Ice added to Water (heats of rxn)

[aqwsde]

Homework Statement

If 24g of ice at -5C is added to a isolated dewar containing 50 mL of water at 10C. What would the temperature of the system be at equilibrium if the heat capacity can be ignored and the system is completely isolated.

Homework Equations

sp ht H2O (l) - 4.18 J/(gC)
sp ht H2O(s) = 2.01 J/(gC)

dHfus = 6.01 kJ/mol
Density H2O = 1g/ml

The Attempt at a Solution

I wasn't sure if the water would start to freeze or the ice would melt or if that is necessary to know. I approached the problem with the assumption that the ice melts and also that the transfer of heat from water to ice during melting is equal and opposite the heat transfer out of the surrounding and thus the terms cancel (I don't know if I can make this assumption). With that in mind

qrxn = -qsurr

qrxn = 24 *2.01 *(0 - (-5)) + 24*4.18*(Tf-0)

-qsurr = -50*4.18*(Tf-10)

Solving for Tf = 5.98C

My main question is if my assumptions are valid and also how would I know without it being said if the ice if going to melt of the water is going to freeze, or does it make a difference?

 

[mgb_phys]

Quick check is how much energy is available before the water gets down to 0C (mcT)
If this is more than the energy to melt the ice then it will be water.

 

[nlightner]

Your approach to the problem is correct. The key assumption you made is that the transfer of heat from water to ice during melting is equal and opposite the heat transfer out of the surrounding. This assumption is valid because the system is completely isolated and there is no external heat source or sink. Therefore, the heat transfer between the water and ice will balance out and the temperature of the system will reach equilibrium.

As for the question of whether the ice will melt or the water will freeze, it is not necessary to know beforehand. In this case, the initial temperatures of the water and ice are such that the final temperature will be between the two. If the initial temperature of the ice was lower than the water, then it would have melted completely. If the initial temperature of the water was lower than the ice, then it would have frozen completely. However, since the initial temperatures are close, both melting and freezing will occur, but the final temperature will be somewhere in between.

Overall, your solution is correct and your assumptions are valid. Keep in mind that this is a simplified approach and in a real-life scenario, other factors such as pressure and non-ideal behavior of the substances may affect the final temperature.