Question

The volume of a quantity of gas at 1.00 atm is compressed from \(3.25 \mathrm{L}\) to \(2.24 \mathrm{L} .\) What is the final pressure of the gas if there is no change in temperature?

Short answer

Answer: The final pressure of the gas is approximately 1.45 atm.

Step-by-step solution

List the given information and write down the formula for Boyle's Law

We are given the following information: Initial pressure, P1 = 1.00 atm Initial volume, V1 = 3.25 L Final volume, V2 = 2.24 L We are asked to find the final pressure, P2. The formula for Boyle's Law is: P1 x V1 = P2 x V2

Rearrange the formula to find the final pressure

We want to solve for the final pressure, P2. Therefore, we need to rearrange Boyle's Law formula: P2 = (P1 x V1) / V2

Plug in the given values and solve for the final pressure

Now, we can substitute the given values into the formula: P2 = (1.00 atm x 3.25 L) / 2.24 L P2 = 3.25 atm/L

Calculate the final pressure

Now, we can calculate the final pressure: P2 = (3.25 atm) / 2.24 L P2 ≈ 1.45 atm The final pressure of the gas after it has been compressed and its volume has decreased to 2.24 L is approximately 1.45 atm.

Key concepts

Pressure calculation

To find the pressure of a gas after compression or expansion, we use specific mathematical formulas. One of these is derived from Boyle's Law, which deals with changes in pressure as volume changes at a constant temperature. In this exercise, we calculated the final pressure of a gas after its volume shifted from 3.25 L to 2.24 L, given an initial pressure of 1.00 atm. The desired quantity, the final pressure \( P_2 \), was found using the equation:

\[ P_2 = \frac{P_1 \times V_1}{V_2} \]

To solve, simply plug the known values into this equation. Math manipulation remains straightforward:

• First, multiply the initial pressure \( P_1 \) by the initial volume \( V_1 \).
• Then divide that product by the final volume \( V_2 \).
• In this problem, the final pressure came out to approximately 1.45 atm.

This formula helps detail the direct relationship between volume and pressure when volume changes.

Gas laws

Gas laws are fundamental to understanding how gases behave under different conditions of temperature, volume, and pressure. An important gas law, Boyle's Law, articulates that if temperature remains constant during a change in volume, the pressure of a gas will adjust inversely.

Key ideas when dealing with gas laws include:

• Boyle's Law: Focuses on pressure and volume at constant temperature.
• Charles's Law: Deals with temperature and volume, keeping pressure constant.
• Avogadro's Law: Relates the volume of gas to the number of molecules.

Gas laws help predict outcomes like the one in the exercise, which used Boyle's Law to calculate pressure after compressing gas without changing temperature, showcasing the intrinsic relationship between these parameters.

Volume and pressure relationship

Understanding the volume and pressure relationship is pivotal in predicting what will happen to a gas when its surroundings change. Boyle's Law aptly illustrates that when the volume of a gas decreases, its pressure increases, provided the temperature remains consistent.

The relationship hinges on a simple inverse proportion:

• If volume decreases, pressure increases, keeping product \( P \times V \) constant.
• This inverse relationship explains why pressing down on a sealed syringe results in a higher internal pressure.

This concept is vital in numerous real-world applications, such as:

• The functioning of our respiratory system where lungs operate under changing pressures.
• Engineering calculations for pressure vessels and storage tanks.

Ultimately, this shows us how finely tuned the interdependent nature of pressure and volume can be under consistent temperature conditions.