Animated Solution for Chemistry - States of Matter: Which one of the following statements is not true about the effect of an increase in temperature on the distribution of molecular speeds in a gas?

Imagine you are standing in a room full of bouncing rubber balls. Some are barely rolling, some are ricocheting off the walls at blinding speeds, but most are moving at a moderate, average pace. This chaotic, beautiful dance is exactly what happens inside a container of gas.

To make sense of this chaos, physicists use the Maxwell-Boltzmann distribution curve. This graph plots the speed of the molecules ($u$) on the x-axis against the fraction of molecules ($N$) moving at that speed on the y-axis. It gives us a perfect snapshot of the gas's kinetic profile at any given temperature $T_1$.

Now, what happens when we turn up the heat and increase the temperature to $T_2$ (where $T_2>T_1$)?

Temperature is essentially a measure of average kinetic energy. By heating the gas, we are injecting thermal energy into the system. The molecules get excited and start moving faster. Because a larger number of molecules are now traveling at higher speeds, the entire distribution curve shifts to the right.

But it doesn't just shift; it also changes shape. The curve spreads out, becoming noticeably broader. This indicates a wider variation in the speeds of the individual molecules.

If you look at the very top of the curve, you will find the peak. The speed corresponding to this peak is known as the most probable speed ($u_{mp}$). It is the specific speed that is possessed by the largest fraction of molecules in the container.

Mathematically, it is given by the formula:

As we can see from the formula and our shifted graph, when the temperature $T$ increases, the most probable speed $u_{mp}$ also increases. The peak moves further down the x-axis.

Why the Peak Drops

Here is where many students fall into a trap. While the most probable speed increases, the fraction of molecules moving at that speed actually decreases.

Look closely at the y-axis value of the peak. When the curve shifts to the right and broadens, the height of the peak drops. Why? Because the molecules are now spread over a much wider range of speeds. The 'crowd' at the most popular speed has thinned out because more molecules have dispersed into the higher speed ranges.

This brings us to the final, crucial constraint: the area under the curve.

The total area under the Maxwell-Boltzmann curve represents the total number of molecules in the gas sample. Since we are merely heating the gas in a closed container—not adding or removing any gas—the total number of molecules remains absolutely constant.

Therefore, the area under the curve at $T_1$ must be exactly equal to the area under the curve at $T_2$. For a curve to become wider (broader) while maintaining the exact same area, it has no choice but to become shorter. This geometric reality perfectly explains why the peak height (the fraction of molecules at $u_{mp}$) must decrease.

Evaluating the given statements against our physical and geometric understanding: - The area under the curve remains the same. (True) - The distribution becomes broader. (True) - The most probable speed increases. (True) - The fraction of molecules with the most probable speed increases. (False)

As we proved, the fraction actually decreases. Therefore, statement (c) is the incorrect statement and the right answer to our problem.