Chapter 11, Problem 25P

The Maxwell-Boltzmann probability density function f(v) is given by:

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

where m (kg) is the mass of each molecule, v (m/s) is the speed, T(K) is the temperature, and k = 1.38×10-²³J/K is Boltzmann’s constant. The most probable speed v_pcorresponds to the maximum value of f(v) and can be determined from $\frac{df(v)}{dv}$ = 0. Create a symbolic expression for f(v), differentiate it with respect to v, and show that v_p= $\sqrt{\frac{2kT}{m}}$ Calculate v_pfor oxygen molecules (m = 5.3×10-²⁶kg) at T = 300 K . Make a plot of f(v) versus v for $0\le v\le 2,500$ m/s for oxygen molecules.