A Kilogram Of Ice In A Kilogram Of Water At Eighty

Q. 1 kg of ice at 0 °C is mixed with 1 kg of water at 80 °C. The final temperature of the mixture is (Take specific heat of water = 4200 J kg⁻¹ K⁻¹, latent heat of ice = 336 kJ/kg) expects A

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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)

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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 Ruby

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

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

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

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)