Problem 5 - Chemistry and Chemical Reactivity

Question

A sample of nitrogen gas has a pressure of \(67.5 \mathrm{mm} \mathrm{Hg}\) in a \(500 .\) mil. flask. What is the pressure of this gas sample when it is transferred to a 125 -mL. flask at the same temperature?

Step-by-Step Solution

Answer

The pressure in the 125 mL flask is 270 mm Hg.

Step 1: Understanding the Problem

We are given a nitrogen gas sample with an initial pressure of \(67.5 \text{ mm Hg}\) in a \(500\text{ mL}\) flask. We have to find out what the pressure will be when the gas is transferred to a \(125\text{ mL}\) flask, assuming the temperature remains constant.

Step 2: Identify the Formula to Use

This problem involves changing the volume of a gas while the temperature remains constant, so we should use Boyle’s Law. Boyle's Law states that the product of the pressure and volume of a gas is constant when the temperature is constant. It can be written as: \(P_1 \times V_1 = P_2 \times V_2\).

Step 3: Assign Known Values

From the problem, we know that \(P_1 = 67.5 \text{ mm Hg}\), \(V_1 = 500 \text{ mL}\), and \(V_2 = 125 \text{ mL}\). We need to solve for \(P_2\), the pressure in the new flask.

Step 4: Plug Values into the Formula

Using the formula from Boyle's Law \(P_1 \times V_1 = P_2 \times V_2\), substitute the known values: \(67.5 \text{ mm Hg} \times 500 \text{ mL} = P_2 \times 125 \text{ mL}\).

Step 5: Solve for \(P_2\)

Rearrange the equation to solve for \(P_2\): \(P_2 = \frac{67.5 \times 500}{125}\). Calculate \(P_2\).

Step 6: Calculate the Pressure

Perform the calculation: \(P_2 = \frac{33750}{125} = 270 \text{ mm Hg}\). This is the pressure of the gas in the \(125 \text{ mL}\) flask.

Key Concepts

Gas Pressure Conversion

When dealing with gas laws and pressure measurements, it's vital to understand different pressure units. The most common units include atmospheres (atm), millimeters of mercury (mm Hg), and pascals (Pa). Each unit is used in various contexts, depending on the region or scientific discipline. Most scientific calculations, like those involving Boyle's Law, often default to mm Hg or atm.
Converting gas pressure between these units is essential when handling exercises or data analysis. For example:

• 1 atm is equivalent to 760 mm Hg.
• 1 mm Hg is equivalent to 133.322 Pascals.

When conducting calculations, ensuring your units are consistent helps prevent errors. In our example, the pressure was given in mm Hg. While conversion isn't necessary in this case, if a conversion were needed, it ensures precision throughout longer calculations.

Volume and Pressure Relationship

Understanding the relationship between volume and pressure is crucial in gas calculations. Boyle's Law captures this relationship beautifully, showing how pressure and volume inversely affect one another when temperature remains constant. Simply put, if you decrease the volume of a gas's container, the pressure increases, provided the temperature doesn't change. This is due to gas molecules moving closer together, colliding more frequently with the walls of the container.

Boyle's Law is mathematically expressed as \(P_1 \times V_1 = P_2 \times V_2\). This equation means that the initial pressure and volume (\(P_1\) and \(V_1\)) product equals the final pressure and volume (\(P_2\) and \(V_2\)) product. In our nitrogen gas example, the flask volume changed from 500 mL to 125 mL, causing pressure to rise. The solving of the equation demonstrated how the decrease in volume led to an increase in pressure, ensuring the product \(P V\) remains constant.

Nitrogen Gas Calculations

Nitrogen gas calculations are common in chemistry due to nitrogen's abundance and significant role in the atmosphere. Performing these calculations, especially applying Boyle's Law, allows for predictions about behavior under different conditions. Even though experiments typically involve nitrogen, Boyle's Law applies to any ideal gas under constant temperature.

When performing these calculations, remember that:

• Initial and final conditions of a single variable can be used to determine changes in other variables.
• Pressure and volume inversely relate — as volume decreases, pressure increases.
• Nitrogen's properties don't change under these conditions, making it ideal for consistent results.

Calculating the transition from one flask to another involved calculating new pressure conditions using known values. These known values were the initial and final volumes and initial pressure, allowing us to find the pressure in the smaller flask. Such calculations emphasize the practical application of gas laws.