A Hot Copper Block on Ice

solved Calorimetry

Q. A copper block of mass 2.5 kg $m_{c} = 2.5$ kg is heated in a furnace to a temperature of 500 °C $T_{1,c} = 500$ °C and then placed on $Q_{L,i} = - Q_{c}$ a large ice block $T_{2,c} = 0$ °C. Specific heat of copper = 0.39 J g⁻¹ K⁻¹ $s_{c} = 0.39$ J g⁻¹ K⁻¹; heat of fusion of water = 335 J g⁻¹ $L_{i} = 335$ J g⁻¹. What is the maximum amount of ice that can melt, in kg?

Inputs

mass

m_{c}

2.5 kg

hot

T_{1,c}

500 °C

copper heat

s_{c}

0.39 J g⁻¹ K⁻¹

fusion

L_{i}

335 J g⁻¹

Outputs

melted

m_{i}

How it's solved

Solved with Calorimetry

Given

$m_{c} = 2.5$ kg
$T_{1,c} = 500$ °C
$s_{c} = 0.39$ J g⁻¹ K⁻¹
$L_{i} = 335$ J g⁻¹
$T_{2,c} = 0$ °C from “a large ice block”
$Q_{L,i} = - Q_{c}$ from “placed on”

Find $m_{i}$ .

Working

$T_{1,c} = 500$ °C $= 773.15$ K
$s_{c} = 0.39$ J g⁻¹ K⁻¹ $= 390$ J kg⁻¹ K⁻¹
$L_{i} = 335$ J g⁻¹ $= 335000$ J/kg
$T_{2,c} = 0$ °C $= 273.15$ K
heat warms a body(copper) $Q_{c} = m_{c} \cdot s_{c} \cdot (T_{2,c} - T_{1,c})$
$Q_{c} = 2.5 \cdot 390 \cdot (273.15 - 773.15)$
$Q_{c} = −487500$ J
placed on $Q_{L,i} = - Q_{c}$
$Q_{L,i} = - (−487500)$
$Q_{L,i} = 487500$ J
heat changes a phase(ice) $Q_{L,i} = m_{i} \cdot L_{i}$
$487500 = m_{i} \cdot 335000$
$m_{i} = 1.455$ kg
$m_{i} \approx 1.5$ kgto 2 significant figures

Conclusion

At most, the copper cools all the way to the ice's melting point, and all the heat it gives up goes into melting ice.

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 :a_hot_copper_block_on_ice do4  kind :numerical5  describes "A copper block of mass %{mass} is heated in a furnace to a temperature of %{hot} and then placed on a " \6            "large ice block. Specific heat of copper = %{copper_heat}; heat of fusion of water = %{fusion}. " \7            "%{melted}, in kg?"8  given mass: "2.5 kg"9  given hot: "500 °C"10  given copper_heat: "0.39 J g-1 K-1"11  given fusion: "335 J g-1"12  asks melted: "What is the maximum amount of ice that can melt"13end

solution.rb

1require_relative "../questions/a_hot_copper_block_on_ice"2require_relative "../science/thermal_properties_of_matter/calorimetry"34solve :a_hot_copper_block_on_ice, using: Calorimetry do5  objects :copper, :ice67  take mass: copper.mass8  take hot: copper.temperature_at_first9  take copper_heat: copper.specific_heat_capacity10  take fusion: ice.latent_heat1112  deduce_from "a large ice block", copper.temperature_after => "0 °C"13  deduce_from "placed on", ice.heat_to_change_phase => -copper.heat_to_warm1415  melted = ice.mass16  answer melted:, in: "kg", figures: 21718  conclusion "At most, the copper cools all the way to the ice's melting point, and all the heat it gives " \19             "up goes into melting ice."20end

playground.rb

1require_relative "../solutions/a_hot_copper_block_on_ice"23play_with :a_hot_copper_block_on_ice do4  title "A Hot Copper Block on Ice"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 copper	$Q_{c}$	J	—
mass of copper	$m_{c}$	kg	0 … ∞
specific heat capacity of copper	$s_{c}$	J kg⁻¹ K⁻¹	—
temperature after of copper	$T_{2,c}$	K	—
temperature at first of copper	$T_{1,c}$	K	—
heat to warm of ice	$Q_{i}$	J	—
mass of ice	$m_{i}$	kg	0 … ∞
specific heat capacity of ice	$s_{i}$	J kg⁻¹ K⁻¹	—
temperature after of ice	$T_{2,i}$	K	—
temperature at first of ice	$T_{1,i}$	K	—
heat to change phase of ground	$Q_{L,g}$	J	—
latent heat of ground	$L_{g}$	J/kg	—
heat to change phase of copper	$Q_{L,c}$	J	—
latent heat of copper	$L_{c}$	J/kg	—
heat to change phase of ice	$Q_{L,i}$	J	—
latent heat of ice	$L_{i}$	J/kg	—
heat taken in of ground	$Q_{in,g}$	J	—
heat taken in of copper	$Q_{in,c}$	J	—
heat taken in of ice	$Q_{in,i}$	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_{c} = m_{c} \cdot s_{c} \cdot (T_{2,c} - T_{1,c})

heat warms a body(copper)

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

heat warms a body(ice)

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

heat changes a phase(ground)

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

heat changes a phase(copper)

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

heat changes a phase(ice)

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

heat taken in warms and changes phase(ground)

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

heat taken in warms and changes phase(copper)

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

heat taken in warms and changes phase(ice)

Q_{L,i} = - Q_{c}

placed on