A Hot Copper Block on Ice

Q. A copper block of mass 2.5 kg is heated in a furnace to a temperature of 500 °C and then placed on a large ice block. Specific heat of copper = 0.39 J g⁻¹ K⁻¹; heat of fusion of water = 335 J g⁻¹. What is the maximum amount of ice that can melt, in kg? expects 1.5 kg

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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?

The playground

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}

The solution

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 Ruby

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

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

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

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