Steam Passed Into Water Till Ninety

Q. Steam is passed into 54 g of water at 30 °C till the temperature of the mixture becomes 90 °C. If the latent heat of steam is 536 cal/g, the mass of the mixture will be, in grams? expects 60 g

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Q. Steam $T_{1,s} = 100$ °C, $s_{s} = s_{w}$ is passed into 54 g $m_{w} = 54$ g of water $s_{w} = 1$ cal/g °C at 30 °C $T_{1,w} = 30$ °C till the temperature of the mixture $T_{2,s} = T_{2,w}$ becomes 90 °C $T_{2,w} = 90$ °C. If the latent heat of steam is 536 cal/g $L_{s} = 536$ cal/g, the mass of the mixture will be, in grams?

The playground

Inputs

water mass

m_{w}

54 g

water temperature

T_{1,w}

30 °C

mixed

T_{2,w}

90 °C

latent heat

L_{s}

536 cal/g

Outputs

total

m_{m}

The solution

Solved with Calorimetry

Given

$m_{w} = 54$ g
$T_{1,w} = 30$ °C
$T_{2,w} = 90$ °C
$L_{s} = 536$ cal/g
$s_{w} = 1$ cal/g °C from “of water”
$T_{1,s} = 100$ °C from “Steam”
$s_{s} = s_{w}$ from “Steam”
$T_{2,s} = T_{2,w}$ from “the temperature of the mixture”
$Q_{w} = Q_{L,s} - Q_{s}$ from “Steam is passed into”
$m_{m} = m_{w} + m_{s}$ from “Steam is passed into”

Find $m_{m}$ .

Working

$m_{w} = 54$ g $= 0.054$ kg
$T_{1,w} = 30$ °C $= 303.15$ K
$T_{2,w} = 90$ °C $= 363.15$ K
$L_{s} = 536$ cal/g $= 2.2441248 \times 10^{6}$ J/kg
$s_{w} = 1$ cal/g °C $= 4186.8$ J kg⁻¹ K⁻¹
$T_{1,s} = 100$ °C $= 373.15$ K
Steam $s_{s} = s_{w}$
$s_{s} = 4186.8$
$s_{s} = 4186.8$ J kg⁻¹ K⁻¹
the temperature of the mixture $T_{2,s} = T_{2,w}$
$T_{2,s} = 363.15$
$T_{2,s} = 363.15$ K
heat warms a body(water) $Q_{w} = m_{w} \cdot s_{w} \cdot (T_{2,w} - T_{1,w})$
$Q_{w} = 0.054 \cdot 4186.8 \cdot (363.15 - 303.15)$
$Q_{w} = 13565.232$ J
heat warms a body(steam) $Q_{s} = m_{s} \cdot s_{s} \cdot (T_{2,s} - T_{1,s})$
Steam is passed into (heat to warm of water) $Q_{w} = Q_{L,s} - Q_{s}$
heat changes a phase(steam) $Q_{L,s} = m_{s} \cdot L_{s}$
$m_{s} = 0.00593406593$ kg
Steam is passed into (mass of mixture) $m_{m} = m_{w} + m_{s}$
$m_{m} = 0.054 + 0.00593406593$
$m_{m} = 59.93$ g
$m_{m} = 0.0599340659$ kg $= 59.9340659$ g
$m_{m} \approx 60$ gto 2 significant figures

Conclusion

Each gram of steam gives up its latent heat as it condenses and ten calories more as it cools to 90 °C; the water takes sixty calories a gram to warm, so about six grams of steam join the 54 g.

The Ruby

1require_relative "../physics"23question :steam_passed_into_water_till_ninety do4  kind :numerical5  describes "Steam is passed into %{water_mass} of water at %{water_temperature} till the temperature of the " \6            "mixture becomes %{mixed}. If the latent heat of steam is %{latent_heat}, %{total}, in grams?"7  given water_mass: "54 g"8  given water_temperature: "30 °C"9  given mixed: "90 °C"10  given latent_heat: "536 cal/g"11  asks total: "the mass of the mixture will be"12end

1require_relative "../questions/steam_passed_into_water_till_ninety"2require_relative "../science/thermal_properties_of_matter/calorimetry"34solve :steam_passed_into_water_till_ninety, using: Calorimetry do5  objects :steam, :water, :mixture67  take water_mass: water.mass, water_temperature: water.temperature_at_first, mixed: water.temperature_after8  take latent_heat: steam.latent_heat910  deduce_from "of water", water.specific_heat_capacity => "1 cal/g °C"11  deduce_from "Steam", steam.temperature_at_first => "100 °C", steam.specific_heat_capacity => water.specific_heat_capacity12  deduce_from "the temperature of the mixture", steam.temperature_after => water.temperature_after13  deduce_from "Steam is passed into", water.heat_to_warm => steam.heat_to_change_phase - steam.heat_to_warm,14                                      mixture.mass => water.mass + steam.mass1516  total = mixture.mass17  answer total:, in: "g", figures: 21819  conclusion "Each gram of steam gives up its latent heat as it condenses and ten calories more as it cools to " \20             "90 °C; the water takes sixty calories a gram to warm, so about six grams of steam join the 54 g."21end

1require_relative "../solutions/steam_passed_into_water_till_ninety"23play_with :steam_passed_into_water_till_ninety do4  title "Steam Passed Into Water Till Ninety"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 steam	$Q_{s}$	J	—
mass of steam	$m_{s}$	kg	0 … ∞
specific heat capacity of steam	$s_{s}$	J kg⁻¹ K⁻¹	—
temperature after of steam	$T_{2,s}$	K	—
temperature at first of steam	$T_{1,s}$	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 warm of mixture	$Q_{m}$	J	—
mass of mixture	$m_{m}$	kg	0 … ∞
specific heat capacity of mixture	$s_{m}$	J kg⁻¹ K⁻¹	—
temperature after of mixture	$T_{2,m}$	K	—
temperature at first of mixture	$T_{1,m}$	K	—
heat to change phase of ground	$Q_{L,g}$	J	—
latent heat of ground	$L_{g}$	J/kg	—
heat to change phase of steam	$Q_{L,s}$	J	—
latent heat of steam	$L_{s}$	J/kg	—
heat to change phase of water	$Q_{L,w}$	J	—
latent heat of water	$L_{w}$	J/kg	—
heat to change phase of mixture	$Q_{L,m}$	J	—
latent heat of mixture	$L_{m}$	J/kg	—
heat taken in of ground	$Q_{in,g}$	J	—
heat taken in of steam	$Q_{in,s}$	J	—
heat taken in of water	$Q_{in,w}$	J	—
heat taken in of mixture	$Q_{in,m}$	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_{s} = m_{s} \cdot s_{s} \cdot (T_{2,s} - T_{1,s})

heat warms a body(steam)

Q_{w} = m_{w} \cdot s_{w} \cdot (T_{2,w} - T_{1,w})

heat warms a body(water)

Q_{m} = m_{m} \cdot s_{m} \cdot (T_{2,m} - T_{1,m})

heat warms a body(mixture)

Q_{L,g} = m_{g} \cdot L_{g}

heat changes a phase(ground)

Q_{L,s} = m_{s} \cdot L_{s}

heat changes a phase(steam)

Q_{L,w} = m_{w} \cdot L_{w}

heat changes a phase(water)

Q_{L,m} = m_{m} \cdot L_{m}

heat changes a phase(mixture)

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

heat taken in warms and changes phase(ground)

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

heat taken in warms and changes phase(steam)

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

heat taken in warms and changes phase(water)

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

heat taken in warms and changes phase(mixture)

s_{s} = s_{w}

Steam

T_{2,s} = T_{2,w}

the temperature of the mixture

Q_{w} = Q_{L,s} - Q_{s}

Steam is passed into (heat to warm of water)

m_{m} = m_{w} + m_{s}

Steam is passed into (mass of mixture)