Ice in water question - what is final temperature?

[Tyler H]

Homework Statement

An ice cube melts in a 10oC glass of water (mass of water is 225g). If
the ice is allowed to melt completely, what will the final temperature
of the water be? (we're told that the mass of ice can be ignored)

Ti = 10oC
Latent heat of fusion = Lf = 334000 J/kg
specific heat capacity of water = C = 4180 J/kg/oC
mass of water = Mw = 0.225kg

Homework Equations

:[/B]
QH = sensible heat = mC(T2-T1)
QE = latent heat = m*Lf

The Attempt at a Solution

I have figured out an equation that requires the mass of ice to be
included, however we've not been given the mass of ice (and are told
to ignore it), so I'm not sure how to create an equation that ignores
the mass of the ice.

Since we can assume this to be a closed system, the heat lost by the
water will equal the latent heat going into melting the ice and
heating this resultant water to the final temperatureQ1 = energy req'd to melt ice = Lf*Mice
Q2 = energy req'd to warm resultant water to final temperature = Mice
*C *(Tf - 0)

Q3 = energy lost by water in the glass = Mwater * C * (Ti - Tf)

Q1 + Q2 = Q3

(Lf*Mice) + (Mice * C * (Tf - 0)) = Mwater * C * (10 - Tf)

 

[Chestermiller]

It looks like you set it up correctly. Obviously, for the problem as stated, the amount of ice matters. Also, the initial temperature of the ice matters. If there were still ice and water present when the system had re-equilibrated, then you could say that the final temperature is 0 C.

 

[Tyler H]

Hi There! Thank you for the tip. Looks like all the ice melts in this case, and we are left with slightly cooler water (answer is Tfinal =8.6oC). Not sure how to solve this without the mass of the ice though

 

[Nathanael]

Not sure how to solve this without the mass of the ice though

You can't. The final temperature would be quite different if it was, say, 100 grams of ice versus 1 gram of ice. I'm not sure why you would be told to ignore the mass of the ice.

 

[Mister T]

(Lf*Mice) + (Mice * C * (Tf - 0)) = Mwater * C * (10 - Tf)

Set M_ice equal to zero and solve for T_f.

 

[Tyler H]

That would give me the initial temperature of the water (10oC). The answer for Tf in the book shows 8.6oC. Worth the try though!

 

[haruspex]

Are you quite sure you have the wording right? Could it be that the ice is given enough time to melt completely, but doesn't? (But even then, you would need to be told that the water does not freeze completely.)

 

[Mister T]

If you had 4 grams of ice at 0 ^oC then you'd get a final temperature of 8.6 ^oC, ignoring the mass of the melted ice.

 

[Borek]

Question is incomplete, simple mistake by whoever prepared it. However, knowing the final answer you can estimate mass of the ice (solving similar, but a different question), as @Mister T did (not that I checked the result).

If the mass of ice is sufficiently low compared to the mass of water, ignoring it won't change substantially the numerical value of the final answer. You don't have data accurate enough for more than two significant digits in the final answer (you are limited by the initial temperature given as 10°C), and as the problem is quite linear the error from ignoring mass of the ice in the final mass of water is around 4/229 - or 2%. Apparently it was considered acceptable.

 

[Mister T]

If the problem were stated this way:

A 4.0 gram ice cube melts in a 10 ^oC glass of water (mass of water is 225 g). If
the ice is allowed to melt completely, what will the final temperature
of the water be? (we're told that the mass of ice can be ignored)

the answer would be 8.6 °C. That one omission is likely the "simple mistake"made by whoever wrote or read the problem.

 

[jbriggs444]

That one omission is likely the "simple mistake"made by whoever wrote or read the problem.

Makes sense. The "ignore the mass of the ice" could then be understood as "ignore the heat required to raise the 4 grams of melted ice from 0 degrees to the final temperature of the original 225 grams of water".

 

[Mister T]

Or simply, the mass of the melted ice can be ignored. Although it need not be mentioned because, as Borek pointed out, it doesn't have a significant effect on the answer. The fact that it was a comment "told" to the students rather than included in the statement of the problem is another hint that the omission of the mass of the ice cube being 4 grams was the mistake. It may be that the instructor left it out on purpose, thinking mistakenly (for that reason) that it didn't need to be included!