Maxwell Boltzmann Distribution Pogil Answer Key Extension Questions [High-Quality]

An advanced extension question modified from standard POGILs:

Question: A soccer ball (mass 0.43 kg) is treated as a "molecule" at 300 K. Calculate its most probable speed. Why does it not appear to move even though the M-B distribution applies?

Answer: Using ( v_p = \sqrt\frac2RTM ) — but here we use ( R = 8.314 , J/(mol·K) ) and mass in kg/mol. Molar mass of soccer ball = ( 0.43 , kg \times 6.022 \times 10^23 = 2.59 \times 10^23 , kg/mol ).

[ v_p = \sqrt\frac2(8.314)(300)2.59 \times 10^23 \approx \sqrt1.93 \times 10^-20 \approx 1.39 \times 10^-10 , m/s ]

This is slower than a nanometer per second. The reason we don't see the ball move is that the velocity is infinitesimally small due to the enormous "molar mass" of a macroscopic object, and the ball is constantly bombarded asymmetrically by air molecules (Brownian motion), but the net thermal velocity is dwarfed by friction and gravity.

Question: Consider two isotopes: (^235\textUF_6) and (^238\textUF_6) at the same temperature. Draw their M-B distributions. Why is the difference in average speeds small, but the difference in effusion rates significant?

Prompt: Why does the Maxwell-Boltzmann distribution start at zero speed but never touch the y-axis (frequency axis) precisely at speed = 0?

Answer: The probability of a molecule having exactly zero velocity is infinitesimally small.

Reasoning: The distribution function ( f(v) ) is proportional to ( v^2 ) for small ( v ). As ( v \to 0 ), ( f(v) \to 0 ). This makes physical sense: in a gas at any temperature above absolute zero, there are no stationary molecules. Every particle possesses some thermal kinetic energy.