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?

For an ideal gas at equilibrium, the distribution of molecular speeds is described by the Maxwell-Boltzmann (M-B) law. On a graph of number-fraction of molecules $$\left(f(v)\right)$$ versus speed $$v$$, the M-B curve

$$f(v)=4\pi\left(\frac{m}{2\pi kT}\right)^{3/2}v^{2}\exp\!\left(-\frac{mv^{2}}{2kT}\right)$$

depends on absolute temperature $$T$$. When $$T$$ is increased, compare the original curve (lower $$T$$) with the new curve (higher $$T$$):

1. Most probable speed
The most probable speed $$v_{\mathrm{mp}}$$ is obtained from $$\frac{df(v)}{dv}=0$$, giving $$v_{\mathrm{mp}}=\sqrt{\frac{2kT}{m}}$$. Since $$T$$ appears under the square root, $$v_{\mathrm{mp}}$$ increases with temperature. Hence Option A is a true statement.

2. Height of the peak (fraction of molecules having $$v_{\mathrm{mp}}$$)
Although the peak shifts to a higher speed, the curve simultaneously flattens. Mathematically, the value of $$f(v)$$ evaluated at the new $$v_{\mathrm{mp}}$$ decreases because the exponential factor drops faster than the $$v^{2}$$ term rises. Therefore, the fraction of molecules possessing the most probable speed actually decreases. Option B is not true.

3. Broadening of the distribution
At higher $$T$$, significant probabilities extend to both much higher and somewhat lower speeds. The curve spreads out, i.e. becomes broader. Option C is true.

4. Area under the curve
The integral $$\int_{0}^{\infty} f(v)\,dv = 1$$ represents the total fraction (or 100 %) of molecules. Temperature does not change the total number of molecules, so the area under the M-B curve remains exactly the same. Option D is also true.

Since Options A, C and D are correct, the only incorrect statement is Option B.

Final answer: Option B which is: The fraction of the molecules with the most probable speed increases