For 1 mole of oxygen at 300K and 1 atm calculate:

a) The number of molecules whose speeds are between 500,000 and 500,001 m/s (since this speed interval is small, the distribution function practically does not vary, and we can consider it infinitesimal).

b) The number of molecules with $v_z$ between 150000 and 150001 m/s.

c) The number of molecules that simultaneously have $v_z$ and $v_x$ between 150,000 and 150,001 m/s.

Solution:

a) $$\frac{dN_v}{N}=G(v)dv \;\rightarrow N_v=NG(v) \Delta v$$ $$G(v)=\ left(\frac{m}{2\pi kT}\right)^{3/2}e^{\frac{-mv^2}{2kT}}4\pi v^2$$

Substituting, $v=500\;m/s$; $k=1.38x10^{-23}J/s$; $T=300K$; $m=\frac{0.032}{6x10^{23}}=5.33x10^{-26}\;kg$, we obtain: $G(v)=1.85x10^{-3}\;s/m$

Therefore, $N_v=6x10^{23}\cdot (1.85x10^{-3}\;s/m)\cdot (0.001\;m/s)=1.1x10^{18}\;molecules$

b) $$\frac{dN_{vz}}{N}=G(v_z)dv_z \;\rightarrow N_{vz}=NG(v_z) \Delta v_z=7.75x10^{17}\;molecules$$ $$g(v_z)=\left(\frac{m}{2\pi kT}\right)^{1/2}e^{\frac{-mv_{z}^ {2}}{2kT}}$$ It is solved analogously to part a)

c) $$N_{v_xv_yv_z}=Ng(v_x)g(v_z) \Delta v_x \Delta v_z =9.22x10^{11}\;molecules$$