Problem 3 - Chemistry Matter and Change

Question

Challenge Air trapped in a cylinder fitted with a piston occupies 145.7 mL at 1.08 atm pressure. What is the new volume when the piston is depressed, increasing the pressure by 25%?

Step-by-Step Solution

Answer

The new volume of air when the piston is depressed, increasing the pressure by 25%, is approximately 116.11 mL. This is calculated using Boyle's Law formula: V2 = (P1 * V1) / P2, where P1 = 1.08 atm, V1 = 145.7 mL, and P2 = 1.35 atm.

Step 1: Identify the initial pressure, initial volume, and final pressure

The initial pressure (P1) is given as 1.08 atm, the initial volume (V1) is 145.7 mL, and the final pressure (P2) is increased by 25%. To find P2, we multiply the initial pressure by 1.25: P2 = 1.08 atm * 1.25 = 1.35 atm.

Step 2: Apply Boyle's Law formula

Now that we have the initial conditions (P1 and V1) and the final pressure (P2), we can use Boyle's Law formula P1V1 = P2V2 to find the new volume (V2) of the gas: V2 = (P1 * V1) / P2.

Step 3: Calculate the new volume

Substitute the given values into the formula: V2 = (1.08 atm * 145.7 mL) / 1.35 atm.

Step 4: Simplify and find the new volume

Perform the calculations: V2 = (156.756 mL) / 1.35 atm = 116.112 mL. The new volume of air when the piston is depressed is approximately 116.11 mL.

Key Concepts

Gas Laws

Gas laws are essential in understanding how gases behave under different conditions. They are a set of scientific laws that describe the relationship between the pressure, volume, and temperature of a gas. These relationships help predict how a gas will respond to changes in one or more of these variables. The three main gas laws are:

• Boyle's Law, which focuses on pressure and volume
• Charles's Law, which relates temperature and volume
• Gay-Lussac's Law, which connects pressure and temperature

Each of these laws can be applied when the other variables are kept constant, allowing us to isolate the effect of one specific change at a time. Understanding gas laws is crucial in fields like chemistry, meteorology, and engineering, where controlling gas interactions is necessary.

Pressure and Volume

The relationship between pressure and volume is foundational in studying gases. As per Boyle's Law, a critical gas law, the volume of a given amount of gas is inversely proportional to its pressure when the temperature is kept constant. This means:

• If the pressure increases, the volume decreases.
• If the pressure decreases, the volume increases.

The formula for this relationship is given as:
\[ P_1V_1 = P_2V_2 \]
This equation shows that the initial pressure multiplied by the initial volume is equal to the final pressure multiplied by the final volume. By using this equation, you can solve for any unknown variable as long as you have the other values. This principle is evident when manipulating gas within a closed environment, such as a piston-cylinder system.

Boyle's Law Example

Let's consider a practical example of Boyle's Law using the provided cylinder exercise. We are given that a gas occupies 145.7 mL at 1.08 atm. When the pressure is increased by 25%, or in this case, multiplied by 1.25, we can find the new pressure:
\[ P_2 = 1.08 \, \text{atm} \times 1.25 = 1.35 \, \text{atm} \]
With these values, we apply Boyle's Law formula to find the new volume:
\[ V_2 = \frac{P_1V_1}{P_2} = \frac{1.08 \, \text{atm} \times 145.7 \, \text{mL}}{1.35 \, \text{atm}} \]
This computes to a new volume of approximately 116.11 mL. By using Boyle's Law, we demonstrated how increasing the pressure inside the cylinder decreases the gas's volume, perfectly illustrating the inverse relationship between pressure and volume.