Problem 88 - 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 \(R_{\mathrm{mfp}}\), like the ideal-gas constant) and define units for \(R_{\mathrm{mfp}}\).

Step-by-Step Solution

Answer

The mean free path (\(\lambda\)) of a gas molecule can be expressed by the formula \(\lambda = R_{\mathrm{mfp}} \times \frac{T}{Pd^2}\), where \(R_{\mathrm{mfp}}\) is the proportionality constant with units \(\frac{M L^2}{T^3}\), \(T\) is temperature, \(P\) is pressure, and \(d\) is the diameter of the gas molecules.

Step 1: Temperature Relationship

The mean free path \((\lambda)\) of a gas molecule is directly proportional to temperature. Mathematically, this means that when temperature increases, the mean free path increases. This can be represented as follows: \[ \lambda \propto T\]

Step 2: Pressure Relationship

At constant temperature, the mean free path \((\lambda)\) is inversely proportional to pressure. This means that when pressure increases, the mean free path decreases. This can be represented as follows: \[ \lambda \propto \frac{1}{P}\]

Step 3: Diameter Relationship

For two different gas molecules at the same temperature and pressure, the mean free path \((\lambda)\) is inversely proportional to the square of the diameter of the gas molecules. If we use \(d\) to represent the diameter, this relationship can be expressed as: \[ \lambda \propto \frac{1}{d^2}\]

Step 4: Combining the Relationships

Now, we will combine the three relationships to create a formula for the mean free path \((\lambda)\). This can be done by multiplying the relationships, since the proportionality is for all the same variable, in our case \(\lambda\): \[ \lambda \propto T \times \frac{1}{P} \times \frac{1}{d^2} \]

Step 5: Introducing the Proportionality Constant and Units

To make the above equation an equality, we need to introduce a proportionality constant \((R_{\mathrm{mfp}})\): \[ \lambda = R_{\mathrm{mfp}} \times \frac{T}{Pd^2} \] The units of \(R_{\mathrm{mfp}}\) must be such that the left side of the equation (\(\lambda\)) has the same units as the right side. Since mean free path has units of length, \([\lambda] = L\), and we know the units of pressure \(P\) are \([P] = M L^{-1} T^{-2}\) and the diameter \([d] = L\), we can find the units of \(R_{\mathrm{mfp}}\) as follows: \[ [R_{\mathrm{mfp}}] = [\lambda] \times \frac{[P][d^2]}{[T]} = L \times \frac{M L^{-1} T^{-2} \times L^2}{T} = \frac{M L^2}{T^3} \] Therefore, the proportionality constant \(R_{\mathrm{mfp}}\) has units of \(\frac{M L^2}{T^3}\). The final formula for the mean free path of a gas molecule and the units for the proportionality constant \(R_{\mathrm{mfp}}\) is: \[ \lambda = R_{\mathrm{mfp}} \times \frac{T}{Pd^2} \quad , \quad [R_{\mathrm{mfp}}] = \frac{M L^2}{T^3} \]

Key Concepts

Gas Molecules

Gas molecules are tiny particles that make up gases. They are always moving rapidly in various directions, bouncing off one another and the walls of their container. This behavior can be described using kinetic theory. Gas molecules do not have strong interactions with each other, which means that they can spread out to fill any container they're put in. This flexibility and motion impact many physical properties of gases, including the mean free path. The mean free path (\(\lambda\)) refers to the average distance a molecule travels before colliding with another molecule. This distance is influenced by the size of the molecules and their speed, a concept that can be tied directly to temperature and pressure. Understanding the behavior of gas molecules is crucial because it helps us grasp concepts like diffusion, effusion, and how gases respond to changes in temperature and pressure.

Temperature Dependence

Temperature plays a vital role in the behavior of gas molecules and their mean free path. As temperature increases, gas molecules move faster because they have more kinetic energy. This increase in speed results in more frequent collisions but surprisingly increases the mean free path as well. Why is that? Faster-moving molecules spread out more when they collide, allowing them to travel further before hitting another molecule.
To put it mathematically, the mean free path is directly proportional to temperature:\[ \lambda \propto T \]This means that if temperature doubles, the mean free path doubles, assuming constant pressure and molecule size. Understanding how temperature affects the mean free path helps us predict how gas will behave under different thermal conditions.

Pressure Dependence

Pressure significantly affects the mean free path because it refers to the number of molecules in a given space. At high pressure, there are more gas molecules in the same volume, leading to more collisions and a shorter mean free path. Conversely, at lower pressure, fewer molecules are in the volume, resulting in fewer collisions and thus a longer mean free path.
Mathematically, this relationship is expressed as:\[ \lambda \propto \frac{1}{P} \]This equation highlights that the mean free path is inversely proportional to pressure. So, if pressure were to double, the mean free path would be halved. This relationship helps scientists determine how to manipulate pressure for various applications, such as in vacuum technology or gas-based chemical reactions.

Collision Theory

Collision theory is a framework that explains how gas molecules interact and the effects of those interactions. It helps us understand chemical reactions, diffusion, and the mean free path. This theory posits that molecules must collide to react, and the frequency and energy of these collisions influence reaction rates.
The mean free path is directly linked to collision theory because it represents the average collision distance for a molecule. Not only is this path affected by temperature and pressure, but it's also influenced by the size of the molecules. The smaller the diameter of a molecule, the longer its mean free path, as there is less surface area for collision.Mathematically, the mean free path is inversely proportional to the square of the molecule's diameter:\[ \lambda \propto \frac{1}{d^2} \]This equation illustrates that if the diameter were to double, the mean free path would decrease to a quarter of its original length. Overall, collision theory provides us with a deeper understanding of molecular interactions and the factors that affect these interactions in gases.