Temperature Fusion (heat gain = heat loss)

aodhowain

1. The problem statement, all variables and given/known data
If 10 kilogram of ice at zero degrees celsius is added to 20 kilogram of steam at 100 degrees celsius, what is the temperature of the resulting mixture?


2. Relevant equations
uhm... perhaps how many solutions are required to arrive at the answer?


3. The attempt at a solution
I and some of my classmates arrived at 16.11 degrees celsius but my instructor said its wrong. He said the answer is around 40 degrees celsius. Now, he gave it as an assignment to us to find the correct solution.

Borek

In no way that's advanced physics.

Show how you got 16 deg answer. What equations have you used?

aodhowain

Q_E = Q_A
Heat Evolve = Heat Absorb
M_EC_Edelta T_E = M_AC_Adelta T_A

Borek

So far so good.

That's not all.

I have asked you to show how you got the wrong answer.

aodhowain

M_EC_Edelta T_E = M_AC_Adelta T_A
(10 kg)(0.50 cal/kg*C)(tf - 0C) = (2 kg)(0.48 cal/kg*C)(100C - tf)
5 cal/C * tf = 96 cal - 0.96 cal/C * tf
5.96 cal/C * tf = 96 cal
tf = (96 cal)/(0.96 cal/C)
tf = 16.11 C

Borek

As I told you - you are missing part of the equation. Two parts, to be precise. What you did would be OK for a mixture of water at 0 deg C and water at 100 deg C - but you start with ice and steam.

fluidistic

I agree. A bit more help, heat from hot water has to be transfered to the ice to melt it, even though the temperature does not change. Quantify this heat.

aodhowain

I think I got the solution to arrive at 40 degrees celsius. Please verify if my answer is correct.

Q out of steam = Q into ice (for c I use kcal/kgC but You can just as easily use J/kgC)

Q out = m*Lv + m*c*deltaT = m*(Lv + c*deltaT) = 2.0kg*(540kcal/kg + 1.0kcal/kg/C*(100-Tf))

Q in = m*Lf +m*c*deltaT = m*(Lf =c*deltaT) = 10kg*(80kcal/kg + 1.0kcal/kgC*(Tf -0))

Setting these equal gives 2.0*540 +2.0*100 -2.0*Tf = 10*80 + 10*Tf

Solving (10+2)*Tf = 2*540 +2*100-10*80 = 480 so Tf = 480/12 = 40degC

Borek

I have just skimmed, looks reasonably.

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