Gaseous state PYQ series

Mastering Gas Dynamics: The Secrets of Mean Free Path and Kinetic Energy

Hey there, science enthusiasts! Today, we’re diving deep into the fascinating world of gas molecules. We have a question that explores two core concepts of gas dynamics: the mean free path and the kinetic energy of gas molecules. Understanding these can really sharpen your problem-solving skills, especially if you're prepping for exams like JEE, NEET, or the Olympiads. Let’s break it down step-by-step!

The Question at Hand

Consider these two statements:

• Statement (I): The mean free path of gas molecules is inversely proportional to the square of the molecular diameter.
• Statement (II): The average kinetic energy of gas molecules is directly proportional to the absolute temperature of the gas.

Which of the following options is correct?

1. Statement I is false but Statement II is true.
2. Statement I is true but Statement II is false.
3. Both Statement I and Statement II are false.
4. Both Statement I and Statement II are true.

Step-by-Step Solution

First, let’s talk about the mean free path (λ) of gas molecules. In simple terms, it’s the average distance a molecule travels before it bumps into another one. The formula for calculating the mean free path is:

λ = \(\frac{RT}{\sqrt{2} \pi d^2 N_A P}\)

Here, d is the diameter of a molecule. Notice how λ is inversely proportional to the square of the diameter (d²)? This means that as the molecular diameter increases, the mean free path gets shorter. So, Statement (I) is absolutely correct!

Next, we look at the average kinetic energy (KE) of gas molecules. Think of kinetic energy as the energy due to motion, which for gas molecules, depends on temperature. The formula is:

KE = \(\frac{f}{2} nRT\)

Where R is the universal gas constant, T is the absolute temperature, and f is the degrees of freedom. Clearly, KE is directly proportional to the absolute temperature (T). So, guess what? Statement (II) is also spot on!

Wrapping It Up

After breaking down both statements, we can see that they are both correct. Therefore, the right answer is:

(4) Both Statement I and Statement II are true.

And there you have it! This is a great example of how understanding the theory behind gas dynamics can help you tackle tricky problems with confidence. Keep practicing, and these concepts will become second nature. Stay curious and keep exploring!