Sigma Percentile
JEE Main 2019
LEVELJEE Main

Animated Solution for Physics - Kinetic Theory: An ideal gas occupies a volume of at a pressure of . The energy of the gas is

Select Answer:

Visualized Solution

  • Given macroscopic properties of the ideal gas:

  • For an ideal gas, the energy refers to its translational kinetic energy.
  • Translational degrees of freedom,

  • Internal energy formula:
  • Using Ideal Gas Law:
  • Master Equation:

  • Substitute the given values into the master equation:

  • Cancel out the in the numerator and denominator:

  • Matches option (c)

  • If the gas was explicitly diatomic and total internal energy was asked:

The Sigma Insight: Degree of Freedom and Law of Equipartition of Energy

Solution Diagram

The Macroscopic Picture Imagine a sturdy chamber filled with an ideal gas

We are given its macroscopic properties: a volume of , and a pressure of . These two values are the only keys we need to unlock the total energy hidden within the chaotic motion of the gas molecules.

The Master Equation Now, what exactly is the "energy" of this gas? For an ideal gas, unless specified otherwise, we are talking about its translational kinetic energy

The gas molecules are zipping around in three-dimensional space, giving them degrees of freedom.
The fundamental formula for the internal energy of an ideal gas is:
Since the degree of freedom is , we get . But wait, we don't know the temperature or the number of moles ! Here is the catch—we can use the ideal gas law, , to elegantly replace with .
So, our master equation transforms into a purely macroscopic form:

The Final Calculation Let's bring back those values we noted earlier

We need to carefully substitute the pressure and the volume into our master equation. Don't rush through this; let's set it up properly.
Look closely at the equation... do you see the magic? The in the denominator perfectly cancels out the from the volume. It's a very simple calculation now. We are simply left with multiplied by .
Our final answer is , which perfectly matches option (c).
A Quick Thought Experiment: What if the question explicitly mentioned a diatomic gas and asked for the total internal energy? Then we would use degrees of freedom, and the formula would be . Always watch out for these subtle hints in JEE questions!

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