Question

A sample of gas at 47°C and 1.03 atm occupies a volume of 2.20 L. What volume would this gas occupy at 107°C and 0.789 atm?

Short answer

The new volume of the gas is approximately 3.41 L.

Step-by-step solution

Understand the Problem

The problem involves a gas sample where temperature, pressure, and volume change. You need to find the new volume of gas at the changed temperature and pressure using the combined gas law.

Write Down the Combined Gas Law

The combined gas law is expressed as: \[ \frac{P_1 \times V_1}{T_1} = \frac{P_2 \times V_2}{T_2} \] where \(P\), \(V\), and \(T\) are the pressure, volume, and temperature respectively. Subscripts 1 and 2 represent the initial and final states, respectively.

Convert Temperatures to Kelvin

Convert the given temperatures from °C to Kelvin: \[ T_1 = 47^\text{°C} + 273.15 = 320.15 \text{ K} \] \[ T_2 = 107^\text{°C} + 273.15 = 380.15 \text{ K} \]

Identify and Plug in the Values

Identify and plug in the known values into the combined gas law: \[ P_1 = 1.03 \text{ atm}, \ V_1 = 2.20 \text{ L}, \ T_1 = 320.15 \text{ K}, \ P_2 = 0.789 \text{ atm}, \ T_2 = 380.15 \text{ K} \] The equation becomes: \[ \frac{1.03 \text{ atm} \times 2.20 \text{ L}}{320.15 \text{ K}} = \frac{0.789 \text{ atm} \times V_2}{380.15 \text{ K}} \]

Rearrange the Equation to Solve for \(V_2\)

Rearrange the equation to solve for the final volume \(V_2\): \[ V_2 = \frac{(1.03 \text{ atm} \times 2.20 \text{ L}) \times 380.15 \text{ K}}{320.15 \text{ K} \times 0.789 \text{ atm}} \]

Calculate the Final Volume \(V_2\)

Perform the calculations: \[ V_2 = \frac{(1.03 \times 2.20) \times 380.15}{320.15 \times 0.789} \] \[ V_2 \approx \frac{2.266 \times 380.15}{252.5} \approx 3.41 \text{ L} \]

Key concepts

Ideal Gas Law

The Ideal Gas Law is a crucial concept in chemistry and physics. It relates the pressure (P), volume (V), temperature (T), and amount of gas (n) using the equation: \[ PV = nRT \]Here, R is the gas constant. This law assumes that the gas behaves ideally. Ideal gases follow certain assumptions: gas particles have no volume, and there are no intermolecular attractions or repulsions.
This assumption works well for many gases under a variety of conditions, although real gases deviate from this behavior at high pressures and low temperatures. The Ideal Gas Law is essential for understanding how gases interact with their environment. It is often used in combination with other gas laws to solve more complex problems. By understanding the Ideal Gas Law, we can predict how a gas will respond to changes in pressure, temperature, and volume, allowing us to perform calculations like the one in the original exercise.

Gas Laws

Gas laws describe the behavior of gases under various conditions. These include Boyle's Law, Charles's Law, Gay-Lussac's Law, and the Combined Gas Law.

• Boyle's Law: States that the volume of a gas is inversely proportional to its pressure at constant temperature.
• Charles's Law: States that the volume of a gas is directly proportional to its temperature at constant pressure.
• Gay-Lussac's Law: States that the pressure of a gas is directly proportional to its temperature at constant volume.
• Combined Gas Law: Combines Boyle's, Charles's, and Gay-Lussac's laws into one equation to calculate changes in pressure, temperature, and volume.

The Combined Gas Law, \[ \frac{P_1 \times V_1}{T_1} = \frac{P_2 \times V_2}{T_2} \]is particularly useful when solving problems involving changes in more than one variable, as seen in the original exercise. By understanding these fundamental laws, you can grasp how different factors influence the behavior of gases.

Temperature and Pressure Conversion

Correctly converting temperature and pressure is vital in solving gas law problems. Temperatures must be in Kelvin, as gas laws derive from absolute zero, the point where particle motion stops.

• To convert Celsius to Kelvin: Add 273.15 to the Celsius temperature. For example, \[ 47^\text{°C} + 273.15 = 320.15 \text{K} \]
• To convert Fahrenheit to Kelvin: \[ T_\text{K} = \frac{5}{9}(T_\text{F} - 32) + 273.15 \]Pressure conversions depend on the units used. Common units include atmospheres (atm), Pascal (Pa), and millimeters of mercury (mmHg).
  • 1 atm = 101,325 Pa
  • 1 atm = 760 mmHg
  It's crucial to keep units consistent throughout your calculations. In the exercise, the temperatures were converted to Kelvin and pressure remained in atm, making it straightforward to use in the Combined Gas Law.

Volume Calculation

Volume calculation is central in gas law problems. Using the Combined Gas Law, you can determine the final volume of a gas when temperature and pressure change. In the exercise, we rearrange the equation to solve for the new volume (\[ V_2 = \frac{(P_1 \times V_1) \times T_2}{T_1 \times P_2} \]). Begin by inserting known values: \[ V_2 = \frac{(1.03 \text{ atm} \times 2.20 \text{ L}) \times 380.15 \text{ K}}{320.15 \text{ K} \times 0.789 \text{ atm}} \]Next, perform the calculations in steps: \[ V_2 = \frac{2.266 \times 380.15}{252.5} \ V_2 \text{(approx)} = 3.41 \text{ L} \]Understanding how to follow steps for volume calculation ensures accurate results. Double-check your work and ensure units are consistent. By practicing these calculations, you'll become more familiar with gas law applications and concepts.