Argon and Chlorine in a Flask

solved RmsSpeed

Q. A flask contains argon and chlorine in the ratio of 2:1 by mass. The temperature of the mixture $T_{a} = T_{c}$ is 27 °C. The atomic mass of argon is 39.9 g/mol $M_{a} = 39.9$ g/mol and the molecular mass of chlorine is 70.9 g/mol $M_{c} = 70.9$ g/mol. What is the ratio of the root mean square speeds of the molecules of argon to chlorine?

Inputs

argon

M_{a}

39.9 g/mol

chlorine

M_{c}

70.9 g/mol

Outputs

ratio?

How it's solved

Solved with RmsSpeed

Given

$M_{a} = 39.9$ g/mol
$M_{c} = 70.9$ g/mol
$T_{a} = T_{c}$ from “The temperature of the mixture”

Find $\frac{v_{rms,a}}{v_{rms,c}}$ .

Working

rms speed of a gas(chlorine) $v_{rms,c} = {(\frac{3 \cdot R \cdot T_{c}}{M_{c}})}^{0.5}$
$v_{rms,c} = {(\frac{24.9433879 \cdot T_{c}}{0.0709})}^{0.5}$
rms speed of a gas(argon) $v_{rms,a} = {(\frac{3 \cdot R \cdot T_{a}}{M_{a}})}^{0.5}$
$v_{rms,a} = {(\frac{24.9433879 \cdot T_{c}}{0.0399})}^{0.5}$
$\frac{v_{rms,a}}{v_{rms,c}} = 1.33302$
$M_{a} = 39.9$ g/mol $= 0.0399$ kg/mol
$M_{c} = 70.9$ g/mol $= 0.0709$ kg/mol

Conclusion

At one temperature every molecule has the same average kinetic energy, so the lighter argon atoms move faster, as the square root of the inverse ratio of the masses; how much of each gas there is does not matter.

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 :argon_and_chlorine_in_a_flask do4  kind :numerical5  describes "A flask contains argon and chlorine in the ratio of 2:1 by mass. The temperature of the mixture is 27 °C. " \6            "The atomic mass of argon is %{argon} and the molecular mass of chlorine is %{chlorine}. %{ratio} of " \7            "argon to chlorine?"8  given argon: "39.9 g/mol"9  given chlorine: "70.9 g/mol"10  asks ratio: "What is the ratio of the root mean square speeds of the molecules"11end

solution.rb

1require_relative "../questions/argon_and_chlorine_in_a_flask"2require_relative "../science/kinetic_theory/rms_speed"34solve :argon_and_chlorine_in_a_flask, using: RmsSpeed do5  objects :argon, :chlorine67  take argon: argon.molar_mass8  take chlorine: chlorine.molar_mass910  deduce_from "The temperature of the mixture", argon.temperature => chlorine.temperature1112  answer find_relation(argon.rms_speed / chlorine.rms_speed), figures: 31314  conclusion "At one temperature every molecule has the same average kinetic energy, so the lighter argon " \15             "atoms move faster, as the square root of the inverse ratio of the masses; how much of each gas " \16             "there is does not matter."17end

playground.rb

1require_relative "../solutions/argon_and_chlorine_in_a_flask"23play_with :argon_and_chlorine_in_a_flask do4  title "Argon and Chlorine in a Flask"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	—	RmsSpeed
rms speed	$v_{rms}$	m/s	—	RmsSpeed
average thermal energy	$E_{avg}$	J	—	RmsSpeed
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 argon	$P_{a}$	Pa	0 … ∞
volume of argon	$V_{a}$	m³	0 … ∞
amount of argon	$n_{a}$	mol	—
temperature of argon	$T_{a}$	K	0 … 1e+09
pressure of chlorine	$P_{c}$	Pa	0 … ∞
volume of chlorine	$V_{c}$	m³	0 … ∞
amount of chlorine	$n_{c}$	mol	—
temperature of chlorine	$T_{c}$	K	0 … 1e+09
density of ground	$ρ_{g}$	kg/m³	0 … ∞
molar mass of ground	$M_{g}$	kg/mol	—
density of argon	$ρ_{a}$	kg/m³	0 … ∞
molar mass of argon	$M_{a}$	kg/mol	—
density of chlorine	$ρ_{c}$	kg/m³	0 … ∞
molar mass of chlorine	$M_{c}$	kg/mol	—
rms speed of ground	$v_{rms,g}$	m/s	—
rms speed of argon	$v_{rms,a}$	m/s	—
rms speed of chlorine	$v_{rms,c}$	m/s	—
average thermal energy of ground	$E_{avg,g}$	J	—
average thermal energy of argon	$E_{avg,a}$	J	—
average thermal energy of chlorine	$E_{avg,c}$	J	—

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

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

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

ideal gas law(ground)

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

ideal gas law(argon)

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

ideal gas law(chlorine)

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

density of a gas(ground)

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

density of a gas(argon)

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

density of a gas(chlorine)

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

rms speed of a gas(ground)

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

rms speed of a gas(argon)

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

rms speed of a gas(chlorine)

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

three halves k t for each molecule(ground)

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

three halves k t for each molecule(argon)

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

three halves k t for each molecule(chlorine)

T_{a} = T_{c}

The temperature of the mixture