Specific Heat Capacity of Ice, Help

[Prototype07]

Homework Statement

The following data is obtained:
Weight of ice cube added 20 g
Weight of copper calorimeter 80.5 g
Weight of water in calorimeter 105 g
Initial temp of ice cube -18 deg C
Initial temp of water 23 deg C
Final temp of water 6 deg C
Specific Heat Capacity copper 385 J kg-1K-1
Specific Heat Capacity water 4185 J kg-1K-1
Specific Latent Heat Fusion of Ice 335 kJ kg-1

Calculate a value for the Specific Heat Capacity of ice in J kg-1K-1.

Homework Equations

Q= Mass x Specific Heat Capacity x change in temperature

The Attempt at a Solution

I've tried countless attempts and literally spent hours trying to work this out, all have pointed to dead ends and I really don't know what else to try...
20g = 0.02kg
335kJ = 335000 J
105g = 0.105kg

(0.02kg x 335000J kg-1) + (0.02kg x c) x (6 - -18) = (0.105kg x 4185J kg-1K-1) x (23 - 6)
6700 + (0.02 x c) x 24 = 439.425 x 17
6700 + (0.02 x c) x 24 = 7470.225
0.02 x c x 24 = 770.225
c = 770.225 / (0.02 x 24)
c = 1604.635 J kg-1K-1

Now, I need to work out the SHC of ice which, pre-calculations, we should know it as being around 2000-2100, right? So surely that should mean 1604 is incorrect?
I've really got no idea what to do anymore, should I have included the SHC of copper and the mass of the calorimeter into the equation somewhere too?
Any help would be appreciated!

 

[Gunthi]

You have to consider that your system doesn't exchange heat with the universe, or in other words $$\delta Q_{U}=0$$, so all you have to do now is consider every component in the system that exchanges heat inside the system.

In a general form it would be something like this:

$$\delta Q_U=\delta Q_{Water}+\delta Q_{Ice} + \delta Q_{Copper}=0$$

Remember the system is in thermal equilibrium in the final stage, so you can solve it for the variable you need.

Hope this helps.

Note: $$\delta Q = Q$$.

 

[Prototype07]

Right, I -think- I made some progress
Assuming the specific heat capacity of copper is calculated to be 536.87
Q=m x c x delta t
therefore: Q = 0.0805 x 385 x 17
Q = 526.87

Would it then make sense to add this value to the current value of 1604.635
resulting in 2131.505?
This would seem to then confirm the assumed value of 2100..?

 

[frognose]

Well what I tried with this problem is:

Q_ice = -(Q_water + Q_copper)

Assuming that the copper changes temperature in the same way as the water inside of it, which is a fairly reasonably assumption or the temperatures wouldn't settle in the ways stated, this gives the heat transfer equation as

(m_ice x dT_ice x C_ice) + ( L_ice x m_ice) + (m_ice x C_water x dT'_ice) = -[(m_water x C_water x dT_water) + (m_copper x C_copper x dT_copper)]

Plugging in the numbers and rearranging gives me a value of C_ice at 2194 J/kgK.

P.S. Sorry for the screwy notation. Underscores indicate subscript, d is delta (or change in), m is mass, T is temperature, C is specific heat capacity, L is latent heat of fusion

EDIT: I forgot to mention, dT_ice is 18 degrees because it goes from -18 to 0 to melt, and dT'_ice is 6 because it then goes from 0 to 6 degrees before reaching equilibrium.