Problem 10 - Chemistry and Chemical Reactivity

Question

You have a sample of \(\mathrm{CO}_{2}\) in flask \(\mathrm{A}\) 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 \(\mathrm{B} ?\)

Step-by-Step Solution

Answer

The volume of flask B is approximately 117.13 mL.

Step 1: Identify the Known Variables

We have the initial conditions for flask A: \(V_1 = 25.0 \mathrm{mL}\), \(P_1 = 436.5 \mathrm{mmHg}\), and \(T_1 = 20.5^{\circ}C\). Convert the temperature to Kelvin: \(T_1 = 20.5 + 273.15 = 293.65 \text{ K}\). We also have the final conditions for flask B: \(P_2 = 94.3 \mathrm{mmHg}\) and \(T_2 = 24.5^{\circ}C\). Convert this temperature to Kelvin: \(T_2 = 24.5 + 273.15 = 297.65 \text{ K}\).

Step 2: Apply the Combined Gas Law

The combined gas law can be applied because the number of moles of gas remains constant: \[\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}\].We need to solve for \(V_2\), the volume of flask B, so rearrange the equation to find: \[V_2 = \frac{P_1 V_1 T_2}{P_2 T_1}\].

Step 3: Insert the Known Values and Solve

Insert the known values into the rearranged formula:\[V_2 = \frac{436.5 \mathrm{mmHg} \times 25.0 \mathrm{mL} \times 297.65 \text{ K}}{94.3 \mathrm{mmHg} \times 293.65 \text{ K}}\].Calculate each part:\[V_2 = \frac{3241264.625 \text{ mL} \cdot \mathrm{K}}{27676.395 \text{ K \cdot mmHg}}\].Simplifying gives:\[V_2 \approx 117.13 \mathrm{mL}\].

Step 4: Conclusion: Determine the Final Volume

The volume of flask B is therefore approximately \(117.13 \mathrm{mL}\).

Key Concepts

Gas Pressure

Gas pressure is a crucial property in understanding how gases behave. It refers to the force exerted by gas molecules when they collide with the walls of their container. The more frequently and forcefully they collide, the higher the pressure.
Factors affecting gas pressure include:

• The number of gas molecules. More molecules mean more collisions, resulting in higher pressure.
• The temperature of the gas. A higher temperature causes molecules to move faster, increasing collision frequency and pressure.
• The volume of the container. A smaller volume means gas molecules are closer together, leading to more collisions.

In the exercise, the pressure of the carbon dioxide (\(\mathrm{CO}_{2}\)) gas in flask A starts at 436.5 mm Hg, but decreases to 94.3 mm Hg in flask B. This change demonstrates how pressure can vary with changes in the environment.

Gas Volume

Gas volume is the space that a gas occupies, and it's subject to change based on pressure and temperature. In general, gases expand to fill their containers and adjust their volume based on surroundings.
When the gas is transferred from flask A to flask B, its volume changes because of the different pressure and temperature conditions. To calculate the new volume using the combined gas law, we keep the amount of gas constant and relate the initial and final conditions of pressure, volume, and temperature. In the formula, volume is adjusted according to pressure levels and temperature changes. The rule of thumb is:

• Lower pressure allows a gas to occupy a larger volume.
• Higher pressure compresses the gas, reducing its volume.

In the exercise, flask B's volume increased to approximately 117.13 mL due to a decrease in pressure while some increase in temperature was accounted for.

Temperature Conversion

Temperature conversion is an essential step in gas law calculations as temperatures must be in Kelvin. The Kelvin scale starts from absolute zero, making it ideal for scientific calculations.
To convert temperature from Celsius to Kelvin, use the formula: \[K = °C + 273.15\]This conversion ensures accurate calculation with gas laws since Kelvin provides an absolute measure of thermal energy.
In the exercise, temperatures are converted from Celsius as follows:

• Flask A: \(T_1 = 20.5^{\circ}\, C \to T_1 = 293.65 \, K\)
• Flask B: \(T_2 = 24.5^{\circ}\, C \to T_2 = 297.65 \, K\)

This step guarantees that changes in thermal conditions are accounted for in solving for the gas’s volume.