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A calorimeter contains 353 ml of water at 57 degrees C and 47 g of ice at 0 degrees C.

Find the final temperature of the system .

The specific heat of water is 1 cal/g* C and the latent heat of fusion of water is 3.33*10^5 J/kg.

Answer in units of degree C

Development of formulas:

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If ice is fully melted, the following formula applies:
Teq = (m_i*(T_m – h_f/c_w) + m_w*Tw1)/(m_w + m_i)

If ice only partially melts, the following formula applies:
m_i = m_w*c_w*(Tw1 – Tm)/h_f

And in such a situation, Teq = Tm because it achieves equilibrium in a two-phase coexistence.

Common data:
Tw1:=57 C; m_w:=353 grams; c_w:=1 cal/gram-C; Tm:=0 C; h_f:=79.6 cal/gram;

Data if ice fully melts:
m_i:=47 grams;

Result if ice fully melts:
Teq = 40.94 Celsius

Result if ice only partially melts:
Teq = 0 C
m_i = 252.8 grams, which is more ice than there existed.

Conclusion:
Teq = 40.94 Celsius and ice fully melts