Question

A sample of nitrogen gas has a pressure of \(67.5 \mathrm{mm} \mathrm{Hg}\) in a 500 -mL. flask. What is the pressure of this gas sample when it is transferred to a 125 -m 1 . flask at the same temperature?

Short answer

The final pressure of the gas is 270.0 mm Hg.

Step-by-step solution

Identify the Known Quantities

We are given that the initial pressure of the nitrogen gas is \(67.5 \mathrm{mm} \mathrm{Hg}\) and the initial volume is \(500 \text{ mL}\). The final volume is \(125 \text{ mL}\). We are to find the final pressure of the gas.

Recall Boyle's Law

Boyle's Law states that for a given mass of gas at constant temperature, the product of pressure and volume is constant. Mathematically, this is represented as \(P_1 V_1 = P_2 V_2\), where \(P_1\) and \(V_1\) are the initial pressure and volume, and \(P_2\) and \(V_2\) are the final pressure and volume.

Plug Known Quantities into Boyle's Law

Using the formula \(P_1 V_1 = P_2 V_2\), we substitute the known values: \(67.5 \mathrm{mm} \mathrm{Hg} \times 500 \text{ mL} = P_2 \times 125 \text{ mL}\).

Solve for the Unknown Quantity

To find \(P_2\), the pressure in the \(125 \text{ mL}\) flask, we rearrange the equation from Step 3: \(P_2 = \frac{67.5 \times 500}{125}\).

Calculate the Final Pressure

Perform the calculation: \(P_2 = \frac{67.5 \times 500}{125} = 270.0 \mathrm{mm} \mathrm{Hg}\).

Key concepts

Gas Pressure

Gas pressure is a fundamental concept in understanding how gases behave. It refers to the force that the gas exerts on the walls of its container. This force results from gas molecules moving and colliding with the walls. The more collisions, the higher the pressure.
Key points to remember about gas pressure:

• The more gas molecules present, the higher the pressure.
• Faster-moving molecules exert more pressure.
• Reducing the container size while keeping the amount of gas the same increases pressure, as shown by Boyle's Law.

Pressure is usually measured in units like atmospheres (atm), millimeters of mercury (mm Hg), or Pascals (Pa). In calculations involving Boyle's Law, pressure is a crucial variable.

Volume

The concept of volume is central to gas laws, including Boyle's Law. Volume is the amount of space that a substance (in this case, gas) occupies. For gases, volume is significant because they expand to fill any available space.
Some important aspects of volume in gas behavior:

• A gas will always expand to fill its container.
• When the volume decreases (and temperature stays constant), the gas molecules are confined in a smaller space.
• This confinement results in more frequent collisions with the container walls, leading to higher pressure.
• Volume is often measured in liters (L) or milliliters (mL).

Understanding how volume changes impacts pressure is vital for solving problems using Boyle's Law.

Constant Temperature

Constant temperature is a critical condition in Boyle's Law, allowing us to focus solely on the variables of pressure and volume. When temperature remains constant (a process referred to as isothermal), the average kinetic energy of the gas molecules doesn't change, ensuring consistent behavior.
Significance of constant temperature:

• The kinetic energy of gas molecules stays stable.
• Ensures only pressure and volume interplay is analyzed.
• Provides predictability in observing how changes in volume affect pressure and vice versa.

Maintaining a constant temperature simplifies calculations and predictions regarding a gas's behavior under volume changes.

Ideal Gas Law

The Ideal Gas Law is an important extension of Boyle's Law. It incorporates additional variables to provide a more comprehensive understanding of gas behavior. The law is expressed mathematically as:\[ PV = nRT \]where:

• \(P\) is the pressure of the gas.
• \(V\) is the volume of the gas.
• \(n\) is the number of moles of gas.
• \(R\) is the ideal gas constant.
• \(T\) is the temperature in Kelvin.

Compared to Boyle’s Law, the Ideal Gas Law considers the number of gas particles via the moles and requires temperature to be in Kelvin. This law allows calculations for changes in the state of a gas when multiple variables are in play and not just pressure and volume, providing a comprehensive understanding of gas behavior under various conditions.