Oxygen Molecules in Their Gas

solved MolecularSize

Q. Take the diameter of an oxygen molecule to be 3 Å $d = 3$ Å. Estimate the fraction of molecular volume to the actual volume occupied by oxygen gas at STP $T = 273$ K, $P = 1$ atm?

Inputs

diameter

d

3 Å

Outputs

fraction

f_{V}

How it's solved

Solved with MolecularSize

Given

$d = 3$ Å
$T = 273$ K from “at STP”
$P = 1$ atm from “at STP”

Find $f_{V}$ .

Working

$d = 3$ Å $= 3 \times 10^{−10}$ m
$P = 1$ atm $= 101325$ Pa
ideal gas law(oxygen) $P \cdot V = n \cdot R \cdot T$
molecules in a gas(oxygen) $N = n \cdot N_{A}$
molecules as little spheres(oxygen) $V_{m} = \frac{N \cdot π \cdot d^{3}}{6}$
share of the space the molecules fill(oxygen) $f_{V} = \frac{V_{m}}{V}$
$f_{V} = \frac{V_{m}}{V}$
$f_{V} = 3.8004329 \times 10^{−4}$
$f_{V} \approx 4 \times 10^{−4}$ to 1 significant figure

Conclusion

The molecules themselves fill only a tiny part of the space a gas takes up; the rest is empty, which is why a gas squeezes so easily.

The web of concepts

given read from the words equation asked

The Ruby that solves it

The question and its solution are small Ruby files. The engine runs them.

question.rb

1require_relative "../physics"23question :oxygen_molecules_in_their_gas do4  kind :numerical5  describes "Take the diameter of an oxygen molecule to be %{diameter}. %{fraction} to the actual volume occupied by " \6            "oxygen gas at STP?"7  given diameter: "3 Å"8  asks fraction: "Estimate the fraction of molecular volume"9end

solution.rb

1require_relative "../questions/oxygen_molecules_in_their_gas"2require_relative "../science/kinetic_theory/molecular_size"34solve :oxygen_molecules_in_their_gas, using: MolecularSize do5  objects :oxygen67  take diameter: oxygen.molecular_diameter89  deduce_from "at STP", oxygen.temperature => "273 K", oxygen.pressure => "1 atm"1011  fraction = oxygen.fraction_filled_by_molecules12  answer fraction:, figures: 11314  conclusion "The molecules themselves fill only a tiny part of the space a gas takes up; the rest is empty, " \15             "which is why a gas squeezes so easily."16end

playground.rb

1require_relative "../solutions/oxygen_molecules_in_their_gas"23play_with :oxygen_molecules_in_their_gas do4  title "Oxygen Molecules in Their Gas"5end

The laws it drew on

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
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 oxygen	$P$	Pa	0 … ∞
volume of oxygen	$V$	m³	0 … ∞
amount of oxygen	$n$	mol	—
temperature of oxygen	$T$	K	0 … 1e+09
density of ground	$ρ_{g}$	kg/m³	0 … ∞
molar mass of ground	$M_{g}$	kg/mol	—
density of oxygen	$ρ$	kg/m³	0 … ∞
molar mass of oxygen	$M$	kg/mol	—
mass of ground	$m_{g}$	kg	0 … ∞
mass of oxygen	$m$	kg	0 … ∞
molecules of ground	$N_{g}$	—	—
molecules of oxygen	$N$	—	—
molecular diameter of ground	$d_{g}$	m	0 … ∞
molecular radius of ground	$r_{g}$	m	—
molecular diameter of oxygen	$d$	m	0 … ∞
molecular radius of oxygen	$r$	m	—
volume of the molecules of ground	$V_{m,g}$	m³	—
volume of the molecules of oxygen	$V_{m}$	m³	—
fraction filled by molecules of ground	$f_{V,g}$	—	—
fraction filled by molecules of oxygen	$f_{V}$	—	—

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?

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(oxygen)

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(oxygen)

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

mass of a gas(ground)

m = n \cdot M

mass of a gas(oxygen)

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

molecules in a gas(ground)

N = n \cdot N_{A}

molecules in a gas(oxygen)

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

diameter is twice the radius(ground)

d = 2 \cdot r

diameter is twice the radius(oxygen)

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(oxygen)

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(oxygen)

n \neq 0

holds some gas?(oxygen)