Question

You have a sample of \(\mathrm{CO}_{2}\) in a flask A with a volume of \(25.0 \mathrm{mL} .\) At \(20.5^{\circ} \mathrm{C},\) the pressure of the gas is \(436.5 \mathrm{mm}\) Hg. To find the volume of another flask \(B\), you move the \(\mathrm{CO}_{2}\) to that flask and find that its pressure is now \(94.3 \mathrm{mm}\) Hg at \(24.5^{\circ} \mathrm{C} .\) What is the volume of flask \(\mathrm{B} ?\)

Short answer

The volume of flask B is approximately 115 mL.

Step-by-step solution

Understand the Problem

We have a sample of \( \mathrm{CO}_2 \) under different conditions in two flasks: flask A and flask B. We know the volume, temperature, and pressure of \( \mathrm{CO}_2 \) in flask A and want to find the volume of flask B after the gas transfer when the pressure and temperature in flask B are known.

Use the Ideal Gas Law for Flask A

The ideal gas law is given by \( PV = nRT \). For flask A, we rearrange to find the number of moles \( n \): \[ n = \frac{P_1 \cdot V_1}{R \cdot T_1} \]Convert the given temperature from Celsius to Kelvin: \[ T_1 = 20.5 + 273.15 = 293.65 \text{ K} \]Substitute the known values: - \( P_1 = 436.5 \text{ mm Hg} = 0.574 \text{ atm} \), using the conversion factor \(1 \text{ atm} = 760 \text{ mm Hg}\).- \( V_1 = 25.0 \text{ mL} = 0.025 \text{ L} \). - \( R = 0.0821 \text{ L atm K}^{-1} \text{ mol}^{-1} \).Calculating, \[ n = \frac{0.574 \times 0.025}{0.0821 \times 293.65} \]

Calculate the Number of Moles of CO2

From Step 2:\[ n = \frac{0.574 \times 0.025}{0.0821 \times 293.65} \approx 0.000582 \text{ mol} \]This is the number of moles of \( \mathrm{CO}_2 \) that will remain constant before and after the transfer between the flasks.

Use the Ideal Gas Law for Flask B

Now let's consider flask B, where the gas has a new pressure and temperature. Using the ideal gas law again:\[ V_2 = \frac{nRT_2}{P_2} \]Convert the new temperature:\[ T_2 = 24.5 + 273.15 = 297.65 \text{ K} \]Convert the new pressure:\[ P_2 = 94.3 \text{ mm Hg} = 0.124 \text{ atm} \]Substitute values: \[ V_2 = \frac{0.000582 \times 0.0821 \times 297.65}{0.124} \]

Calculate the Volume of Flask B

From Step 4:\[ V_2 = \frac{0.000582 \times 0.0821 \times 297.65}{0.124} \approx 0.115 \text{ L} \]Convert this into milliliters (mL):\[ 0.115 \text{ L} = 115 \text{ mL} \] Therefore, the volume of flask B is approximately 115 mL.

Key concepts

Gas Laws

Gas laws are mathematical relationships that describe the behavior of gases. One of the most important gas laws is the Ideal Gas Law, which is a combination of Boyle's, Charles's, and Avogadro's laws. These individual laws explain how pressure (P), volume (V), and temperature (T) are related to each other.
The Ideal Gas Law is given by the formula:

• \( PV = nRT \)

This formula allows you to calculate any one of the variables if the other three are known. In this equation, \( n \) is the number of moles of gas, and \( R \) is the universal gas constant, which is approximately \( 0.0821 \ \text{L atm K}^{-1} \ \text{mol}^{-1} \).
By understanding the Ideal Gas Law, we can solve equations involving different conditions for gas in a container as we did with the flasks containing CO2.

Gas Pressure

Pressure is a measure of the force that gas molecules exert as they collide with the walls of their container. In our problem, we encountered two different pressures for the CO2 gas in two separate flasks.
Pressure can be expressed in various units such as atmospheres (atm), millimeters of mercury (mm Hg), and pascals (Pa). It is crucial to convert all pressure measurements to the same unit when using the Ideal Gas Law.
In our exercise, the pressures were initially given in mm Hg. We converted them to atm using the conversion factor:

• 1 atm = 760 mm Hg

By understanding how pressure influences gas behavior, we can use changes in pressure to determine the volume of a different container holding the gas.

Temperature Conversion

Temperature has a significant effect on the behavior of gases, as described by the gas laws.
In the Ideal Gas Law, it is crucial to use the temperature in Kelvin, the absolute temperature scale, rather than Celsius or Fahrenheit. This is because gas laws are based on the absolute temperature scale.
To convert from Celsius to Kelvin, use the formula:

• \( T_{K} = T_{C} + 273.15 \)

In our problem, we converted the temperatures in Celsius to Kelvin for accurate calculations. Understanding temperature conversion is essential when solving problems that involve changes in gas conditions.

Volume Calculations

Volume is the amount of space that a substance or object occupies. In gas law problems, volume can change depending on pressure and temperature. We calculated the volume of flask B by rearranging the Ideal Gas Law:

• \( V_2 = \frac{nRT_2}{P_2} \)

The known values, including the previously calculated number of moles (n) and the constant (R), allowed us to find the new volume under the new conditions.
We then converted the final answer from liters (L) into milliliters (mL), as this is a standard unit for measuring smaller volumes. Proper volume calculations ensure that we understand the changes in container sizes when gases are transferred from one to another.