Copper Block On Ice

Part of eval multi-llm branch, agent version multi-llm-servers · 2b11b97

solved Calorimetry

Q. A copper block of 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 a large ice block $T_{1,i} = 0$ °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 $T_{2,c} = 0$ °C, $T_{2,i} = 0$ °C, $Q_{c} = - Q_{L,i}$ , in kg?

Inputs

copper mass

m_{c}

2.5 kg

copper temp

T_{1,c}

500 °C

copper specific heat

s_{c}

0.39 J g⁻¹ K⁻¹

fusion

L_{i}

335 J g⁻¹

Outputs

ice melt

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
$T_{1,i} = 0$ °C from “placed on a large ice block”
$T_{2,c} = 0$ °C from “maximum amount of ice that can melt”
$T_{2,i} = 0$ °C from “maximum amount of ice that can melt”
$Q_{c} = - Q_{L,i}$ from “maximum amount of ice that can melt”

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
$T_{1,i} = 0$ °C $= 273.15$ K
$T_{2,c} = 0$ °C $= 273.15$ K
$T_{2,i} = 0$ °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
maximum amount of ice that can melt $Q_{c} = - Q_{L,i}$
$(−487500) = - Q_{L,i}$
$Q_{L,i} = 487500$ 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

The copper cools to 0 °C, giving up heat that melts ice; the maximum ice melted is when all the copper's heat goes into melting.

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

1question :copper_block_on_ice do2  kind :numerical3  describes "A copper block of %{copper_mass} is heated in a furnace to a temperature of %{copper_temp} and then placed on a large ice block. Specific heat of copper = %{copper_specific_heat}; heat of fusion of water = %{fusion}. %{ice_melt}, in kg?"4  given copper_mass: "2.5 kg"5  given copper_temp: "500 °C"6  given copper_specific_heat: "0.39 J g-1 K-1"7  given fusion: "335 J g-1"8  asks ice_melt: "What is the maximum amount of ice that can melt"9end

solution.rb

1solve :copper_block_on_ice, using: Calorimetry do2  objects :copper, :ice34  take copper_mass: copper.mass5  take copper_temp: copper.temperature_at_first6  take copper_specific_heat: copper.specific_heat_capacity7  take fusion: ice.latent_heat89  deduce_from "placed on a large ice block", ice.temperature_at_first => "0 °C"10  deduce_from "maximum amount of ice that can melt", copper.heat_to_warm => -(ice.heat_to_change_phase), copper.temperature_after => "0 °C", ice.temperature_after => "0 °C"11121314151617  1819  ice_melt = ice.mass20  answer ice_melt:, in: "kg", figures: 22122  conclusion "The copper cools to 0 °C, giving up heat that melts ice; the maximum ice melted is when all the copper's heat goes into melting."23end

playground.rb

1play_with :copper_block_on_ice do2  title "Copper Block On Ice"3end

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

maximum amount of ice that can melt