The Specific Heat of a Metal

solved Calorimetry

Q. In an experiment on the specific heat of a metal, a 0.20 kg $m_{m} = 0.20$ kg block of the metal at 150 °C $T_{1,m} = 150$ °C is dropped in a copper calorimeter $T_{1,c} = T_{1,w}$ , $T_{2,c} = T_{2,w}$ (of water equivalent $s_{c} = s_{w}$ 0.025 kg $m_{c} = 0.025$ kg) containing 150 cm³ of water $m_{w} = 0.150$ kg at 27 °C $T_{1,w} = 27$ °C. The final temperature $T_{2,m} = T_{2,w}$ , $Q_{m} = - (Q_{w} + Q_{c})$ is 40 °C $T_{2,w} = 40$ °C. Taking the specific heat capacity of water as 4.186 J g⁻¹ K⁻¹ $s_{w} = 4.186$ J g⁻¹ K⁻¹, what is the specific heat of the metal, in J g-1 K-1?

Inputs

metal mass

m_{m}

0.2 kg

hot

T_{1,m}

150 °C

equivalent

m_{c}

0.025 kg

cool

T_{1,w}

27 °C

final

T_{2,w}

40 °C

water heat

s_{w}

4.186 J g⁻¹ K⁻¹

Outputs

heat capacity

s_{m}

How it's solved

Solved with Calorimetry

Given

$m_{m} = 0.20$ kg
$T_{1,m} = 150$ °C
$m_{c} = 0.025$ kg
$T_{1,w} = 27$ °C
$T_{2,w} = 40$ °C
$s_{w} = 4.186$ J g⁻¹ K⁻¹
$m_{w} = 0.150$ kg from “150 cm³ of water”
$s_{c} = s_{w}$ from “water equivalent”
$T_{1,c} = T_{1,w}$ from “copper calorimeter”
$T_{2,c} = T_{2,w}$ from “copper calorimeter”
$T_{2,m} = T_{2,w}$ from “The final temperature”
$Q_{m} = - (Q_{w} + Q_{c})$ from “The final temperature”

Find $s_{m}$ .

Working

$T_{1,m} = 150$ °C $= 423.15$ K
$T_{1,w} = 27$ °C $= 300.15$ K
$T_{2,w} = 40$ °C $= 313.15$ K
$s_{w} = 4.186$ J g⁻¹ K⁻¹ $= 4186$ J kg⁻¹ K⁻¹
heat warms a body(water) $Q_{w} = m_{w} \cdot s_{w} \cdot (T_{2,w} - T_{1,w})$
$Q_{w} = 0.15 \cdot 4186 \cdot (313.15 - 300.15)$
$Q_{w} = 8162.7$ J
water equivalent $s_{c} = s_{w}$
$s_{c} = 4186$
$s_{c} = 4186$ J kg⁻¹ K⁻¹
copper calorimeter (temperature after of calorimeter) $T_{2,c} = T_{2,w}$
$T_{2,c} = 313.15$
$T_{2,c} = 313.15$ K
copper calorimeter (temperature at first of calorimeter) $T_{1,c} = T_{1,w}$
$T_{1,c} = 300.15$
$T_{1,c} = 300.15$ K
heat warms a body(calorimeter) $Q_{c} = m_{c} \cdot s_{c} \cdot (T_{2,c} - T_{1,c})$
$Q_{c} = 0.025 \cdot 4186 \cdot (313.15 - 300.15)$
$Q_{c} = 1360.45$ J
The final temperature (heat to warm of metal) $Q_{m} = - (Q_{w} + Q_{c})$
$Q_{m} = - (8162.7 + 1360.45)$
$Q_{m} = −9523.15$ J
The final temperature (temperature after of metal) $T_{2,m} = T_{2,w}$
$T_{2,m} = 313.15$
$T_{2,m} = 313.15$ K
heat warms a body(metal) $Q_{m} = m_{m} \cdot s_{m} \cdot (T_{2,m} - T_{1,m})$
$(−9523.15) = 0.2 \cdot s_{m} \cdot (313.15 - 423.15)$
$s_{m} = 0.4329$ J g⁻¹ K⁻¹
$s_{m} = 432.870455$ J kg⁻¹ K⁻¹ $= 0.432870455$ J g⁻¹ K⁻¹
$s_{m} \approx 0.43$ J g⁻¹ K⁻¹to 2 significant figures

Conclusion

The heat the metal gives up warms the water and the calorimeter, which acts like a little extra water; that heat over the metal's mass and its fall in temperature is its specific heat.

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_specific_heat_of_a_metal do4  kind :numerical5  describes "In an experiment on the specific heat of a metal, a %{metal_mass} block of the metal at %{hot} is " \6            "dropped in a copper calorimeter (of water equivalent %{equivalent}) containing 150 cm³ of water at " \7            "%{cool}. The final temperature is %{final}. Taking the specific heat capacity of water as " \8            "%{water_heat}, %{heat_capacity}, in J g-1 K-1?"9  given metal_mass: "0.20 kg"10  given hot: "150 °C"11  given equivalent: "0.025 kg"12  given cool: "27 °C"13  given final: "40 °C"14  given water_heat: "4.186 J g-1 K-1"15  asks heat_capacity: "what is the specific heat of the metal"16end

solution.rb

1require_relative "../questions/the_specific_heat_of_a_metal"2require_relative "../science/thermal_properties_of_matter/calorimetry"34solve :the_specific_heat_of_a_metal, using: Calorimetry do5  objects :metal, :calorimeter, :water67  take metal_mass: metal.mass8  take hot: metal.temperature_at_first9  take equivalent: calorimeter.mass10  take cool: water.temperature_at_first11  take final: water.temperature_after12  take water_heat: water.specific_heat_capacity1314  deduce_from "150 cm³ of water", water.mass => "0.150 kg"15  deduce_from "water equivalent", calorimeter.specific_heat_capacity => water.specific_heat_capacity16  deduce_from "copper calorimeter", calorimeter.temperature_at_first => water.temperature_at_first,17                                    calorimeter.temperature_after => water.temperature_after18  deduce_from "The final temperature", metal.temperature_after => water.temperature_after,19                                       metal.heat_to_warm => -(water.heat_to_warm + calorimeter.heat_to_warm)2021  heat_capacity = metal.specific_heat_capacity22  answer heat_capacity:, in: "J g-1 K-1", figures: 22324  conclusion "The heat the metal gives up warms the water and the calorimeter, which acts like a little " \25             "extra water; that heat over the metal's mass and its fall in temperature is its specific heat."26end

playground.rb

1require_relative "../solutions/the_specific_heat_of_a_metal"23play_with :the_specific_heat_of_a_metal do4  title "The Specific Heat of a Metal"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 metal	$Q_{m}$	J	—
mass of metal	$m_{m}$	kg	0 … ∞
specific heat capacity of metal	$s_{m}$	J kg⁻¹ K⁻¹	—
temperature after of metal	$T_{2,m}$	K	—
temperature at first of metal	$T_{1,m}$	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 metal	$Q_{L,m}$	J	—
latent heat of metal	$L_{m}$	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 metal	$Q_{in,m}$	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_{m} = m_{m} \cdot s_{m} \cdot (T_{2,m} - T_{1,m})

heat warms a body(metal)

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,m} = m_{m} \cdot L_{m}

heat changes a phase(metal)

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,m} = Q_{m} + Q_{L,m}

heat taken in warms and changes phase(metal)

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)

s_{c} = s_{w}

water equivalent

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

copper calorimeter (temperature at first of calorimeter)

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

copper calorimeter (temperature after of calorimeter)

T_{2,m} = T_{2,w}

The final temperature (temperature after of metal)

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

The final temperature (heat to warm of metal)