# Calculate the number of collisions per second of one hydrogen molecule at 24 °C and 2.00 bar. The diameter of a hydrogen molecule is 270 pm?

Nov 19, 2016

WARNING! Long answer! There are 3.46 × 10^10 collisions per second.

#### Explanation:

According to Kinetic Molecular Theory, the collision frequency is equal to the root-mean-square velocity of the molecules divided by their mean free path.

color(blue)(bar(ul(|color(white)(a/a)ν = v_"rms"/λcolor(white)(a/a)|)))" "

Root-mean-square velocity

The formula relating the rms velocity to the temperature and molar mass is:

color(blue)(bar(ul(|color(white)(a/a) v_"rms" = sqrt((3RT)/M)color(white)(a/a)|)))" "

where

$R$ = the Universal Gas Constant
$T$ = the temperature
$M$ = the molar mass

For ${\text{H}}_{2}$ at 24 °C,

$T = \text{(24 + 273.15) K" = "297.15 K}$
$M = \text{2.016 g·mol"^"-1" = 2.016 × 10^"-3"color(white)(l) "kg·mol"^"-1}$

${v}_{\text{rms" = sqrt((3RT)/M) = sqrt((3 × 8.314 color(red)(cancel(color(black)("J·K"^"-1""mol"^"-1"))) × 297.15 color(red)(cancel(color(black)("K"))))/( 2.016 × 10^"-3" color(red)(cancel(color(black)("kg·mol"^"-1")))) × ( 1 color(red)(cancel(color(black)("kg")))·"m"^2"s"^"-2")/(1 color(red)(cancel(color(black)("J"))))) = = "1917 m·s"^"-1}}$

The mean free path

If the molecules have diameter d, then we can use a circle of diameter σ = 2d to represent a molecule's effective collision area.

For a hydrogen molecule, σ = "289 pm".

The formula for the mean free path is

color(blue)(bar(ul(|color(white)(a/a) λ = (RT)/(sqrt2πσ^2N_"A"P)color(white)(a/a)|)))" "

$R = 0 \text{.083 14 bar·L·K"^"-1""mol"^"-1" = 8.314 × 10^"-5"color(white)(l)"bar·m"^3·"K"^"-1""mol"^"-1}$
$T = \text{297.15 K}$
σ = "289 pm" = 289 × 10^"-12"color(white)(l) "m"
${N}_{\text{A" = 6.022 × 10^23color(white)(l) "mol"^"-1}}$
$P = \text{2.00 bar}$

λ = (RT)/(sqrt2πσ^2N_"A"P) = (8.314 × 10^"-5"color(red)(cancel(color(black)("bar")))·stackrelcolor(blue)("m")(color(red)(cancel(color(black)("m"^3))))·color(red)(cancel(color(black)("K"^"-1""mol"^"-1"))) × 297.15 color(red)(cancel(color(black)("K"))))/(sqrt2π × (289 × 10^"-12" color(red)(cancel(color(black)("m"))))^2 × 6.022 × 10^23 color(red)(cancel(color(black)("mol"^"-1"))) × 2.00 color(red)(cancel(color(black)("bar"))))

= 5.52 × 10^"-8"color(white)(l) "m" = "55.2 nm"

Collision frequency

ν = v_"rms"/λ = (1917 color(red)(cancel(color(black)("m")))·"s"^"-1")/(5.52 × 10^"-8" color(red)(cancel(color(black)("m")))) = 3.46 × 10^10color(white)(l) "s"^"-1"