Question

A McLeod gauge measures low gas pressures by compressing a known volume of the gas at constant temperature. If \(315 \mathrm{~cm}^{3}\) of gas is compressed to a volume of 0.0457 \(\mathrm{cm}^{3}\) under a pressure of \(2.51 \mathrm{kPa}\), what was the original gas pressure?

Short answer

The original gas pressure was approximately 0.000364 kPa.

Step-by-step solution

Identify Known Values

We know the initial volume of the gas, \( V_1 = 315 \, \mathrm{cm}^3 \), and the final volume after compression, \( V_2 = 0.0457 \, \mathrm{cm}^3 \). The final pressure is given as \( P_2 = 2.51 \, \mathrm{kPa} \). We need to find the initial pressure \( P_1 \).

Use the Boyle's Law Formula

According to Boyle's Law, for a given mass of gas at constant temperature, the pressure of the gas multiplied by its volume is constant. This can be expressed as \( P_1 \times V_1 = P_2 \times V_2 \).

Substitute Known Values into the Equation

Substitute the known values into the equation: \( P_1 \times 315 \, \mathrm{cm}^3 = 2.51 \, \mathrm{kPa} \times 0.0457 \, \mathrm{cm}^3 \).

Solve for the Initial Pressure \( P_1 \)

Rearrange the equation to solve for \( P_1 \): \[ P_1 = \frac{2.51 \, \mathrm{kPa} \times 0.0457 \, \mathrm{cm}^3}{315 \, \mathrm{cm}^3} \]. Calculate this to find \( P_1 \).

Calculate the Result

Calculate \( P_1 = \frac{2.51 \, \mathrm{kPa} \times 0.0457}{315} \). This simplifies to \( P_1 = 0.000364356 \, \mathrm{kPa} \).

Key concepts

Gas Pressure Measurement

Measuring gas pressure is a crucial task in understanding the behavior of gases. Various instruments are designed for such measurements, with the McLeod gauge being one of them. The McLeod gauge is particularly adept at measuring low gas pressures by utilizing the principle of compressing a known gas volume at constant temperature. This method relies on Boyle's Law, where the pressure and volume of a gas are inversely proportional.

• The initial gas volume is known and is compressed to a much smaller volume.
• The gauge measures the final pressure after compression.
• Using these details, we can calculate the original pressure of the gas.

Understanding how these gauges work helps us accurately gauge pressures in scientific and industrial processes, essential for maintaining proper conditions in gas-related applications.

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in chemistry and physics that describes the state of an ideal gas. It combines Boyle's Law, Charles's Law, and Avogadro's Law. However, for the specific calculation of initial gas pressure using a McLeod gauge, Boyle's Law suffices.
The Ideal Gas Law is expressed 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 universal gas constant
• \(T\) is the temperature in Kelvin

For an ideal gas, the relationship between these variables helps predict how a gas will behave under different conditions. While the Ideal Gas Law gives a more comprehensive picture when temperature, moles of gas, and other factors vary, Boyle's Law isolates pressure and volume as the primary factors for specific gas behaviors at constant temperature.

Volume and Pressure Relationship

The relationship between volume and pressure is at the heart of understanding gas behavior. According to Boyle's Law, when the temperature of a gas is held constant, its pressure and volume are inversely related. This means:

• If the volume decreases, the pressure increases, provided the temperature stays the same.
• If the volume increases, the pressure decreases, again maintaining constant temperature.

Boyle's Law can be mathematically expressed as:
\[ P_1 \times V_1 = P_2 \times V_2 \]
This simple proportionality allows scientists and engineers to predict how a change in volume could affect pressure. In our example, when using a McLeod gauge, the gas's initial volume is compressed to a known smaller volume, dramatically increasing the pressure. Knowing this relationship helps solve for unknown variables, like the initial pressure of the gas.

Chemical Education

Chemical education delves deeply into understanding the properties and behaviors of gases, which are common topics in both high school and college levels. Boyle's Law and the Ideal Gas Law are foundational concepts that help students comprehend more complex subjects of gas dynamics and physical chemistry.
Effective chemistry education focuses on:

• Illustrating essential gas laws with real-world applications, like the use of McLeod gauges.
• Providing hands-on experiences to see these principles in action.
• Simulating conditions to visualize how gases react to pressure changes.

These educational strategies help demystify the laws governing gases, making abstract concepts like pressure and volume relationships more tangible for learners. Understanding these core principles is crucial for students, as they form the basis for advanced topics in chemical engineering, atmospheric sciences, and various technological applications.