# What is the final equilibrium temperature when 20.00 grams of ice at -15.0°C is mixed with 5.000 grams of steam at 120.0°C?

## The specific heat of ice is 2.100 kJ/kg °C, the heat of fusion for ice at 0°C is 333.7 kJ/kg, the specific heat of water 4.186 kJ/kg °C, the heat of vaporization of water at 100.0°C is 2,256 kJ/kg, and the specific heat of steam is 2.020 kJ/kg °C.

Jul 13, 2018

59.92 color(white)(l)°C

#### Explanation:

The question doesn't state the phase of the mixture at equilibrium. Due to the presence of the latent heat of phase change, it is necessary to start with an assumption of the final state of the mixture and verify the accuracy of the assumption after determining the temperature of the final mixture.

Assuming that the mixture is in its liquid state at its equilibrium temperature $\textcolor{n a v y}{t}$. Under STP color(navy)(t) in (0 °C, 100 °C) between the melting point and boiling point of water at sea level for the condition to hold.

A well-insulated system shall see no heat exchange with the surroundings. Thus energy conserves within the system.

"E"("released") = "E"("absorbed")

for which

• "E"("released") = "E"("condensation") + "E"("cooling")
$\textcolor{w h i t e}{\text{E"("released")) = "L"_color(purple)(v) * m("steam}}$
color(white)("E"("released") =) + c("steam") * m("steam") * (120 °C - 100 °C)
color(white)("E"("released") =) + c("water") * m("steam") * (100 °C -color(navy)( t) °C)

• "E"("absorbed") = bb(-) bb("(")"E"("fusion") + "E"("heating")bb(")")
$\textcolor{w h i t e}{\text{E"("absorbed")) = bb(-) bb("(")-"L"_color(purple)(f) * m("ice}}$
color(white)("E"("absorbed") =) + c("ice") * m("ice") * ((-15) °C - 0 °C)
$\textcolor{w h i t e}{\text{E"("absorbed") =) + c("water") * m("ice") * (0 °C -color(navy)( t) °C)bb(")}}$

"E"("fusion") + "E"("heating") is expected to yield a negative value given the endothermic nature of this process. The pair of brackets and the negative sign enclosing this expression (as shown in bold typeface) ensures a positive value for "E"("absorbed").

Equating "E"("released")  and solving for $t$ gives

t = 59.92 color(white)(l) °C

The value of $t$ is in the range (0 °C, 100 °C) and is consistent with the assumption. Therefore the equilibrium temperature of the mixture would be 59.92 color(white)(L) °C