Question

For each of the following sets of pressure/volume data, calculate the missing quantity. Assume that the temperature and the amount of gas remain constant. a. \(V=123 \mathrm{L}\) at 4.56 atm; \(V=?\) at \(1002 \mathrm{mm} \mathrm{Hg}\) b. \(V=634 \mathrm{mL}\) at \(25.2 \mathrm{mm} \mathrm{Hg} ; V=166 \mathrm{mL}\) at? \(\mathrm{atm}\) c. \(V=443 \mathrm{L}\) at 511 torr; \(V=?\) at \(1.05 \mathrm{kPa}\)

Short answer

a. \(V_2 = 451 \ \text{L}\) b. \(P_2 = 0.126 \ \text{atm}\) c. \(V_2 = 28612 \ \text{L}\)

Step-by-step solution

Conversion of pressure units

Convert 1002 mm Hg to atm. 1 atm = 760 mm Hg: \(P_2 = \frac{1002 \ \text{mm Hg}}{760 \ \text{mm Hg/atm}} = 1.319 \ \text{atm}\)

Apply Boyle's Law

Use the relationship \(P_1V_1 = P_2V_2\): \(V_2 = \frac{P_1V_1}{P_2} = \frac{4.56 \ \text{atm} \times 123 \ \text{L}}{1.319 \ \text{atm}} = 451 \ \text{L}\) b. Find the missing pressure:

Conversion of volume and pressure units

Convert 634 mL to L and 25.2 mm Hg to atm: \(V_1 = \frac{634 \ \text{mL}}{1000 \ \text{mL/L}} = 0.634 \ \text{L}\) \(P_1 = \frac{25.2 \ \text{mm Hg}}{760 \ \text{mm Hg/atm}} = 0.0332 \ \text{atm}\)

Apply Boyle's Law

Use the relationship \(P_1V_1 = P_2V_2\): \(P_2 = \frac{P_1V_1}{V_2} = \frac{0.0332 \ \text{atm} \times 0.634 \ \text{L}}{0.166 \ \text{L}} = 0.126 \ \text{atm}\) c. Find the missing volume:

Conversion of pressure units

Convert 511 torr to atm and 1.05 kPa to atm: \(P_1 = \frac{511 \ \text{torr}}{760 \ \text{torr/atm}} = 0.672 \ \text{atm}\) \(P_2 = \frac{1.05 \ \text{kPa}}{101.325 \ \text{kPa/atm}} = 0.0104 \ \text{atm}\)

Apply Boyle's Law

Use the relationship \(P_1V_1 = P_2V_2\): \(V_2 = \frac{P_1V_1}{P_2} = \frac{0.672 \ \text{atm} \times 443 \ \text{L}}{0.0104 \ \text{atm}} = 28612 \ \text{L}\)

Key concepts

Pressure Conversion

Pressure conversion is an essential step in problems that involve the behavior of gases, such as Boyle's Law. Different units like atm, mm Hg, torr, and kPa are used to measure pressure. To solve gas law problems, it's crucial to convert these units appropriately so calculations remain consistent.

- **mm Hg to atm**: 1 atm is equal to 760 mm Hg. So, to convert mm Hg to atm, simply divide the pressure in mm Hg by 760.
- **torr to atm**: The conversion is the same as mm Hg since 1 torr equals 1 mm Hg. Use the same formula for conversion.
- **kPa to atm**: Use the conversion factor, where 1 atm equals 101.325 kPa. Understanding these basics helps solve problems more efficiently and ensure the use of accurate measurements in your calculations.

Volume Calculation

Volume calculation in gas law problems often requires adjusting the volume units to ensure they align with units in other components of the problem. This is crucial for applying Boyle's Law, which relates pressure and volume. Standard units are typically given in liters (L) or milliliters (mL).

- **mL to L**: To convert from milliliters to liters, divide the milliliters by 1000, since 1 liter equals 1000 milliliters.
- **Using Boyle's Law**: The relationship is expressed as \(P_1V_1 = P_2V_2\). Once the units are consistent, plug in the numbers to solve for the unknown volume.
Getting the units right ensures that the calculations for volume changes or confirmations reflect real physical scenarios and offer practical answers.

Gas Laws

Gas laws, such as Boyle's Law, are fundamental principles that describe the behavior of gases under various conditions. Boyle's Law specifically addresses the relationship between pressure and volume at a constant temperature and amount of gas. It can be expressed with the formula \(P_1V_1 = P_2V_2\), indicating that for a given amount of gas, the product of the initial pressure and volume equals the product of their final values when measured in the same units.

The concept of gas laws is important because it helps predict how gases will respond to changes in their environment. A firm understanding of these laws can streamline problem-solving involving gas behaviors in chemistry, physics, and engineering.

Ideal Gas Behavior

Ideal gas behavior is an assumption used in gas law problems to simplify calculations. It assumes gases consist of many small particles moving randomly and their interactions are perfectly elastic. While real gases don't adhere perfectly to these conditions, they do behave in a manner close enough to the ideal gas under many conditions.

In applying our calculations here, we consider the interactions and movements of gas particles ideal. Hence, principles like Boyle's Law can seamlessly describe the relationship between pressure and volume as the particles aren't attracted or repelled by each other.
Understanding ideal gas behavior helps predict the conditions under which gas laws will most accurately describe real gases, especially under low-pressure and high-temperature scenarios.