principles through experiments and problem-solving exercises, similar to the given exercise of calculating the final pressure of a gas given its initial states. Such exercises reinforce their grasp on how mathematical principles apply to real-world situations.

Building a strong understanding of gas laws aids in the comprehension of more complex chemical concepts that follow. New concepts like kinetic molecular theory and thermodynamics are introduced, broadening students' understanding of how gases behave under various conditions. Mastery of these initial topics is critical for students’ success in advanced sciences.