The Specific Heat of a Metal

Q. In an experiment on the specific heat of a metal, a 0.20 kg block of the metal at 150 °C is dropped in a copper calorimeter (of water equivalent 0.025 kg) containing 150 cm³ of water at 27 °C. The final temperature is 40 °C. Taking the specific heat capacity of water as 4.186 J g⁻¹ K⁻¹, what is the specific heat of the metal, in J g-1 K-1? expects 0.43 J g⁻¹ K⁻¹

What the draft establishes

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The web of concepts

given read from the words equation asked

The equations it used

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Check an answer

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?

The playground

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}

The solution

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 Ruby

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

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

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

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)