Problem 84 - Chemistry: The Central Science in SI Units, Global Edition

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 formula for the mean free path (λ) of a gas molecule, taking into account its direct proportionality to temperature (T), inverse proportionality to pressure (P), and inverse proportionality to the square of the diameter of the gas molecules (d), is given by: \[λ = R_{\mathrm{mfp}} \times T \times \frac{1}{P} \times \frac{1}{d^2}\] where the proportionality constant R_mfp has the units of \(L/(K \times Pa)\).

Step 1: Relation between λ and Temperature

According to the problem, at constant pressure, the mean free path (λ) of a gas molecule is directly proportional to temperature (T). So, we can write this as: \[λ ∝ T\]

Step 2: Relation between λ and Pressure

Next, at constant temperature, the problem states that λ is inversely proportional to pressure (P). So, we can write this as: \[λ ∝ \frac{1}{P}\]

Step 3: Relation between λ and Diameter of Gas Molecules

Finally, if we compare two different gas molecules at the same temperature and pressure, λ is inversely proportional to the square of the diameter of the gas molecules (d). Therefore, we can write this as: \[λ ∝ \frac{1}{d^2}\]

Step 4: Combining the Relations

Now, we will combine the three proportional relationships to form a single formula for λ: \[λ ∝ T \times \frac{1}{P} \times \frac{1}{d^2}\]

Step 5: Introducing the Proportionality Constant

To convert this proportionality relation into an equation, we need to introduce a proportionality constant, called R_mfp, which will have the same functionality as the ideal-gas constant in the ideal gas equation. So the final formula will look like this: \[λ = R_{\mathrm{mfp}} \times T \times \frac{1}{P} \times \frac{1}{d^2}\]

Step 6: Defining the Units for R_mfp

Since the units for λ should be length (L), for temperature (T) it's Kelvin (K), for pressure (P) it's Pascal (Pa), and for the diameter of gas molecules (d) it's also length (L), we need to define the units for R_mfp in such a way that the units in the formula balance out properly. Therefore R_mfp should have units: \[[R_{\mathrm{mfp}}] = \frac{L^3}{K \times Pa \times L^2} = \frac{L}{K \times Pa}\] So, the required formula for the mean free path of a gas molecule is given by: \[λ = R_{\mathrm{mfp}} \times T \times \frac{1}{P} \times \frac{1}{d^2}\] where R_mfp has the units of \(L/(K \times Pa)\).

Key Concepts

Gas Molecules

Gas molecules are tiny particles that make up gases. They move around freely and independently, often colliding with each other and the walls of their container. This movement forms the basis of the kinetic theory of gases. The term "mean free path" refers to the average distance a molecule travels between these collisions.

• Gas molecules are constantly in motion.
• Their motion is random and they often interact with one another.

The mean free path is a critical concept because it helps us understand how gas molecules behave under different conditions. This distance depends on a variety of factors, including temperature, pressure, and the size of the molecules themselves.

Temperature

Temperature is a measure of the average kinetic energy of gas molecules. In simpler terms, it indicates how fast the molecules are moving. The higher the temperature, the faster the molecules move, which impacts the mean free path. At constant pressure, as temperature increases, gas molecules move more rapidly and collide less often, leading to a longer mean free path.

• Higher temperature generally means faster-moving molecules.
• Temperature is measured in Kelvin (K) in scientific contexts.

Understanding the relationship between temperature and the mean free path is crucial for predicting gas behavior in different thermal conditions.

Pressure

Pressure is the force exerted by gas molecules on the walls of their container. It is caused by the frequent collisions of gas molecules within the container. When the temperature remains constant, an increase in pressure indicates more frequent collisions, resulting in a shorter mean free path.

• Pressure is commonly measured in Pascals (Pa).
• More collisions mean a shorter mean free path at constant temperature.

By exploring the inverse relationship between pressure and the mean free path, we can better understand how gases will behave under different pressures and use this knowledge to develop systems and processes involving gases.

Diameter of Molecules

The diameter of gas molecules is a key factor in determining the mean free path. Larger molecules are more likely to collide with each other, which reduces the mean free path. At constant temperature and pressure, the mean free path is inversely proportional to the square of the molecular diameter.

• Larger diameter results in more frequent collisions.
• The mean free path decreases with increasing diameter size.

This relationship is crucial for comparing different gases or understanding how a change in molecular size might affect the behavior of a gas.

Proportionality Constant

The proportionality constant, denoted as \(R_{\mathrm{mfp}}\), helps convert the qualitative proportional relationships into a quantitative equation. This constant plays a similar role to the ideal-gas constant in the ideal gas law, providing a precise relation that incorporates temperature, pressure, and molecular diameter to calculate the mean free path. For the mean free path formula, \(R_{\mathrm{mfp}}\) has units of \(L/(K \times Pa)\).

• Transforms proportionality into an exact mathematical expression.
• Ensures dimensional consistency and accurate calculations.

By understanding the role of this constant, one can apply it effectively to model gas behavior and predict how changes in environmental conditions will affect gases.