Animated Solution for Physics - Thermodynamics: Consider a sample of oxygen behaving like an ideal gas. At $300\text{ K}$, the ratio of root mean square (rms) velocity to the average velocity of gas molecule would be (Molecular weight of oxygen is $32\text{ g/mol}$; $R=8.3{\text{ J K}}^{-1}{\text{ mol}}^{-1}$)

When you first read this problem, your brain might immediately start crunching numbers. You see a temperature of $300\text{ K}$, a molar mass of $32\text{ g/mol}$, and the universal gas constant $R=8.3{\text{ J K}}^{-1}{\text{ mol}}^{-1}$. The natural instinct is to plug these values into the formulas and calculate the exact speeds before taking their ratio.

But wait! This is a classic JEE trap designed to test your algebraic intuition and save you precious time. Before you touch a calculator, always write down the symbolic expressions. You will often find that the specific numbers are completely irrelevant.

According to the Kinetic Theory of Gases, the molecules of an ideal gas are in constant, random motion. Their speeds follow the Maxwell-Boltzmann distribution. From this distribution, we define three characteristic speeds:

1. Root Mean Square Velocity ($v_{rms}$): This is the square root of the average of the squares of the velocities. It is directly related to the kinetic energy of the gas.

2. Average Velocity ($v_{av}$): This is the simple arithmetic mean of the speeds of all molecules.

Let's set up the ratio of $v_{rms}$ to $v_{av}$ exactly as requested:

Notice the beauty of this expression. The term $\sqrt{\frac{RT}{M}}$ is present in both the numerator and the denominator. This means the ratio is completely independent of the temperature $T$ and the molar mass $M$. Whether it's oxygen at $300\text{ K}$ or hydrogen at $1000\text{ K}$, the ratio remains exactly the same!

Canceling out the common terms, we get:

Rearranging the fraction by bringing $\pi$ to the numerator, we arrive at our final, elegant answer:

Maxwell-Boltzmann Distribution

This problem is a great opportunity to visualize the Maxwell-Boltzmann distribution curve. If you plot the probability density function $f(v)$ against velocity $v$, you will see a skewed bell curve.

On this curve, the Most Probable Speed ($v_{mp}=\sqrt{\frac{2RT}{M}}$) sits at the very peak. Slightly to the right of the peak is the Average Speed ($v_{av}$), and even further to the right is the Root Mean Square Speed ($v_{rms}$).

Always remember the order of these speeds: $v_{rms}>v_{av}>v_{mp}$. A quick mnemonic to remember this order is RAM (RMS, Average, Most probable).