PX154 - C7 - maxwell-boltzmann distribution

molecular speeds and energy

the molecules in a gas have different speeds, and we define a distribution function [YF eqn 18.32]

f (v) = 4 π {(\frac{m}{2 π k_{B} T})}^{\frac{3}{2}} v^{2} \exp (\frac{- m v^{2}}{2 k_{B} T})

for $N_{2}$ at three temperatures:
as $T$ increases:
- peak of the distribution moves to higher speeds
- the distribution also becomes broader
- some slow molecules at high $T$ , conversely some fast molecules at a low $T$
if we have $N$ molecules, the number whose speeds between $v$ and $v + d v$ is: $d N = N f (v) d v$ [YF p627 ]
$v_{p r o b}$ is the most probable speed $= \sqrt{\frac{2}{m} k_{B} T}$
$v_{a v g}$ is the average speed = $= \sqrt{\frac{8}{π m} k_{B} T}$
$v_{r m s} = \sqrt{{\bar{v}}^{2}} = \sqrt{\frac{3}{m} k_{B} T}$
- fits with our kinetic model for the ideal gas and its prediction for ${\bar{v}}^{2}$
- $v_{r m s}$ at $S T P$ for $N_{2} = 484 m s^{-} 1$
we can find the energy distribution too:
$d N = N f (v) d v$ using $E = \frac{1}{2} m v^{2} ⟹ d E = m v d v$ , we get: $d N = N f (E) d E$

f (E) = \frac{2}{\sqrt{π}} {(\frac{1}{k_{B} T})}^{\frac{3}{2}} E^{\frac{1}{2}} \exp (- \frac{E}{k_{B} T})

- $p e a k : E \approx k_{B} T$

$k_{B} T$ gives the energy associated with a given temperature
- at $R T P (T = 290 K), k_{B} T = 4 \times 10^{- 21} J \approx 25 m e V$