A Nitrogen Molecule's Collisions

Q. Take the radius of a nitrogen molecule to be roughly 1.0 Å. The molecular mass of N2 is 28.0 g/mol. Estimate the collision frequency of a nitrogen molecule in a cylinder containing nitrogen at 2.0 atm and temperature 17 °C, in collisions per second? expects 4.6 × 10⁹ s⁻¹

What the draft establishes

Fills in once a draft is checked.

The web of concepts

given read from the words equation asked

The equations it used

None yet.

Check an answer

Q. Take the radius of a nitrogen molecule to be roughly 1.0 Å $r = 1.0$ Å. The molecular mass of N2 is 28.0 g/mol $M = 28.0$ g/mol. Estimate the collision frequency of a nitrogen molecule in a cylinder containing nitrogen at 2.0 atm $P = 2.0$ atm and temperature 17 °C $T = 290$ K, in collisions per second?

The playground

Inputs

radius

r

1 Å

molar mass

M

28 g/mol

pressure

P

2 atm

Outputs

frequency

ν_{c}

The solution

Solved with MeanFreePath

Given

$r = 1.0$ Å
$M = 28.0$ g/mol
$P = 2.0$ atm
$T = 290$ K from “temperature 17 °C”

Find $ν_{c}$ .

Working

$r = 1.0$ Å $= 1 \times 10^{−10}$ m
$M = 28.0$ g/mol $= 0.028$ kg/mol
$P = 2.0$ atm $= 202650$ Pa
rms speed of a gas(nitrogen) $v_{rms} = {(\frac{3 \cdot R \cdot T}{M})}^{0.5}$
$v_{rms} = {(\frac{3 \cdot 8.31446262 \cdot 290}{0.028})}^{0.5}$
$v_{rms} = 508.273776$ m/s
diameter is twice the radius(nitrogen) $d = 2 \cdot r$
$d = 2 \cdot 1 \times 10^{−10}$
$d = 2 \times 10^{−10}$ m
path between collisions(nitrogen) $l = \frac{R \cdot T}{\sqrt{2} \cdot π \cdot d^{2} \cdot P \cdot N_{A}}$
$l = \frac{8.31446262 \cdot 290}{\sqrt{2} \cdot 3.14159265 \cdot {(2 \times 10^{−10})}^{2} \cdot 202650 \cdot 6.02214076 \times 10^{23}}$
$l = 1.11175684 \times 10^{−7}$ m
collisions each second(nitrogen) $ν_{c} = \frac{v_{rms}}{l}$
$ν_{c} = \frac{508.273776}{1.11175684 \times 10^{−7}}$
$ν_{c} = 4.572 \times 10^{9}$ s⁻¹
$ν_{c} = 4.57180704 \times 10^{9}$ Hz $= 4.57180704 \times 10^{9}$ s⁻¹
$ν_{c} \approx 4.6 \times 10^{9}$ s⁻¹to 2 significant figures

Conclusion

Each collision comes after a run of one mean free path, so a molecule moving at its rms speed collides as often as that speed covers the path.

The Ruby

1require_relative "../physics"23question :a_nitrogen_molecule_s_collisions do4  kind :numerical5  describes "Take the radius of a nitrogen molecule to be roughly %{radius}. The molecular mass of N2 is " \6            "%{molar_mass}. %{frequency} of a nitrogen molecule in a cylinder containing nitrogen at %{pressure} and " \7            "temperature 17 °C, in collisions per second?"8  given radius: "1.0 Å"9  given molar_mass: "28.0 g/mol"10  given pressure: "2.0 atm"11  asks frequency: "Estimate the collision frequency"12end

1require_relative "../questions/a_nitrogen_molecule_s_collisions"2require_relative "../science/kinetic_theory/mean_free_path"34solve :a_nitrogen_molecule_s_collisions, using: MeanFreePath do5  objects :nitrogen67  take radius: nitrogen.molecular_radius8  take molar_mass: nitrogen.molar_mass9  take pressure: nitrogen.pressure1011  deduce_from "temperature 17 °C", nitrogen.temperature => "290 K"1213  frequency = nitrogen.collision_frequency14  answer frequency:, in: "s-1", figures: 21516  conclusion "Each collision comes after a run of one mean free path, so a molecule moving at its rms speed " \17             "collides as often as that speed covers the path."18end

1require_relative "../solutions/a_nitrogen_molecule_s_collisions"23play_with :a_nitrogen_molecule_s_collisions do4  title "A Nitrogen Molecule's Collisions"5end

The laws

quantity	written	units	within	from
pi	$π$	—	0 … ∞	MolecularSize
gas constant	$R$	J mol⁻¹ K⁻¹	—	IdealGas
pressure	$P$	Pa	0 … ∞	IdealGas
volume	$V$	m³	0 … ∞	IdealGas
amount	$n$	mol	—	IdealGas
temperature	$T$	K	0 … 1e+09	IdealGas
molar mass	$M$	kg/mol	—	IdealGas
density	$ρ$	kg/m³	0 … ∞	IdealGas
avogadro constant	$N_{A}$	mol⁻¹	—	Moles
mass	$m$	kg	0 … ∞	Moles
molecules	$N$	—	—	Moles
molecular diameter	$d$	m	0 … ∞	MolecularSize
molecular radius	$r$	m	—	MolecularSize
volume of the molecules	$V_{m}$	m³	—	MolecularSize
fraction filled by molecules	$f_{V}$	—	—	MolecularSize
rms speed	$v_{rms}$	m/s	—	RmsSpeed
average thermal energy	$E_{avg}$	J	—	RmsSpeed
mean free path	$l$	m	—	MeanFreePath
collision frequency	$ν_{c}$	Hz	—	MeanFreePath
pressure of ground	$P_{g}$	Pa	0 … ∞
volume of ground	$V_{g}$	m³	0 … ∞
amount of ground	$n_{g}$	mol	—
temperature of ground	$T_{g}$	K	0 … 1e+09
pressure of nitrogen	$P$	Pa	0 … ∞
volume of nitrogen	$V$	m³	0 … ∞
amount of nitrogen	$n$	mol	—
temperature of nitrogen	$T$	K	0 … 1e+09
density of ground	$ρ_{g}$	kg/m³	0 … ∞
molar mass of ground	$M_{g}$	kg/mol	—
density of nitrogen	$ρ$	kg/m³	0 … ∞
molar mass of nitrogen	$M$	kg/mol	—
mass of ground	$m_{g}$	kg	0 … ∞
mass of nitrogen	$m$	kg	0 … ∞
molecules of ground	$N_{g}$	—	—
molecules of nitrogen	$N$	—	—
molecular diameter of ground	$d_{g}$	m	0 … ∞
molecular radius of ground	$r_{g}$	m	—
molecular diameter of nitrogen	$d$	m	0 … ∞
molecular radius of nitrogen	$r$	m	—
volume of the molecules of ground	$V_{m,g}$	m³	—
volume of the molecules of nitrogen	$V_{m}$	m³	—
fraction filled by molecules of ground	$f_{V,g}$	—	—
fraction filled by molecules of nitrogen	$f_{V}$	—	—
rms speed of ground	$v_{rms,g}$	m/s	—
rms speed of nitrogen	$v_{rms}$	m/s	—
average thermal energy of ground	$E_{avg,g}$	J	—
average thermal energy of nitrogen	$E_{avg}$	J	—
mean free path of ground	$l_{g}$	m	—
mean free path of nitrogen	$l$	m	—
collision frequency of ground	$ν_{c,g}$	Hz	—
collision frequency of nitrogen	$ν_{c}$	Hz	—

P_{g} \cdot V_{g} = n_{g} \cdot R \cdot T_{g}

ideal gas law

ρ_{g} = \frac{n_{g} \cdot M_{g}}{V_{g}}

density of a gas

m_{g} = n_{g} \cdot M_{g}

mass of a gas

N_{g} = n_{g} \cdot N_{A}

molecules in a gas

d_{g} = 2 \cdot r_{g}

diameter is twice the radius

V_{m,g} = \frac{N_{g} \cdot π \cdot {d_{g}}^{3}}{6}

molecules as little spheres

f_{V,g} = \frac{V_{m,g}}{V_{g}}

share of the space the molecules fill

n_{g} \neq 0

holds some gas?

v_{rms,g} = {(\frac{3 \cdot R \cdot T_{g}}{M_{g}})}^{0.5}

rms speed of a gas

E_{avg,g} = \frac{3 \cdot R \cdot T_{g}}{2 \cdot N_{A}}

three halves k t for each molecule

l_{g} = \frac{R \cdot T_{g}}{\sqrt{2} \cdot π \cdot {d_{g}}^{2} \cdot P_{g} \cdot N_{A}}

path between collisions

ν_{c,g} = \frac{v_{rms,g}}{l_{g}}

collisions each second

P_{g} \cdot V_{g} = n_{g} \cdot R \cdot T_{g}

ideal gas law(ground)

P \cdot V = n \cdot R \cdot T

ideal gas law(nitrogen)

ρ_{g} = \frac{n_{g} \cdot M_{g}}{V_{g}}

density of a gas(ground)

ρ = \frac{n \cdot M}{V}

density of a gas(nitrogen)

m_{g} = n_{g} \cdot M_{g}

mass of a gas(ground)

m = n \cdot M

mass of a gas(nitrogen)

N_{g} = n_{g} \cdot N_{A}

molecules in a gas(ground)

N = n \cdot N_{A}

molecules in a gas(nitrogen)

d_{g} = 2 \cdot r_{g}

diameter is twice the radius(ground)

d = 2 \cdot r

diameter is twice the radius(nitrogen)

V_{m,g} = \frac{N_{g} \cdot π \cdot {d_{g}}^{3}}{6}

molecules as little spheres(ground)

V_{m} = \frac{N \cdot π \cdot d^{3}}{6}

molecules as little spheres(nitrogen)

f_{V,g} = \frac{V_{m,g}}{V_{g}}

share of the space the molecules fill(ground)

f_{V} = \frac{V_{m}}{V}

share of the space the molecules fill(nitrogen)

v_{rms,g} = {(\frac{3 \cdot R \cdot T_{g}}{M_{g}})}^{0.5}

rms speed of a gas(ground)

v_{rms} = {(\frac{3 \cdot R \cdot T}{M})}^{0.5}

rms speed of a gas(nitrogen)

E_{avg,g} = \frac{3 \cdot R \cdot T_{g}}{2 \cdot N_{A}}

three halves k t for each molecule(ground)

E_{avg} = \frac{3 \cdot R \cdot T}{2 \cdot N_{A}}

three halves k t for each molecule(nitrogen)

l_{g} = \frac{R \cdot T_{g}}{\sqrt{2} \cdot π \cdot {d_{g}}^{2} \cdot P_{g} \cdot N_{A}}

path between collisions(ground)

l = \frac{R \cdot T}{\sqrt{2} \cdot π \cdot d^{2} \cdot P \cdot N_{A}}

path between collisions(nitrogen)

ν_{c,g} = \frac{v_{rms,g}}{l_{g}}

collisions each second(ground)

ν_{c} = \frac{v_{rms}}{l}

collisions each second(nitrogen)

n \neq 0

holds some gas?(nitrogen)