Question

At constant temperature, what would be the percentage increase in pressure for a \(5 \%\) decrease in the volume of gas? (1) \(5 \%\) (2) \(10 \%\) (3) \(5.26 \%\) (4) \(4.26 \%\)

Short answer

The percentage increase in pressure is approximately 5.26%, so the correct answer is (3) 5.26%.

Step-by-step solution

Understand the Relationship

At constant temperature, the relationship between pressure (P) and volume (V) of a gas is given by Boyle's Law: \[ P_1 V_1 = P_2 V_2 \]where \(P_1\) and \(V_1\) are the initial pressure and volume, and \(P_2\) and \(V_2\) are the final pressure and volume.

Express Volume Decrease

Given a 5% decrease in volume, we write \[ V_2 = V_1 - 0.05V_1 = 0.95V_1 \].

Substitute into Boyle's Law

Substitute the values into Boyle's Law:\[ P_1 V_1 = P_2 (0.95V_1) \].

Simplify the Equation

Cancel out the \(V_1\) terms:\[ P_1 = 0.95 P_2 \].

Solve for \(P_2\)

Rearrange to find the new pressure \(P_2\):\[ P_2 = \frac{P_1}{0.95} \]which simplifies to\[ P_2 \approx 1.0526 P_1 \].

Calculate Percentage Increase

The percentage increase in pressure is given by:\[ \left( \frac{P_2 - P_1}{P_1} \right) \times 100 \%\approx \left( \frac{1.0526 P_1 - P_1}{P_1} \right) \times 100 \%\approx 5.26\% \].

Key concepts

Pressure and Volume Relationship

The relationship between pressure and volume in a gas, when the temperature is held constant, is defined by Boyle's Law. Boyle's Law tells us that the product of the initial pressure and initial volume of a gas (\(P_1 V_1\)) is equal to the product of the final pressure and final volume (\(P_2 V_2\)). This means when the volume of a gas decreases, the pressure increases proportionally to keep the product constant, and vice versa. The formula for Boyle's Law is simple:

Here, the temperature must remain unchanged for these relationships to hold true. An important thing to note is that Boyle's Law is applicable to ideal gases, allowing us to predict how changes in volume will affect pressure.

Percentage Change Calculation

Understanding how to calculate percentage changes can help us determine the effect on gas pressure when its volume changes. In the given problem, we look at a 5% decrease in volume. Initially, we express this decrease as follows:

Now, we substitute this reduction into Boyle's Law: Cancel out the identical terms and simplify to find the new pressure equation:

To express the percentage change in pressure, use the formula: For a 5% decrease in volume, the pressure increase is calculated as around 5.26%. This calculation means if volume decreases by 5%, pressure increases by approximately 5.26%, demonstrating the inverse relationship described by Boyle's Law.

Gas Laws

Gas laws, such as Boyle's Law, describe the behavior of gases under varying conditions of pressure, volume, and temperature. These fundamental laws help us comprehend and predict how gases will react in different scenarios. Boyle's Law focuses on the inverse relationship between pressure and volume at constant temperature, but it is part of a larger set of gas laws, including Charles's Law and Avogadro's Law.
Below are key points about gas laws in general:

• Charles's Law: Relates volume and temperature, holding pressure constant. Volume varies directly with temperature.
• Avogadro's Law: States that equal volumes of all gases, at constant temperature and pressure, contain equal numbers of molecules.
• Ideal Gas Law: Combines Boyle's, Charles's, and Avogadro's laws into one equation: The ideal gas law gives a comprehensive description of gas behavior under different conditions.

Understanding these principles allows for deeper insight into the physical properties and behavior of gases, crucial for both theoretical and practical applications in various fields such as chemistry, physics, and engineering.