The Heat of Aluminium

Q. A sphere of 0.047 kg aluminium is placed for sufficient time in a vessel containing boiling water, so that the sphere is at 100 °C. It is then immediately transferred to a 0.14 kg copper calorimeter containing 0.25 kg water at 20 °C. The temperature of water rises and attains a steady state at 23 °C. Taking the specific heat capacity of water as 4.18 × 10³ J kg⁻¹ K⁻¹ and of copper as 0.386 × 10³ J kg⁻¹ K⁻¹, what is the specific heat capacity of aluminium, in J kg⁻¹ K⁻¹? expects 911 J kg⁻¹ K⁻¹

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Q. A sphere of 0.047 kg $m_{s} = 0.047$ kg aluminium is placed for sufficient time in a vessel containing boiling water, so that the sphere is at 100 °C $T_{1,s} = 100$ °C. It is then immediately transferred to a 0.14 kg $m_{c} = 0.14$ kg copper calorimeter containing $T_{1,c} = T_{1,w}$ 0.25 kg $m_{w} = 0.25$ kg water at 20 °C $T_{1,w} = 20$ °C. The temperature of water rises $Q_{s} = - (Q_{w} + Q_{c})$ and attains a steady state $T_{2,s} = T_{2,w}$ , $T_{2,c} = T_{2,w}$ at 23 °C $T_{2,w} = 23$ °C. Taking the specific heat capacity of water as 4.18 × 10³ J kg⁻¹ K⁻¹ $s_{w} = 4.18 \times 10^{3}$ J kg⁻¹ K⁻¹ and of copper as 0.386 × 10³ J kg⁻¹ K⁻¹ $s_{c} = 0.386 \times 10^{3}$ J kg⁻¹ K⁻¹, what is the specific heat capacity of aluminium, in J kg⁻¹ K⁻¹?

The playground

Inputs

sphere mass

m_{s}

0.047 kg

hot

T_{1,s}

100 °C

calorimeter mass

m_{c}

0.14 kg

water mass

m_{w}

0.25 kg

cool

T_{1,w}

20 °C

steady

T_{2,w}

23 °C

water heat

s_{w}

4180 J kg⁻¹ K⁻¹

copper heat

s_{c}

386 J kg⁻¹ K⁻¹

Outputs

heat capacity

s_{s}

The solution

Solved with Calorimetry

Given

$m_{s} = 0.047$ kg
$T_{1,s} = 100$ °C
$m_{c} = 0.14$ kg
$m_{w} = 0.25$ kg
$T_{1,w} = 20$ °C
$T_{2,w} = 23$ °C
$s_{w} = 4.18 \times 10^{3}$ J kg⁻¹ K⁻¹
$s_{c} = 0.386 \times 10^{3}$ J kg⁻¹ K⁻¹
$T_{1,c} = T_{1,w}$ from “copper calorimeter containing”
$T_{2,s} = T_{2,w}$ from “attains a steady state”
$T_{2,c} = T_{2,w}$ from “attains a steady state”
$Q_{s} = - (Q_{w} + Q_{c})$ from “The temperature of water rises”

Find $s_{s}$ .

Working

$T_{1,s} = 100$ °C $= 373.15$ K
$T_{1,w} = 20$ °C $= 293.15$ K
$T_{2,w} = 23$ °C $= 296.15$ K
heat warms a body(water) $Q_{w} = m_{w} \cdot s_{w} \cdot (T_{2,w} - T_{1,w})$
$Q_{w} = 0.25 \cdot 4180 \cdot (296.15 - 293.15)$
$Q_{w} = 3135$ J
attains a steady state (temperature after of calorimeter) $T_{2,c} = T_{2,w}$
$T_{2,c} = 296.15$
$T_{2,c} = 296.15$ K
copper calorimeter containing $T_{1,c} = T_{1,w}$
$T_{1,c} = 293.15$
$T_{1,c} = 293.15$ K
heat warms a body(calorimeter) $Q_{c} = m_{c} \cdot s_{c} \cdot (T_{2,c} - T_{1,c})$
$Q_{c} = 0.14 \cdot 386 \cdot (296.15 - 293.15)$
$Q_{c} = 162.12$ J
The temperature of water rises $Q_{s} = - (Q_{w} + Q_{c})$
$Q_{s} = - (3135 + 162.12)$
$Q_{s} = −3297.12$ J
attains a steady state (temperature after of sphere) $T_{2,s} = T_{2,w}$
$T_{2,s} = 296.15$
$T_{2,s} = 296.15$ K
heat warms a body(sphere) $Q_{s} = m_{s} \cdot s_{s} \cdot (T_{2,s} - T_{1,s})$
$(−3297.12) = 0.047 \cdot s_{s} \cdot (296.15 - 373.15)$
$s_{s} = 911.1$ J kg⁻¹ K⁻¹
$s_{s} \approx 911$ J kg⁻¹ K⁻¹to 3 significant figures

Conclusion

In the steady state the heat the sphere gave up is the heat the water and the calorimeter took in, and that fixes how much heat a kilogram of aluminium holds per degree.

The Ruby

1require_relative "../physics"23question :the_heat_of_aluminium do4  kind :numerical5  describes "A sphere of %{sphere_mass} aluminium is placed for sufficient time in a vessel containing boiling " \6            "water, so that the sphere is at %{hot}. It is then immediately transferred to a %{calorimeter_mass} " \7            "copper calorimeter containing %{water_mass} water at %{cool}. The temperature of water rises and " \8            "attains a steady state at %{steady}. Taking the specific heat capacity of water as %{water_heat} and of " \9            "copper as %{copper_heat}, %{heat_capacity}, in J kg⁻¹ K⁻¹?"10  given sphere_mass: "0.047 kg"11  given hot: "100 °C"12  given calorimeter_mass: "0.14 kg"13  given water_mass: "0.25 kg"14  given cool: "20 °C"15  given steady: "23 °C"16  given water_heat: "4.18e3 J kg-1 K-1"17  given copper_heat: "0.386e3 J kg-1 K-1"18  asks heat_capacity: "what is the specific heat capacity of aluminium"19end

1require_relative "../questions/the_heat_of_aluminium"2require_relative "../science/thermal_properties_of_matter/calorimetry"34solve :the_heat_of_aluminium, using: Calorimetry do5  objects :sphere, :calorimeter, :water67  take sphere_mass: sphere.mass8  take hot: sphere.temperature_at_first9  take calorimeter_mass: calorimeter.mass10  take water_mass: water.mass11  take cool: water.temperature_at_first12  take steady: water.temperature_after13  take water_heat: water.specific_heat_capacity14  take copper_heat: calorimeter.specific_heat_capacity1516  deduce_from "copper calorimeter containing", calorimeter.temperature_at_first => water.temperature_at_first17  deduce_from "attains a steady state", sphere.temperature_after => water.temperature_after,18                                        calorimeter.temperature_after => water.temperature_after19  deduce_from "The temperature of water rises",20              sphere.heat_to_warm => -(water.heat_to_warm + calorimeter.heat_to_warm)2122  heat_capacity = sphere.specific_heat_capacity23  answer heat_capacity:, in: "J kg-1 K-1", figures: 32425  conclusion "In the steady state the heat the sphere gave up is the heat the water and the calorimeter " \26             "took in, and that fixes how much heat a kilogram of aluminium holds per degree."27end

1require_relative "../solutions/the_heat_of_aluminium"23play_with :the_heat_of_aluminium do4  title "The Heat of Aluminium"5end

The laws

quantity	written	units	within	from
mass	$m$	kg	0 … ∞	Calorimetry
specific heat capacity	$s$	J kg⁻¹ K⁻¹	—	Calorimetry
latent heat	$L$	J/kg	—	Calorimetry
temperature at first	$T_{1}$	K	—	Calorimetry
temperature after	$T_{2}$	K	—	Calorimetry
heat to warm	$Q$	J	—	Calorimetry
heat to change phase	$Q_{L}$	J	—	Calorimetry
heat taken in	$Q_{in}$	J	—	Calorimetry
heat to warm of ground	$Q_{g}$	J	—
mass of ground	$m_{g}$	kg	0 … ∞
specific heat capacity of ground	$s_{g}$	J kg⁻¹ K⁻¹	—
temperature after of ground	$T_{2,g}$	K	—
temperature at first of ground	$T_{1,g}$	K	—
heat to warm of sphere	$Q_{s}$	J	—
mass of sphere	$m_{s}$	kg	0 … ∞
specific heat capacity of sphere	$s_{s}$	J kg⁻¹ K⁻¹	—
temperature after of sphere	$T_{2,s}$	K	—
temperature at first of sphere	$T_{1,s}$	K	—
heat to warm of calorimeter	$Q_{c}$	J	—
mass of calorimeter	$m_{c}$	kg	0 … ∞
specific heat capacity of calorimeter	$s_{c}$	J kg⁻¹ K⁻¹	—
temperature after of calorimeter	$T_{2,c}$	K	—
temperature at first of calorimeter	$T_{1,c}$	K	—
heat to warm of water	$Q_{w}$	J	—
mass of water	$m_{w}$	kg	0 … ∞
specific heat capacity of water	$s_{w}$	J kg⁻¹ K⁻¹	—
temperature after of water	$T_{2,w}$	K	—
temperature at first of water	$T_{1,w}$	K	—
heat to change phase of ground	$Q_{L,g}$	J	—
latent heat of ground	$L_{g}$	J/kg	—
heat to change phase of sphere	$Q_{L,s}$	J	—
latent heat of sphere	$L_{s}$	J/kg	—
heat to change phase of calorimeter	$Q_{L,c}$	J	—
latent heat of calorimeter	$L_{c}$	J/kg	—
heat to change phase of water	$Q_{L,w}$	J	—
latent heat of water	$L_{w}$	J/kg	—
heat taken in of ground	$Q_{in,g}$	J	—
heat taken in of sphere	$Q_{in,s}$	J	—
heat taken in of calorimeter	$Q_{in,c}$	J	—
heat taken in of water	$Q_{in,w}$	J	—

Q_{b} = m_{b} \cdot s_{b} \cdot (T_{2,b} - T_{1,b})

heat warms a body

Q_{L,b} = m_{b} \cdot L_{b}

heat changes a phase

Q_{in,b} = Q_{b} + Q_{L,b}

heat taken in warms and changes phase

Q_{g} = m_{g} \cdot s_{g} \cdot (T_{2,g} - T_{1,g})

heat warms a body(ground)

Q_{s} = m_{s} \cdot s_{s} \cdot (T_{2,s} - T_{1,s})

heat warms a body(sphere)

Q_{c} = m_{c} \cdot s_{c} \cdot (T_{2,c} - T_{1,c})

heat warms a body(calorimeter)

Q_{w} = m_{w} \cdot s_{w} \cdot (T_{2,w} - T_{1,w})

heat warms a body(water)

Q_{L,g} = m_{g} \cdot L_{g}

heat changes a phase(ground)

Q_{L,s} = m_{s} \cdot L_{s}

heat changes a phase(sphere)

Q_{L,c} = m_{c} \cdot L_{c}

heat changes a phase(calorimeter)

Q_{L,w} = m_{w} \cdot L_{w}

heat changes a phase(water)

Q_{in,g} = Q_{g} + Q_{L,g}

heat taken in warms and changes phase(ground)

Q_{in,s} = Q_{s} + Q_{L,s}

heat taken in warms and changes phase(sphere)

Q_{in,c} = Q_{c} + Q_{L,c}

heat taken in warms and changes phase(calorimeter)

Q_{in,w} = Q_{w} + Q_{L,w}

heat taken in warms and changes phase(water)

T_{1,c} = T_{1,w}

copper calorimeter containing

T_{2,s} = T_{2,w}

attains a steady state (temperature after of sphere)

T_{2,c} = T_{2,w}

attains a steady state (temperature after of calorimeter)

Q_{s} = - (Q_{w} + Q_{c})

The temperature of water rises