Animated Solution for Physics - Thermodynamics: Nitrogen gas is at $300^{\circ }\text{C}$ temperature. The temperature (in K) at which the rms speed of a ${\text{H}}_2$ molecule would be equal to the rms speed of a ${\text{N}}_2$ molecule is ......... (Molar mass of ${\text{N}}_2$ gas is $28\text{ g}$.)

Imagine you are observing two different gases in separate containers: Nitrogen ($N_2$) and Hydrogen ($H_2$). The molecules are zipping around, colliding with each other and the walls of their containers. The speed at which they move is directly related to their temperature. But here's the catch—they don't all move at the exact same speed. Instead, we talk about their root mean square (rms) speed, which gives us a reliable measure of their average kinetic energy.

The formula for the rms speed is a beautiful piece of physics:

Here, $R$ is the universal gas constant, $T$ is the absolute temperature in Kelvin, and $M$ is the molar mass of the gas. Notice how the speed depends on two things: it increases with temperature ($T$) and decreases with molar mass ($M$). Heavier molecules are sluggish, while lighter molecules are nimble and fast.

In our problem, we are told that the rms speed of the heavy Nitrogen molecules is exactly equal to the rms speed of the light Hydrogen molecules.

Let's substitute our formula into this equality:

By squaring both sides, the square roots vanish. The constant $3R$ is present on both sides, so it gracefully cancels out, leaving us with a remarkably simple and elegant ratio:

Before we rush into plugging in numbers, there is a classic trap waiting for us. The temperature of Nitrogen is given as $300^{\circ }\text{C}$. In thermodynamics, we must always use the absolute temperature scale (Kelvin).

Let's convert it:

Now we are ready. We know the molar mass of Nitrogen ($M_1$) is $28\text{ g/mol}$, and the molar mass of Hydrogen ($M_2$) is $2\text{ g/mol}$.

We need to find $T_2$, the temperature of the Hydrogen gas. Rearranging our ratio gives:

Substitute the values we've gathered:

Rounding to the nearest integer, we get $41\text{ K}$.

Think about what this means physically. Nitrogen is 14 times heavier than Hydrogen. To get those heavy Nitrogen molecules moving as fast as the light Hydrogen molecules, you have to heat the Nitrogen up to a scorching $573\text{ K}$. Meanwhile, the nimble Hydrogen molecules can achieve that exact same speed while chilling at a freezing $41\text{ K}$! Physics is beautifully consistent.