Problem 1 - Chemistry Matter and Change

Question

Assume that the temperature and the amount of gas are constant in the following problems. The volume of a gas at 99.0 \(\mathrm{kPa}\) is 300.0 \(\mathrm{mL} .\) If the pressure is increased to 188 \(\mathrm{kPa}\) , what will be the new volume?

Step-by-Step Solution

Answer

The new volume of the gas is approximately 158.0 mL.

Step 1: Identify the Initial Conditions

The initial conditions are given as follows: the pressure of the gas is 99.0 \( \text{kPa} \) and the volume is 300.0 \( \text{mL} \).

Step 2: Identify the Final Pressure

The final pressure of the gas is given as 188 \( \text{kPa} \).

Step 3: Use Boyle's Law

Boyle's Law states that for a constant temperature, the pressure of a gas times its volume is constant: \( P_1 V_1 = P_2 V_2 \).

Step 4: Substitute Known Values into Boyle's Law

Substitute the initial and final pressures and the initial volume into Boyle's Law: \( (99.0 \text{kPa}) \times (300.0 \text{mL}) = (188 \text{kPa}) \times V_2 \).

Step 5: Solve for the New Volume

Rearrange the equation to solve for the new volume \( V_2 \): \[ V_2 = \frac{99.0 \text{kPa} \times 300.0 \text{mL}}{188 \text{kPa}} \]. Calculate \( V_2 \).

Step 6: Perform the Calculation

Perform the calculation: \[ V_2 = \frac{99.0 \times 300.0}{188} = \frac{29700}{188} \approx 158.0 \text{ mL} \].

Key Concepts

Gas Laws

Gas laws help us understand how gases behave under different conditions. They form an integral part of thermodynamics and can predict changes in volume, pressure, temperature, and the amount of gas when one or more of these factors changes. One of the foundational gas laws is Boyle's Law, which is particularly useful when the temperature and the amount of gas are constant.
Here's a breakdown of Boyle's Law:

• For a fixed quantity of gas at constant temperature, the product of pressure and volume is constant, expressed as \(P_1 V_1 = P_2 V_2\).
• If you know the initial and final pressures and the initial volume, you can calculate the new volume once the pressure changes, as long as the temperature does not vary.

These laws help scientists and engineers make accurate predictions about gas behaviors in many applications, from industrial processes to weather forecasting.

Pressure-Volume Relationship

The pressure-volume relationship in gases is reciprocal. This means when pressure increases, volume decreases, and conversely, when pressure decreases, volume increases. This relationship is the basis of Boyle's Law. Imagining a balloon can help illustrate this concept:
If you squeeze the balloon, the pressure inside increases, causing the volume of the balloon to decrease. This demonstrates the inverse relationship.In mathematical terms:

• Starting with Boyle's Law, \(P_1 V_1 = P_2 V_2\), you see that if \(P_1 < P_2\), then \(V_1 > V_2\); conversely, if \(P_1 > P_2\), then \(V_1 < V_2\).
• Boyle’s Law allows you to solve for unknown pressures or volumes when the other is altered.

Understanding how pressure and volume relate is crucial for designing and operating equipment relying on gas compression and expansion, like engines and refrigeration systems.

Thermodynamics

Thermodynamics focuses on how heat, work, and energy affect matter. In the context of gases, thermodynamics offers a framework to analyze how gas laws, such as Boyle's Law, work together in natural and engineered systems.
There are a few key principles:

• Energy can neither be created nor destroyed, only transformed from one form to another (First Law of Thermodynamics).
• The total energy of a system, including its internal energy, is influenced by changes in pressure and volume, as well as temperature and amount of gas.

For gases at constant temperature, as seen in Boyle's Law exercises, the energy related to pressure and volume changes while keeping thermal energy stable. This aspect simplifies studying gas behavior in practical situations, such as in pistons, syringes, or airflows. Understanding thermodynamics, including its principles and variables, is essential for students and professionals in sciences and engineering fields to design complex systems efficiently.