Problem 12 - General Chemistry: Principles and Modern Applications

Question

We want to change the volume of a fixed amount of gas from \(725 \mathrm{mL}\) to 2.25 L while holding the temperature constant. To what value must we change the pressure if the initial pressure is \(105 \mathrm{kPa} ?\)

Step-by-Step Solution

Answer

The pressure needs to be changed to approximately \(33.375 \mathrm{kPa}\).

Step 1: Make sure the units of volume are consistent

Before applying Boyle's law, check the units of volume. Here, the initial volume is given in milliliter (mL) and the final volume in liter (L). As 1 L = 1000 mL, convert 2.25 L to 2250 mL for consistency.

Step 2: Apply Boyle's Law

Using \( P_1V_1=P_2V_2 \), where \( P_1 = 105 \mathrm{kPa} \), \( V_1 = 725 \mathrm{mL} \), and \( V_2 = 2250 \mathrm{mL} \), you can solve for \( P_2 \), the final pressure.

Step 3: Solve Algebraically

Rearrange the Boyle's Law equation to isolate \( P_2 \): \( P_2 = P_1V_1/V_2 \). Substituting the known values gives \( P_2 = 105 \mathrm{kPa} \times 725 \mathrm{mL} / 2250 \mathrm{mL} \)

Step 4: Solve Numerically

Performing the calculation gives \( P_2 \approx 33.375 \mathrm{kPa} \).

Key Concepts

Gas laws

Gas laws are a set of principles that explain the behaviors of gases in different conditions. These laws help us predict how gases will react when we change their temperature, volume, or pressure. The core gas laws include Boyle's Law, Charles' Law, and Avogadro's Law. We often use these in chemistry and physics to understand gas reactions.

• Boyle's Law looks at pressure and volume relations.
• Charles' Law focuses on temperature and volume changes.
• Avogadro's Law deals with volume and moles of gas.

For our exercise involving changing gas volume, Boyle's Law is most relevant since it discusses how pressure and volume influence each other when temperature remains constant. Other gas laws also play crucial roles when we involve temperature or different amounts of gas.

Pressure-volume relationship

The pressure-volume relationship is essential for understanding how gases behave. This relationship is the core of Boyle's Law, which states that, for a fixed amount of gas at a constant temperature, the pressure of the gas is inversely proportional to its volume.In simple terms, if you decrease the volume of the gas, its pressure increases. If you increase the volume, the pressure decreases.
Mathematically, Boyle's Law is expressed 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.

By keeping the temperature constant, we understand that any change in volume can only affect pressure in our closed system.

Kinetic molecular theory

The kinetic molecular theory provides another layer of understanding about gas behavior. It explains how particles in gases move and interact, giving insight into why gas laws like Boyle's Law work. This theory suggests:

• Gas molecules are in constant random motion.
• They frequently collide with each other and the walls of their container.
• The pressure exerted by gases is due to these collisions.

The speed and energy of these molecules are affected by temperature changes, but Boyle's Law holds temperature constant. When we alter the volume of the container, we change the frequency of these collisions, thereby affecting pressure. This theory backs up the intuitive understanding of the pressure-volume relationship from a microscopic level.