A 23 g block of ice is cooled to -84°C. It is added to 521 g of water in an 99 g copper calorimeter at a temperature of 27°C.
Find the final temperature. The specific heat of copper is 387 J/kg °C and of ice is
2090 J/kg °C. The latent heat of fusion of water is 3.33 × 10% J/kg and its specific heat is 4186 J/kg -°C.
Answer in units of °C.
To find the final temperature, we need to calculate the heat lost by the ice in cooling from -84°C to the final temperature, the heat gained by the copper calorimeter and water in warming up to the final temperature, and the heat gained by the ice in melting.
Step 1: Heat lost by the ice in cooling from -84°C to the final temperature:
q = mcΔT
q = 23g * 2090 J/kg °C * (T - (-84°C))
q = 48070J (T + 84)
Step 2: Heat gained by the copper calorimeter and water in warming up to the final temperature:
q = mcΔT
q = 99g * 387 J/kg °C * (T - 27°C) + 521g * 4186 J/kg °C * (T - 27°C)
q = 387*(T-27)*99 + 4186*(T-27)*521
q = 38133T - 101769 (J)
Step 3: Heat gained by the ice in melting:
q = mLf
q = 23g * 3.33 × 10^5J/kg
q = 7679.1J
Since the total heat lost by the ice is equal to the sum of the heat gained by the copper calorimeter and water and the heat gained by the ice in melting:
48070(T + 84) = 38133T - 101769 + 7679.1
Solving for T:
48070T + 4037480 = 38133T - 101769 + 7679.1
9949T = - 4022770.1
T ≈ -405.08°C
Therefore, the final temperature is approximately -405.08°C.