I try to solve a thermodynamics problem on heat transfer

[Salmone]

Homework Statement

Find the thermal capacity of a container with ice and water

Relevant Equations

$Q=C \Delta T$

I have some doubts about a thermodynamics exercise I did-I'll write down the text and show how I tried to solve it to see if I reasoned it out right.

In a container of thermal capacity $C$ containing $1.0 kg$ of ice at a temperature of $-4.0 °C$, a mass of $3.0 kg$ of water is poured into it at a temperature of $40.0 °C$. There is no heat exchange with the surroundings. The specific heats of water and ice are $4186 J/kgK$ and $2090 J/kgK$, respectively, while the latent heat of fusion of ice is $3.33 × 105 J/kg$.

The exercise asks to calculate the thermal capacity $C$ that the container must have if we want that at the equilibrium temperature just $\frac{1}{3}$ of the mass of the ice is melted.

I reasoned like this:

at the equilibrium temperature if the ice is not completely melted, it means the equilibrium temperature is 0 degrees. This means that the 3 kilograms of water will give up a total amount of heat equal to $Q=cm \Delta T=(4186)(3)(40)=502320J$. This amount of heat will be absorbed initially by the ice to be brought to zero degrees and later to melt $\frac{1}{3}$ of the mass.

The heat absorbed by the ice for reach $0$ degrees can be calculated as: $Q=cm(4 °C)=8360J$.

The heat absorbed by the ice in order to melt $\frac{1}{3}$ of the mass can be calculated with the ice latent heat of fusion as: $\frac{3.33 x 10^5}{3}=111000J$
.
At this point, I can calculate the amount of total heat that the ice absorbs, which is the sum of $8360J$ and $111000J$ which is equal to $119360J$, I can subtract it from the amount of total heat that the water gives up, and I get the amount of heat given up by the water that is not absorbed by the ice and must therefore be absorbed by the container. This quantity is $502320J-119360J=382960J$.

I imagined that the initial temperature of the container was -4 degrees so this must absorb that amount of heat and have a heat capacity that allows it to raise its T by 4 degrees, absorbing that amount of heat.

From here I can calculate the final $C=\frac{382960J}{4}=95740 J/K$

We are at the end of the problem, is this the right way of reasoning?

 

[Chestermiller]

Please show more details of your equations and calculations. Your answer looks about right.

 

[TSny]

Your reasoning looks good to me. Your calculations also look correct except for this subtraction:

$502320J-119360J=383785J$.

 

[Salmone]

Your reasoning looks good to me. Your calculations also look correct except for this subtraction:

Thank you, corrected.