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 \(R_{\text {mfpp }}\), like the ideal-gas constant) and define units for \(R_{\text {mip. }}\).

Short answer

The mean free path of a gas molecule can be expressed as \(\lambda = R_{\text{mfpp}} \frac{T}{Pd^2}\), where the proportionality constant \(R_{\text{mfpp}}\) has the units of \(\frac{\text{kg}\cdot \text{m}^3}{\text{s}^2\cdot \text{K}}\).

Step-by-step solution

Identifying the Proportional Relationships

We can rewrite the given relationships using proportionality symbols: 1. \(\lambda \propto T\) (constant \(P\)) 2. \(\lambda \propto \frac{1}{P}\) (constant \(T\)) 3. \(\lambda \propto \frac{1}{d^2}\) (constant \(T\) and \(P\))

Combining the Relationships

Now we will combine these relationships into a single expression. Since \(\lambda\) is proportional to the variables in each of these relationships, we can write: $$ \lambda \propto \frac{T}{Pd^2}. $$

Introducing the Proportionality Constant

To convert the proportionality to an equation, we need to introduce a proportionality constant \(R_{\text{mfpp}}\): $$ \lambda = R_{\text{mfpp}} \frac{T}{Pd^2}. $$ Here, \(R_{\text{mfpp}}\) is the mean free path proportionality constant.

Defining Units for \(R_{\text{mfpp}}\)

To define the units for \(R_{\text{mfpp}}\), let's look at the equation above and the units of the variables involved. The units of the mean free path are length (L). For temperature, we have the Kelvin (K) as a unit. The pressure unit is Pascal (Pa) or N/m\(^2\). The diameter unit is also length (L). Inserting the units in the equation gives us: $$ [\text{L}] = [R_{\text{mfpp}}] \frac{[\text{K}]}{[\text{N/m}^2][\text{L}]^2}. $$ We know that 1 N = 1 kg m/s\(^2\), so replacing N with kg and m/s\(^2\) in the equation above yields: $$ [\text{L}] = [R_{\text{mfpp}}] \frac{[\text{K}]}{[\text{kg}][\text{m/s}^2][\text{m}^{-2}][\text{L}]^2}. $$ Simplifying the units on the right side of the equation, we find the units for \(R_{\text{mfpp}}\): $$ [R_{\text{mfpp}}] = \frac{\text{kg}\cdot \text{m}^3}{\text{s}^2\cdot \text{K}}. $$ Therefore, the mean free path of a gas molecule can be expressed as: $$ \lambda = R_{\text{mfpp}} \frac{T}{Pd^2}, $$ where the proportionality constant \(R_{\text{mfpp}}\) has the units of \(\frac{\text{kg}\cdot \text{m}^3}{\text{s}^2\cdot \text{K}}\).

Key concepts

Proportional Relationships in Gases

Understanding how different variables affect each other is crucial when studying gases. In the context of mean free path (\(ddotdiamonddiamonddiamonddot\)), several proportional relationships come into play. One fundamental aspect is that \(ddotdiamonddiamonddiamonddot\) is directly proportional to the temperature (\(T\)) when pressure (\(P\)) holds steady, meaning if temperature rises, so does the mean free path, assuming pressure is unchanged.

Conversely, at a constant temperature, the mean free path is inversely proportional to pressure; thus, as pressure increases, the mean free path decreases. Finally, for two different gas molecules under the same conditions of temperature and pressure, the mean free path is inversely proportional to the square of their diameters. In simple terms, smaller gas molecules tend to have a longer mean free path than larger ones under identical conditions. These proportional relationships allow us to predict how gas molecules behave under varying conditions and are fundamental concepts in kinetic molecular theory.

Introduction of Proportionality Constant

In physics, to convert a proportional relationship into an equation we use a proportionality constant. This constant provides a precise connection between variables that are otherwise subjected to a simple 'more or less' relationship. For the mean free path calculation, this constant is symbolized as \(R_{\text{mfpp}}\), analogous to how the ideal gas law has its own proportionality constant, \(R\).

This constant enables us to derive a useful and applicable formula: \(ddotdiamonddiamonddiamonddot = R_{\text{mfpp}} \frac{T}{Pd^2}\). Here, each variable's impact on the mean free path is quantifiable, making it a powerful tool to predict gas molecule behavior. By introducing \(R_{\text{mfpp}}\), we bridge the gap between abstract proportional relations and concrete, calculable phenomena.

Units of Measurement in Gas Laws

Units are as fundamental to gas laws as recipes are to cooking; they give us the framework to quantify and understand the measurements and calculations we work with. For the mean free path proportionality constant \(R_{\text{mfpp}}\), determining its units requires an examination of the formula \(ddotdiamonddiamonddiamonddot = R_{\text{mfpp}} \frac{T}{Pd^2}\) and the units for each variable within it. Since mean free path is a distance, it is measured in meters (m).

Temperature is measured in Kelvin (K), pressure in Pascals (Pa), which effectively boils down to Newtons per square meter (\(text{N/m}^2\)), and diameter, like the mean free path, is also in meters (m). Combining these into the gas law formula and finding consistency across the units, we derive that \(R_{\text{mfpp}}\)'s units are \(\frac{\text{kg}\cdot \text{m}^3}{\text{s}^2\cdot \text{K}}\). It's important for students to understand these units to properly apply the formula and ensure accurate calculations in any gas-related problem.