The Heat of Aluminium

solved Calorimetry

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?

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}

How it's solved

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 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 :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

solution.rb

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

playground.rb

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

The laws it drew on

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