https://r2.erweima.ai/mathbot/image/1727035672103jj6spho2.jpg

### The questions are as follows:

**2.**  
a) What are the corrections in the kinetic theory of gases that led to the equation of state for a real gas?  
b) Define mean free path. Show that the mean free path is inversely proportional to the macroscopic collision cross section.  
c) The mean free path of the molecules of a gas is \(2 \times 10^3 \, \text{cm}\), when there are \(1.25 \times 10^{19} \, \text{molecules per cm}^3\). Compute the diameter of the molecules.

---

### Solution:

#### **(a) Corrections in the Kinetic Theory of Gases for Real Gas Equation of State**

In the ideal gas law, gases are assumed to have the following properties:
1. The gas molecules have no volume.
2. There are no intermolecular forces between the gas molecules.

However, real gases deviate from ideal behavior because:
- **Finite Volume of Molecules:** Real gas molecules occupy a finite volume, so the available volume for movement is less than the actual container volume.
- **Intermolecular Forces:** Real gas molecules experience attractive and repulsive forces, which alter their behavior.

These corrections lead to the **Van der Waals equation** for a real gas:

\[
\left( P + \frac{a}{V_m^2} \right) (V_m - b) = RT
\]

Where:
- \(P\) is the pressure,
- \(V_m\) is the molar volume,
- \(R\) is the gas constant,
- \(T\) is the temperature,
- \(a\) and \(b\) are constants specific to the gas (representing intermolecular attractions and the finite volume of gas molecules, respectively).

Thus, the **corrections** are:
1. A correction for intermolecular attractions (represented by \(a\)),
2. A correction for the finite volume of the molecules (represented by \(b\)).

#### **(b) Define Mean Free Path and Prove Inverse Proportionality**

**Mean Free Path** (\(\lambda\)) is the average distance a molecule travels between collisions with other molecules. It is given by the formula:

\[
\lambda = \frac{1}{\sqrt{2} \, n \sigma}
\]

Where:
- \(n\) is the number density of molecules (number of molecules per unit volume),
- \(\sigma\) is the **collision cross section**, which represents the effective area where collisions can occur.

To show that the mean free path is inversely proportional to the macroscopic collision cross section:
- The collision cross section, \(\sigma\), is related to the size of the molecules.
- Since \(\lambda \propto \frac{1}{\sigma}\), as \(\sigma\) increases (i.e., larger molecules or more frequent collisions), the mean free path decreases. Therefore, the mean free path is **inversely proportional** to the macroscopic collision cross section.

#### **(c) Compute the Diameter of the Molecules**

We are given:
- Mean free path, \(\lambda = 2 \times 10^3 \, \text{cm}\),
- Number density of molecules, \(n = 1.25 \times 10^{19} \, \text{molecules/cm}^3\).

Using the formula for the mean free path:

\[
\lambda = \frac{1}{\sqrt{2} \, n \sigma}
\]

The collision cross section, \(\sigma\), is related to the diameter \(d\) of the molecules as:

\[
\sigma = \pi d^2
\]

Rearranging the mean free path equation to solve for \(d\):

\[
d = \sqrt{\frac{1}{\sqrt{2} \pi n \lambda}}
\]

Substituting the values:

\[
d = \sqrt{\frac{1}{\sqrt{2} \pi (1.25 \times 10^{19}) (2 \times 10^3)}}
\]

Let me compute this for you.

The diameter of the molecules is approximately \( 3.00 \times 10^{-4} \, \text{angstroms} \).

---

### Would you like more details or clarifications on any part?

Here are five related questions for further exploration:
1. What is the physical significance of the Van der Waals constants \(a\) and \(b\)?
2. How does the mean free path change with temperature and pressure in a gas?
3. Can the Van der Waals equation be used at all temperatures and pressures?
4. What assumptions are made when deriving the mean free path formula?
5. How would the mean free path change if the gas molecules were smaller?

**Tip:** The mean free path is longer in gases with lower densities and shorter in denser gases.

2. a) What are the corrections in the kinetic theory of gases that led to the equation of state for a real gas? b) Define mean free path. Show that the mean free path is inversely proportional to the macroscopic collision cross section. c) The mean free path of the molecules of a gas is 2×10^3 cm, when there are 1.25×10^19 molecules per cm³. Compute the diameter of the molecules.

Solve questions on the kinetic theory of gases, corrections leading to the Van der Waals equation, and calculation of the diameter of molecules using mean free path and collision cross section. This detailed solution is ideal for college physics students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation