Warming Helium in a Cylinder

Q. A cylinder of fixed capacity 44.8 L contains helium gas at standard temperature and pressure. Taking R as 8.31 J mol⁻¹ K⁻¹, what is the amount of heat needed to raise the temperature of the gas in the cylinder by 15.0 °C, in joules? expects 374 J

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Q. A cylinder of fixed capacity 44.8 L $V = 44.8$ L contains helium $f = 3$ gas at standard temperature and pressure $T = 273$ K, $P = 1$ atm. Taking R as 8.31 J mol⁻¹ K⁻¹ $R = 8.31$ J mol⁻¹ K⁻¹, what is the amount of heat needed to raise the temperature of the gas in the cylinder by 15.0 °C $Δ T = 15.0$ °C, in joules?

The playground

Inputs

volume

V

44.8 L

constant

R

8.31 J mol⁻¹ K⁻¹

rise

Δ T

15 °C

Outputs

heat

Q

The solution

Solved with SpecificHeatOfGases

Given

$V = 44.8$ L
$R = 8.31$ J mol⁻¹ K⁻¹
$Δ T = 15.0$ °C
$T = 273$ K from “standard temperature and pressure”
$P = 1$ atm from “standard temperature and pressure”
$f = 3$ from “helium”

Find $Q$ .

Working

$V = 44.8$ L $= 0.0448$ m³
$Δ T = 15.0$ °C $= 15$ K
$P = 1$ atm $= 101325$ Pa
ideal gas law(helium) $P \cdot V = n \cdot R \cdot T$
$101325 \cdot 0.0448 = n \cdot 8.31 \cdot 273$
$n = 2.00092567$ mol
energy shared among the degrees of freedom(helium) $C_{v} = \frac{f \cdot R}{2}$
$C_{v} = \frac{3 \cdot 8.31}{2}$
$C_{v} = 12.465$ J mol⁻¹ K⁻¹
heat to warm a gas at fixed volume(helium) $Q = n \cdot C_{v} \cdot Δ T$
$Q = 2.00092567 \cdot 12.465 \cdot 15$
$Q = 374.1$ J
$Q \approx 374$ Jto 3 significant figures

Conclusion

At standard conditions the cylinder holds two moles; helium's atoms only move, three ways, so each mole takes three halves of R for every kelvin.

The Ruby

1require_relative "../physics"23question :warming_helium_in_a_cylinder do4  kind :numerical5  describes "A cylinder of fixed capacity %{volume} contains helium gas at standard temperature and pressure. " \6            "Taking R as %{constant}, %{heat} to raise the temperature of the gas in the cylinder by %{rise}, " \7            "in joules?"8  given volume: "44.8 L"9  given constant: "8.31 J mol-1 K-1"10  given rise: "15.0 °C"11  asks heat: "what is the amount of heat needed"12end

1require_relative "../questions/warming_helium_in_a_cylinder"2require_relative "../science/kinetic_theory/specific_heat_of_gases"34solve :warming_helium_in_a_cylinder, using: SpecificHeatOfGases do5  objects :helium67  take volume: helium.volume8  take constant: R9  take rise: helium.temperature_rise1011  deduce_from "standard temperature and pressure", helium.temperature => "273 K", helium.pressure => "1 atm"12  deduce_from "helium", helium.degrees_of_freedom => "3"1314  heat = helium.heat_at_fixed_volume15  answer heat:, in: "J", figures: 31617  conclusion "At standard conditions the cylinder holds two moles; helium's atoms only move, three ways, " \18             "so each mole takes three halves of R for every kelvin."19end

1require_relative "../solutions/warming_helium_in_a_cylinder"23play_with :warming_helium_in_a_cylinder do4  title "Warming Helium in a Cylinder"5end

The laws

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
degrees of freedom	$f$	—	—	SpecificHeatOfGases
molar heat at fixed volume	$C_{v}$	J mol⁻¹ K⁻¹	—	SpecificHeatOfGases
temperature rise	$Δ T$	K	—	SpecificHeatOfGases
heat at fixed volume	$Q$	J	—	SpecificHeatOfGases
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 helium	$P$	Pa	0 … ∞
volume of helium	$V$	m³	0 … ∞
amount of helium	$n$	mol	—
temperature of helium	$T$	K	0 … 1e+09
density of ground	$ρ_{g}$	kg/m³	0 … ∞
molar mass of ground	$M_{g}$	kg/mol	—
density of helium	$ρ$	kg/m³	0 … ∞
molar mass of helium	$M$	kg/mol	—
molar heat at fixed volume of ground	$C_{v,g}$	J mol⁻¹ K⁻¹	—
degrees of freedom of ground	$f_{g}$	—	—
molar heat at fixed volume of helium	$C_{v}$	J mol⁻¹ K⁻¹	—
degrees of freedom of helium	$f$	—	—
heat at fixed volume of ground	$Q_{g}$	J	—
temperature rise of ground	$Δ T_{g}$	K	—
heat at fixed volume of helium	$Q$	J	—
temperature rise of helium	$Δ T$	K	—

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

C_{v,g} = \frac{f_{g} \cdot R}{2}

energy shared among the degrees of freedom

Q_{g} = n_{g} \cdot C_{v,g} \cdot Δ T_{g}

heat to warm a gas at fixed volume

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(helium)

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(helium)

C_{v,g} = \frac{f_{g} \cdot R}{2}

energy shared among the degrees of freedom(ground)

C_{v} = \frac{f \cdot R}{2}

energy shared among the degrees of freedom(helium)

Q_{g} = n_{g} \cdot C_{v,g} \cdot Δ T_{g}

heat to warm a gas at fixed volume(ground)

Q = n \cdot C_{v} \cdot Δ T

heat to warm a gas at fixed volume(helium)