[OTI]

The background goes like this:

Jello making is adding a packet of sugar and gelatin to 4 cups of near boiling water and stirring till the gelatin dissolves. Then, the mixture is put in the fridge which would take hours.

Home econ majors know how to cool them faster. They add 3 cups of water (0.795 L) at 80.°C with stirring. (Added with gelatin and sugar stuff but those variables aren't included in the solving of the problem) Then, the mixture is cooled by adding 250. grams of ice (at 0.0°C) and stirring till they melt. The mixture takes one hour to cool in the fridge. (Don't know if this part is relevant or not)

Key Info:

0.795 L of water or 795 g
Water's starting temp even after sugar and gelatin = 80.°C
250. g of ice
Ice's starting temp = 0.0

The first problem is this:
What temp is the mixture at when the ice has finished melting? ΔH_fus = 334 J/g

The second problem is this:
What is the final temp of the mixture after it comes to thermal equilibrium?

TBH, I have only a little idea of where to start.

I learned several equations that should lead me to the right answers, but I don't know which one to apply them too.

I learned the -q_surr = q_syst and vice versa.

I also learned the q=mcΔT thing.

I also learned another equation that I'm not that sure about. It goes kinda like this (Though I'll probably make some errors)

(m of something)(c of something) + (m of something)(ΔT)(c of something) = -[(m of something)(ΔT)(c of something)]
Would be glad for an explanation for this too?

note:
m= mass
c= heat capacity
T = temp

One of the troubles that I have with the problem is with the "when the ice has finished melting" part. I'm pretty sure none of the equations I've named so far can describe that. Also, like I said before, I don't understand the 3rd equation I named and it wasn't in the textbook, though the third equation is probably the key to the second problem.

That's my struggles....
Yup. Would be glad for help.

[Borek]

Then, the mixture is put in the fridge which would take hours.

Then, the mixture is cooled by adding 250. grams of ice (at 0.0°C) and stirring till they melt. The mixture takes one hour to cool in the fridge.

While adding ice will definitely speed up the cooling, finding the time required to cool something is in general not a trivial task. Actually it is quite difficult.

The first problem is this:
What temp is the mixture at when the ice has finished melting? ΔH_fus = 334 J/g

The second problem is this:
What is the final temp of the mixture after it comes to thermal equilibrium?

These are classical calorimetry problems, see if these pages don't help:

http://www.physicsclassroom.com/class/thermalP/Lesson-2/Calorimeters-and-Calorimetry

http://chemwiki.ucdavis.edu/Physical_Chemistry/Thermodynamics/Calorimetry/Constant_Pressure_Calorimetry

I learned several equations that should lead me to the right answers, but I don't know which one to apply them too.

You are on the right track in general, you need a mix of these equations. I believe you are missing one important and general equation describing heat balance:

heat lost = heat gained

which is a basis for all calorimetry calculations. It is just a matter of finding what loses and what gains energy.

(m of something)(c of something) + (m of something)(ΔT)(c of something) = -[(m of something)(ΔT)(c of something)]

This is easily derived from the above and equations you listed earlier for a specific case, for example two objects being cooled by putting them into cold water.

One of the troubles that I have with the problem is with the "when the ice has finished melting" part. I'm pretty sure none of the equations I've named so far can describe that.

Yes, you need another formula for so called latent heat. This one is actually quite simple

q=mc_l

where m is mass (as in all other cases) and c_l is the latent heat - it can be a heat of melting or heat of boiling. It is called "latent" as there is no accompanying temperature change (compare to q=mcΔT).

[OTI]

Ty so much! I kinda get it a bit more now. For some reason, I can't find this latent heat thing anywhere in my textbook.