Problem 84 - Chemistry The Central Science In Si Units

Question

At constant pressure, the mean free path \((\lambda)\) of a gas \(\mathrm{mol}-\) ecule is directly proportional to temperature. At constant temperature, \(\lambda\) is inversely proportional to pressure. If you compare two different gas molecules at the same temperature and pressure, \(\lambda\) is inversely proportional to the square of the diameter of the gas molecules. Put these facts together to create a formula for the mean free path of a gas molecule with a proportionality constant (call it \(R_{\mathrm{mfp}}\), like the ideal- gas constant) and define units for \(R_{m}\)

Step-by-Step Solution

Answer

\( \lambda = R_{\mathrm{mfp}} \frac{T}{Pd^2} \); units of \( R_{\mathrm{mfp}} \) are \( \text{m}^3\cdot\text{Pa}^{-1}\cdot \text{K}^{-1} \).

Step 1: Express Mean Free Path in Terms of Temperature

At constant pressure, the mean free path \( \lambda \) is directly proportional to temperature. We can write this as \( \lambda \propto T \), where \( T \) is the temperature.

Step 2: Express Mean Free Path in Terms of Pressure

At constant temperature, the mean free path \( \lambda \) is inversely proportional to pressure. We can write this as \( \lambda \propto \frac{1}{P} \), where \( P \) is the pressure.

Step 3: Combine Temperature and Pressure Effects

By combining the direct proportionality with temperature and inverse proportionality with pressure, the relationship can be expressed as \( \lambda \propto \frac{T}{P} \).

Step 4: Incorporate the Effect of Molecular Diameter

When comparing two different gas molecules, \( \lambda \) is inversely proportional to the square of the diameter \( d \) of the gas molecules. This can be represented as \( \lambda \propto \frac{1}{d^2} \).

Step 5: Combine All Relations into a Single Formula

By combining all proportionalities, we can write \( \lambda \propto \frac{T}{Pd^2} \). To convert this into an equation, introduce a proportionality constant \( R_{\mathrm{mfp}} \) to get \( \lambda = R_{\mathrm{mfp}} \frac{T}{Pd^2} \).

Step 6: Define the Units for Proportionality Constant

The units of \( \lambda \) (mean free path) are typically meters (m). \( T \) is in Kelvin (K), \( P \) is in Pascal (Pa), and \( d \) is in meters (m). Therefore, the units for \( R_{\mathrm{mfp}} \) should be such that they cancel to give \( m \), leading to \( R_{\mathrm{mfp}} \) having units of \( \text{m}^3\cdot\text{Pa}^{-1}\cdot \text{K}^{-1} \).

Key Concepts

Temperature Dependence

Understanding how temperature affects the mean free path \((\lambda)\) of gas molecules is crucial in many scientific fields. At constant pressure, as the temperature increases, molecules move faster and farther apart. Thus, the mean free path, or the average distance a molecule travels before colliding with another, becomes longer. This relationship is expressed as \( \lambda \propto T \).

In simple terms, imagine a room filled with small balls. If the balls are moving slowly, they collide more often. However, if you increase their speed using a fan, they collide less frequently because they are zipping across the room faster. Similarly, for gas molecules, a higher temperature means they have more energy and are less likely to collide often.

Pressure Dependence

Pressure plays a significant role in determining the mean free path of gas molecules. At a constant temperature, increasing the pressure effectively compresses the gas molecules, bringing them closer together. This means the mean free path becomes shorter, as molecules are more likely to collide. This relation is captured by \( \lambda \propto \frac{1}{P} \).

Think of it like inflating a balloon. When you add more air, the pressure inside increases, and the molecules have less space to move around without bumping into each other. By contrast, a partially inflated balloon provides more room for the molecules, increasing the mean free path. Thus, higher pressure leads to more frequent collisions and a shorter mean free path.

Molecular Diameter

The size of the gas molecules also impacts the mean free path significantly. If you compare different gases under the same conditions of temperature and pressure, the mean free path is inversely proportional to the square of the molecular diameter, expressed as \( \lambda \propto \frac{1}{d^2} \).

Imagine juggling different-sized balls in a confined space. Larger balls will be more likely to collide simply because they occupy more space. Similarly, larger molecular diameters mean less space for movement, resulting in a reduced mean free path.

• Larger diameter: Shorter \( \lambda \)
• Smaller diameter: Longer \( \lambda \)

This concept helps explain why larger molecules in gases tend to have shorter paths between collisions compared to smaller molecules.