Question

Longest mean free path stands for (1) Nitrogen \(\left(\mathrm{N}_{2}\right)\) (2) Oxygen \(\left(\mathrm{O}_{2}\right)\) (3) Hydrogen \(\left(\mathrm{H}_{2}\right)\) (4) Chlorine \(\left(\mathrm{Cl}_{2}\right)\)

Short answer

Hydrogen \(\text{H}_2\) has the longest mean free path.

Step-by-step solution

- Understand the Concept of Mean Free Path

The mean free path is the average distance a molecule travels before colliding with another molecule. It is inversely proportional to the molecular diameter and the number density of molecules.

- Evaluate Molecular Sizes

Among the given options, hydrogen \(\text{H}_2\) has the smallest molecular size compared to nitrogen \(\text{N}_2\), oxygen \(\text{O}_2\), and chlorine \(\text{Cl}_2\).

- Relate Size to Mean Free Path

Since the mean free path is inversely proportional to the molecular diameter, the smaller the molecule, the longer the mean free path.

- Determine the Result

Given that hydrogen \(\text{H}_2\) has the smallest molecular diameter, it will have the longest mean free path.

Key concepts

molecular size

Molecules come in different sizes. The term 'molecular size' usually refers to the diameter of a molecule. When dealing with gases, smaller molecules typically mean a smaller diameter. For example, in our exercise, hydrogen \(\text{H}_2\) has a smaller molecular size compared to nitrogen \(\text{N}_2\), oxygen \(\text{O}_2\), and chlorine \(\text{Cl}_2\).
The size of a molecule affects how it behaves in a gas. Smaller molecules are lighter and can move faster, making them collide with other molecules more frequently.
Let's summarize that smaller molecular size means hydrogen \(\text{H}_2\) will have an advantage in terms of mean free path as explained in the next sections.

inverse proportionality

Inverse proportionality is a relationship between two quantities where one quantity increases as the other decreases. When we talk about the mean free path, it is inversely proportional to the molecular size and the number density of molecules.
This means:

• The smaller the molecular size, the longer the mean free path.
• The higher the number density of molecules, the shorter the mean free path.

In our exercise, hydrogen's small molecular size leads to a longer mean free path. Why? Because as per inverse proportionality, a smaller molecule will travel a longer distance on average before bumping into another molecule.
Understanding inverse proportionality helps to find out why hydrogen \(\text{H}_2\) has the longest mean free path compared to other gases listed.

collision frequency

Collision frequency refers to how often molecules collide with each other in a given time period. For example, in gases, molecules are in constant motion and crash into each other regularly.
The collision frequency is influenced by factors like:

• Molecular size: Smaller molecules tend to collide more frequently.
• Temperature: Higher temperatures increase the speed of molecules, leading to more collisions.
• Number density: More molecules in a given volume lead to a higher collision frequency.

In the case of hydrogen \(\text{H}_2\), it has a small molecular size which means it collides more frequently with other hydrogen molecules. However, due to its small size and the way mean free path works, it travels a longer distance on average before each collision.
Understanding collision frequency helps us grasp why hydrogen travels further, making its mean free path the longest among the listed gases.