[jdicu812]

I tutor chemistry at a college where I am also a student (this is my first semester turtoring chem)  I was posed a question Friday that I just can not find the answer to .  If an amount of an element is at a one temp and is dropped in ice water  and a given amount of the ice melts and ice and water remain how do we find the specific heat of the element in J/g-C with the imformation of
specific heat capacity of ice and water and the heat of fusion (6.02 KJ/mol) of ice to water?

[lavoisier]

The element doesn't undergo phase transitions, it only changes its temperature from T0 to the equilibrium temperature Te (in this case Te is 0°C for the reason explained later).
Therefore the heat transferred is:
Q_element = m_element C_element DT_element = m_element C_element (T0_element - Te)
(I used m for the mass of element, C for its specific heat)

The water/ice mixture is (and stays) biphasic, therefore at standard pressure its temperature is constant and it is the melting point of ice (0°C).
Therefore the heat transferred is the latent heat of fusion of ice at 0°C:
Q_water = L_water n_water
(I used L for the heat of fusion of ice, n for the moles of ice melted)

If the system is adiabatic, all the heat exchange occurs between the element and the water/ice mixture, therefore you can equate the two Q's:

m_element C_element (T0_element - Te) = L_water n_water

In this equation there is only one unknown, C_element, which is what you want to find.

I hope I didn't make any macroscopic mistake with the maths, but this should be the procedure.