Air Molecules in a Room

solved Moles

Q. Estimate the total number of air molecules (inclusive of oxygen, nitrogen, water vapour and other constituents) in a room of capacity 25.0 m³ $V = 25.0$ m³ at a temperature of 27 °C $T = 300$ K and 1 atm $P = 1$ atm pressure?

Inputs

volume

V

25 m³

pressure

P

1 atm

Outputs

molecules

N

How it's solved

Solved with Moles

Given

$V = 25.0$ m³
$P = 1$ atm
$T = 300$ K from “a temperature of 27 °C”

Find $N$ .

Working

$P = 1$ atm $= 101325$ Pa
ideal gas law(air) $P \cdot V = n \cdot R \cdot T$
$101325 \cdot 25 = n \cdot 8.31446262 \cdot 300$
$n = 1015.5497$ mol
molecules in a gas(air) $N = n \cdot N_{A}$
$N = 1015.5497 \cdot 6.02214076 \times 10^{23}$
$N = 6.11578323 \times 10^{26}$
$N \approx 6.1 \times 10^{26}$ to 2 significant figures

Conclusion

The ideal gas law counts the moles from the pressure, volume and temperature alone, whatever mix of molecules the air is, and each mole holds Avogadro's number of them.

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 :air_molecules_in_a_room do4  kind :numerical5  describes "%{molecules} (inclusive of oxygen, nitrogen, water vapour and other constituents) in a room of capacity " \6            "%{volume} at a temperature of 27 °C and %{pressure} pressure?"7  given volume: "25.0 m3"8  given pressure: "1 atm"9  asks molecules: "Estimate the total number of air molecules"10end

solution.rb

1require_relative "../questions/air_molecules_in_a_room"2require_relative "../science/kinetic_theory/moles"34solve :air_molecules_in_a_room, using: Moles do5  objects :air67  take volume: air.volume8  take pressure: air.pressure910  deduce_from "a temperature of 27 °C", air.temperature => "300 K"1112  molecules = air.molecules13  answer molecules:, figures: 21415  conclusion "The ideal gas law counts the moles from the pressure, volume and temperature alone, whatever mix of " \16             "molecules the air is, and each mole holds Avogadro's number of them."17end

playground.rb

1require_relative "../solutions/air_molecules_in_a_room"23play_with :air_molecules_in_a_room do4  title "Air Molecules in a Room"5end

The laws it drew on

quantity	written	units	within	from
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
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 air	$P$	Pa	0 … ∞
volume of air	$V$	m³	0 … ∞
amount of air	$n$	mol	—
temperature of air	$T$	K	0 … 1e+09
density of ground	$ρ_{g}$	kg/m³	0 … ∞
molar mass of ground	$M_{g}$	kg/mol	—
density of air	$ρ$	kg/m³	0 … ∞
molar mass of air	$M$	kg/mol	—
mass of ground	$m_{g}$	kg	0 … ∞
mass of air	$m$	kg	0 … ∞
molecules of ground	$N_{g}$	—	—
molecules of air	$N$	—	—

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

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(air)

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(air)

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

mass of a gas(ground)

m = n \cdot M

mass of a gas(air)

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

molecules in a gas(ground)

N = n \cdot N_{A}

molecules in a gas(air)