Problem 84 - Chemistry The Central Science

Question

At constant pressure, the mean free path \((\lambda)\) of a gas molecule 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 \(R_{\text { mfp }}\) , like the ideal-gas constant) and define units for \(R_{\operatorname{mfp}}\) .

Step-by-Step Solution

Answer

The formula for the mean free path of a gas molecule, considering the given relationships, is: \[ \lambda = R_{\text { mfp }} \frac{T}{Pd^2} \] where \(\lambda\) is the mean free path, \(R_{\text { mfp }}\) is the proportionality constant with the unit \(\frac{\text{Pa}\cdot\text{m}^2}{\text{K}}\), \(T\) is the temperature in Kelvin, \(P\) is the pressure in pascals, and \(d\) is the diameter of the gas molecules in meters.

Step 1: Relationship 1: Mean free path and temperature

According to the first relationship, the mean free path of a gas molecule is directly proportional to temperature at constant pressure. Mathematically, this can be represented as: \[ \lambda \propto T \]

Step 2: Relationship 2: Mean free path and pressure

According to the second relationship, the mean free path of a gas molecule is inversely proportional to pressure at constant temperature. Mathematically, this can be represented as: \[ \lambda \propto \frac{1}{P} \]

Step 3: Relationship 3: Mean free path and diameter

According to the third relationship, the mean free path of a gas molecule is inversely proportional to the square of the diameter of the gas molecules at the same temperature and pressure. Let's represent the diameter of the gas molecules as \(d\). Therefore, we have: \[ \lambda \propto \frac{1}{d^2} \]

Step 4: Combining the relationships

To combine these three relationships into a single formula, we will multiply the proportionalities together: \[ \lambda \propto \frac{T}{Pd^2} \]

Step 5: Introducing proportionality constant \(R_{\text{mfp}}\)

To convert the proportionality into an equation, we need to introduce a proportionality constant \(R_{\text { mfp }}\): \[ \lambda = R_{\text { mfp }} \frac{T}{Pd^2} \]

Step 6: Units of \(R_{\text{mfp}}\)

Now let's analyze the units of \(R_{\text { mfp }}\) to get the final answer. We will use SI units: - Mean free path \(\lambda\) has the unit of length: meters (m). - Temperature \(T\) is measured in Kelvin (K). - Pressure \(P\) is measured in pascals (Pa). - Diameter \(d\) is measured in meters (m). Thus, the units for \(R_{\text { mfp }}\) can be written as: \[ \text{meters} \cdot \frac{\text{pascals} \cdot \text{meters}^2}{\text{Kelvin}} = \frac{\text{Pa} \cdot \text{m}^2}{\text{K}} \] So, the formula for the mean free path of a gas molecule is: \[ \lambda = R_{\text { mfp }} \frac{T}{Pd^2} \] where \(R_{\text { mfp }}\) is the proportionality constant with the unit \(\frac{\text{Pa}\cdot\text{m}^2}{\text{K}}\).

Key Concepts

Gas Molecules

Gas molecules are tiny particles that make up gases. They are in constant motion, bouncing off each other and the walls of their container. This motion is random, and because molecules are so small, their interactions are frequent. The path a single molecule travels before it collides with another is termed as its "mean free path."
The mean free path is an average distance that a gas molecule travels before colliding with another molecule. This concept helps understand how gases behave and interact under different conditions. Since gas molecules are always moving, they are spread out and have the freedom to move across distances without restraint, unless hindered by collisions. Understanding the mean free path of gas molecules is crucial when analyzing any gas's pressure, temperature, and volume behaviors.

Temperature Dependence

The temperature of a gas has a significant impact on its mean free path. When the temperature increases, gas molecules move faster. This increased speed spreads the molecules apart, leading to fewer collisions. As a result, at higher temperatures, the mean free path—the average distance a molecule travels before a collision—also increases. Mathematically, this is expressed as:

• \( \lambda \propto T \)

At constant pressure, the mean free path is directly proportional to the temperature. Essentially, as temperature rises, the mean free path does the same, leading to more unpredictable and dynamic gas behavior.

Pressure Dependence

Unlike temperature, pressure has an inverse relationship with the mean free path. Pressure is defined as the force per unit area that the gas molecules exert on the walls of the container. When pressure is increased while the temperature is kept constant, it implies that there are more molecules in a given volume, or they are packed closer together.This makes the likelihood of collisions more frequent, which means gas molecules have a shorter mean free path. Mathematically, it can be shown as:

• \( \lambda \propto \frac{1}{P} \)

Therefore, at constant temperature, an increase in pressure decreases the mean free path, as the molecules collide more often when crowded in a smaller space.

Molecular Diameter

The diameter of a gas molecule is crucial when considering its mean free path. Larger molecules have a bigger diameter and therefore, occupy more space in a given volume of gas. This results in more frequent collisions since these larger molecules are more likely to run into each other.The relationship between the mean free path and molecular diameter is described by the inverse square relationship:

• \( \lambda \propto \frac{1}{d^2} \)

Where \(d\) represents the diameter of the molecule. Smaller molecules will have longer mean free paths as they can travel further without colliding, due to their relatively small size, making them less likely to encounter other molecules.

Proportionality Constant

To describe the relationships involving mean free path comprehensively, we use a proportionality constant, named \( R_{\text{mfp}} \). This constant allows for a more precise calculation that accounts for temperature, pressure, and molecular diameter to determine the mean free path:\[ \lambda = R_{\text{mfp}} \frac{T}{Pd^2} \]Where:

• \( \lambda \) is the mean free path.
• \( R_{\text{mfp}} \) is the proportionality constant.
• \( T \) is the temperature in Kelvin.
• \( P \) is the pressure in pascals.
• \( d \) is the molecular diameter in meters.

This constant's units are determined through dimensional analysis as \( \frac{\text{Pa} \cdot \text{m}^2}{\text{K}} \). Using \( R_{\text{mfp}} \), the equation seamlessly combines all these relationships into one comprehensive formula descriptive of a gas's behavior under various conditions.