Oxygen Molecules in Their Gas

Q. Take the diameter of an oxygen molecule to be 3 Å. Estimate the fraction of molecular volume to the actual volume occupied by oxygen gas at STP? expects 4 × 10⁻⁴

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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?

The playground

Inputs

diameter

d

3 Å

Outputs

fraction

f_{V}

The solution

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 Ruby

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

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

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

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)