Animated Solution for Physics - Thermodynamics: For a given gas at $1\text{ atm}$ pressure, rms speed of the molecules is $200\text{ m/s}$ at $127^{\circ }\text{C}$. At $2\text{ atm}$ pressure and at $227^{\circ }\text{C}$, the rms speed of the molecules will be

A Tale of Two States

Imagine you are observing a sealed container of gas. In the first state, the gas is at a pressure of $1\text{ atm}$ and a temperature of $127^{\circ }\text{C}$. At this state, the molecules are zipping around with a Root Mean Square (RMS) speed of $200\text{ m/s}$.

Then, the conditions are cranked up! The pressure is doubled to $2\text{ atm}$, and the temperature is raised to $227^{\circ }\text{C}$. The question asks us to find the new RMS speed of these gas molecules.

At first glance, you might think, "Oh, the pressure doubled, so that must affect the speed, right?" This is a classic trap designed to test your conceptual clarity.

Does Pressure Matter?

Let's look at the master equation for the RMS speed of an ideal gas:

Where $R$ is the universal gas constant, $T$ is the absolute temperature, and $M$ is the molar mass of the gas. Notice anything missing? There is absolutely no pressure term $P$ in this equation!

As long as the gas behaves ideally, changing the pressure does not directly change the RMS speed. The average kinetic energy of the gas molecules is strictly a function of their absolute temperature. Therefore, we can completely ignore the pressure values given in the problem. They are just decoys!

Absolute Temperature

Before we do any math, we must address the most common silly mistake in thermodynamics: using Celsius instead of Kelvin. The formula requires absolute temperature $T$.

Let's convert our given temperatures:

Now we are ready to set up our mathematical relationship.

Setting up the Ratio

Since we are dealing with the same "given gas" in both states, its molar mass $M$ remains constant. The universal gas constant $R$ and the number $3$ are also constants.

This means the RMS speed is directly proportional to the square root of the absolute temperature:

We can express this proportionality as a neat ratio between the two states:

Bringing It Home

Now, let's substitute our known values into the ratio. We know $v_1=200\text{ m/s}$, $T_1=400\text{ K}$, and $T_2=500\text{ K}$.

The zeros inside the square root cancel out beautifully:

Finally, we isolate $v_2$ by multiplying both sides by $200$:

And there we have it! The new RMS speed is $100\sqrt{5}\text{ m/s}$. By staying focused on the core dependencies and avoiding the pressure trap, the math becomes incredibly elegant and straightforward.