A Kilogram Of Ice In A Kilogram Of Water At Eighty

solved Calorimetry

Q. 1 kg $m_{i} = 1$ kg of ice at 0 °C $T_{1,i} = 0$ °C is mixed with $s_{i} = s_{w}$ , $T_{2,i} = T_{2,w}$ , $Q_{in,i} = - Q_{in,w}$ 1 kg $m_{w} = 1$ kg of water at $Q_{L,w} = 0$ J 80 °C $T_{1,w} = 80$ °C. The final temperature of the mixture is (Take specific heat of water = 4200 J kg⁻¹ K⁻¹ $s_{w} = 4200$ J kg⁻¹ K⁻¹, latent heat of ice = 336 kJ/kg $L_{i} = 336$ kJ/kg)

How it's solved

Solved with Calorimetry

Given

$m_{i} = 1$ kg
$T_{1,i} = 0$ °C
$m_{w} = 1$ kg
$T_{1,w} = 80$ °C
$s_{w} = 4200$ J kg⁻¹ K⁻¹
$L_{i} = 336$ kJ/kg
$Q_{L,w} = 0$ J from “of water at”
$s_{i} = s_{w}$ from “is mixed with”
$T_{2,i} = T_{2,w}$ from “is mixed with”
$Q_{in,i} = - Q_{in,w}$ from “is mixed with”

Find $T_{2,w}$ .

Working

$T_{1,i} = 0$ °C $= 273.15$ K
$T_{1,w} = 80$ °C $= 353.15$ K
$L_{i} = 336$ kJ/kg $= 336000$ J/kg
heat changes a phase(ice) $Q_{L,i} = m_{i} \cdot L_{i}$
$Q_{L,i} = 1 \cdot 336000$
$Q_{L,i} = 336000$ J
is mixed with (specific heat capacity of ice) $s_{i} = s_{w}$
$s_{i} = 4200$
$s_{i} = 4200$ J kg⁻¹ K⁻¹
heat taken in warms and changes phase(ice) $Q_{in,i} = Q_{i} + Q_{L,i}$
is mixed with (heat taken in of ice) $Q_{in,i} = - Q_{in,w}$
heat taken in warms and changes phase(water) $Q_{in,w} = Q_{w} + Q_{L,w}$
heat warms a body(water) $Q_{w} = m_{w} \cdot s_{w} \cdot (T_{2,w} - T_{1,w})$
heat warms a body(ice) $Q_{i} = m_{i} \cdot s_{i} \cdot (T_{2,i} - T_{1,i})$
is mixed with (temperature after of ice) $T_{2,i} = T_{2,w}$
$T_{2,w} = 0$ °C
$T_{2,w} = 273.15$ K $= 0$ °C

Conclusion

Cooling the water from 80 °C to 0 °C gives up 336 kJ, just what melting the ice takes, so nothing is left to warm the mixture and it stays at 0 °C.

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_kilogram_of_ice_in_a_kilogram_of_water_at_eighty do4  kind :scq5  describes "%{ice_mass} of ice at %{ice_temperature} is mixed with %{water_mass} of water at %{water_temperature}. " \6            "The final temperature of the mixture is (Take specific heat of water = %{water_heat}, latent heat of " \7            "ice = %{fusion})"8  given ice_mass: "1 kg"9  given ice_temperature: "0 °C"10  given water_mass: "1 kg"11  given water_temperature: "80 °C"12  given water_heat: "4200 J kg-1 K-1"13  given fusion: "336 kJ/kg"1415  option "0 °C"16  option "53.3 °C"17  option "40 °C"18  option "60 °C"19end

solution.rb

1require_relative "../questions/a_kilogram_of_ice_in_a_kilogram_of_water_at_eighty"2require_relative "../science/thermal_properties_of_matter/calorimetry"34solve :a_kilogram_of_ice_in_a_kilogram_of_water_at_eighty, using: Calorimetry do5  objects :ice, :water67  take ice_mass: ice.mass, ice_temperature: ice.temperature_at_first8  take water_mass: water.mass, water_temperature: water.temperature_at_first9  take water_heat: water.specific_heat_capacity, fusion: ice.latent_heat1011  deduce_from "of water at", water.heat_to_change_phase => "0 J"12  deduce_from "is mixed with", ice.specific_heat_capacity => water.specific_heat_capacity,13                               ice.temperature_after => water.temperature_after,14                               ice.heat_taken_in => -water.heat_taken_in1516  answer find_value(water.temperature_after, in: "°C")1718  conclusion "Cooling the water from 80 °C to 0 °C gives up 336 kJ, just what melting the ice takes, so " \19             "nothing is left to warm the mixture and it stays at 0 °C."20end

playground.rb

1require_relative "../solutions/a_kilogram_of_ice_in_a_kilogram_of_water_at_eighty"23play_with :a_kilogram_of_ice_in_a_kilogram_of_water_at_eighty do4  title "A Kilogram Of Ice In A Kilogram Of Water At Eighty"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 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 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 ice	$Q_{L,i}$	J	—
latent heat of ice	$L_{i}$	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 ice	$Q_{in,i}$	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_{i} = m_{i} \cdot s_{i} \cdot (T_{2,i} - T_{1,i})

heat warms a body(ice)

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

heat changes a phase(ice)

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

heat taken in warms and changes phase(ice)

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

heat taken in warms and changes phase(water)

s_{i} = s_{w}

is mixed with (specific heat capacity of ice)

T_{2,i} = T_{2,w}

is mixed with (temperature after of ice)

Q_{in,i} = - Q_{in,w}

is mixed with (heat taken in of ice)