Question

A sample of helium gas with a volume of \(29.2 \mathrm{~mL}\) at \(785 \mathrm{~mm} \mathrm{Hg}\) is compressed at constant temperature until its volume is \(15.1 \mathrm{~mL}\). What will be the new pressure in the sample?

Short answer

The final pressure of the helium gas sample after compression, at constant temperature, is approximately \(1.996 \mathrm{~atm}\).

Step-by-step solution

Identify the given values and the unknown value

Given: Initial volume, V1 = 29.2 mL, Initial pressure, P1 = 785 mm Hg, Final volume, V2 = 15.1 mL Unknown: Final pressure, P2

Convert pressure from mm Hg to atm

To calculate the final pressure (P2) in atm, we need to first convert the initial pressure (P1) from mm Hg to atm. The conversion factor is: 1 atm = 760 mm Hg So, we will divide the initial pressure by 760 to get the value in atm: P1(atm) = P1(mm Hg) ÷ 760 P1(atm) = 785 ÷ 760 P1(atm) ≈ 1.033 atm Now, we have the initial pressure in atm.

Apply Boyle's Law

We will now apply Boyle's Law to find the final pressure (P2). According to Boyle's Law: P1 * V1 = P2 * V2 We already have all the values except P2.

Solve for the final pressure (P2)

We can rearrange Boyle's Law to solve for P2: P2 = P1 * V1 ÷ V2 Now, we plug in the given values: P2 = (1.033 atm) * (29.2 mL) ÷ (15.1 mL) P2 ≈ 1.996 atm So, the final pressure of the helium gas sample after compression is approximately 1.996 atm.

Key concepts

Exploring Gas Laws

Gas laws are essential principles in chemistry that describe the behavior of gases under various conditions. They provide the mathematical relationships between pressure, volume, temperature, and the amount of gas. Understanding these laws helps explain how gases respond to changes in their environment. Here are the main gas laws to know:

• Boyle's Law: This law focuses on pressure and volume while keeping the temperature constant. It states that the pressure of a gas increases as the volume decreases, and vice versa, assuming the temperature and number of gas molecules remain unchanged.
• Charles's Law: It describes how the volume of a gas expands with a rise in temperature, assuming pressure and the amount of gas are constant.
• Avogadro's Law: This law states that the volume of a gas is directly proportional to the number of moles of the gas when pressure and temperature are constant.

Each of these laws is useful in different scenarios, but Boyle's Law is particularly applicable to our exercise as it explores the relationship between volume and pressure of a gas in a fixed temperature environment. Understanding these relationships is crucial when predicting how a gas will behave when conditions change.

Mastering Pressure Conversion

Converting pressure units is an important skill when working with gas laws. In many cases, measurements need to be in specific units to apply formulas properly. For instance, in the exercise, the pressure needed to be converted from mm Hg to atm. When converting from mm Hg to atm, remember this simple conversion factor:

• 1 atm = 760 mm Hg

To execute this conversion:

• Divide the given pressure in mm Hg by 760 to get the pressure in atm.

In our exercise:

• The initial pressure was 785 mm Hg.
• The conversion to atm was done by calculating: 785 ÷ 760 = 1.033 atm.

Converting to atm ensures compatibility with gas law formulas that often use atm as the standard unit. Being comfortable with such conversions is essential for success in chemistry.

Understanding Volume and Pressure Relationship

The relationship between volume and pressure of a gas is the crux of Boyle's Law. It illustrates how these variables affect one another when temperature remains constant. Here is how to understand this relationship:Boyle's Law formula is:\[ P_1 \times V_1 = P_2 \times V_2 \]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.

This law implies that the product of pressure and volume at one state will equal the product at another state as long as the gas's temperature does not change.From the exercise, we know:

• The initial conditions were 785 mm Hg (converted to 1.033 atm) and 29.2 mL.
• The final volume desired was 15.1 mL.
• Using Boyle's Law, the final pressure is calculated as: \( P_2 = \frac{(1.033 \times 29.2)}{15.1} \approx 1.996 \) atm.

Recognizing that as volume decreases, pressure increases, helps visualize how gases behave under compression. Boyle's Law is closely followed whenever you see gas compression scenarios.