Fifty Grams Of Ice In Water At Eighty

solved Calorimetry

Q. 50 g $m_{i} = 50$ g of ice at $L_{i} = 80$ cal/g, $s_{i} = s_{w}$ 0 °C $T_{1,i} = 0$ °C is mixed with 50 g $m_{w} = 50$ g of water at $s_{w} = 1$ cal/g °C, $Q_{L,w} = 0$ J 80 °C $T_{1,w} = 80$ °C, final temperature of mixture $T_{2,i} = T_{2,w}$ , $Q_{in,i} = - Q_{in,w}$ will be

How it's solved

Solved with Calorimetry

Given

$m_{i} = 50$ g
$T_{1,i} = 0$ °C
$m_{w} = 50$ g
$T_{1,w} = 80$ °C
$s_{w} = 1$ cal/g °C from “of water at”
$Q_{L,w} = 0$ J from “of water at”
$L_{i} = 80$ cal/g from “of ice at”
$s_{i} = s_{w}$ from “of ice at”
$T_{2,i} = T_{2,w}$ from “final temperature of mixture”
$Q_{in,i} = - Q_{in,w}$ from “final temperature of mixture”

Find $T_{2,w}$ .

Working

$m_{i} = 50$ g $= 0.05$ kg
$T_{1,i} = 0$ °C $= 273.15$ K
$m_{w} = 50$ g $= 0.05$ kg
$T_{1,w} = 80$ °C $= 353.15$ K
$s_{w} = 1$ cal/g °C $= 4186.8$ J kg⁻¹ K⁻¹
$L_{i} = 80$ cal/g $= 334944$ J/kg
heat changes a phase(ice) $Q_{L,i} = m_{i} \cdot L_{i}$
$Q_{L,i} = 0.05 \cdot 334944$
$Q_{L,i} = 16747.2$ J
of ice at $s_{i} = s_{w}$
$s_{i} = 4186.8$
$s_{i} = 4186.8$ J kg⁻¹ K⁻¹
heat taken in warms and changes phase(ice) $Q_{in,i} = Q_{i} + Q_{L,i}$
final temperature of mixture (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) $Q_{i} = m_{i} \cdot s_{i} \cdot (T_{2,i} - T_{1,i})$
final temperature of mixture (temperature after of ice) $T_{2,i} = T_{2,w}$
$T_{2,w} = 0$ °C
$T_{2,w} = 273.15$ K $= 0$ °C

Conclusion

Each gram of water cooling 80 degrees gives the 80 cal a gram of ice needs to melt, and the masses are equal, so the ice just melts and the mixture 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 :fifty_grams_of_ice_in_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            "final temperature of mixture will be"7  given ice_mass: "50 g"8  given ice_temperature: "0 °C"9  given water_mass: "50 g"10  given water_temperature: "80 °C"1112  option "0 °C"13  option "40 °C"14  option "30 °C"15  option "4 °C"16end

solution.rb

1require_relative "../questions/fifty_grams_of_ice_in_water_at_eighty"2require_relative "../science/thermal_properties_of_matter/calorimetry"34solve :fifty_grams_of_ice_in_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_first910  deduce_from "of water at", water.specific_heat_capacity => "1 cal/g °C", water.heat_to_change_phase => "0 J"11  deduce_from "of ice at", ice.latent_heat => "80 cal/g", ice.specific_heat_capacity => water.specific_heat_capacity12  deduce_from "final temperature of mixture", ice.temperature_after => water.temperature_after,13                                              ice.heat_taken_in => -water.heat_taken_in1415  answer find_value(water.temperature_after, in: "°C")1617  conclusion "Each gram of water cooling 80 degrees gives the 80 cal a gram of ice needs to melt, and the masses " \18             "are equal, so the ice just melts and the mixture stays at 0 °C."19end

playground.rb

1require_relative "../solutions/fifty_grams_of_ice_in_water_at_eighty"23play_with :fifty_grams_of_ice_in_water_at_eighty do4  title "Fifty Grams Of Ice In 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}

of ice at

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

final temperature of mixture (temperature after of ice)

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

final temperature of mixture (heat taken in of ice)