A container contains 15 liters af water at 96 celsius. A 1 kg of ice cube at -20 celsius is dropped into the water. No heat is lost to the surroundings. What will the final mixing temperature be?
I do not remember heats of fusion and specific heats of ice and water, look them up
assuming all ice melts and water does not freeze:
15 liters = 15 kg of water
ice warms , heat in = (sh ice)(20)(1kg)
ice melts, heat in = (Hfusion)(1 kg)
ice water warms, heat in = (sh water) (T)(1 kg)
then
water cools, heat out = (sh water)(96-T)(15)
so if heat out = heat in
(sh water)(96-T)(15) = (sh ice)(20)(1kg) +(Hfusion)(1 kg)+ (sh water) (T)(1 kg)
s
To find the final mixing temperature, we need to apply the principle of conservation of energy. The heat lost by the hot water will be equal to the heat gained by the ice cube during the mixing process. This can be calculated using the formula:
Q1 = mcΔT1 (Equation 1)
Q2 = mcΔT2 (Equation 2)
where Q1 is the heat lost by hot water, Q2 is the heat gained by the ice cube, m is the mass, c is the specific heat capacity, and ΔT is the change in temperature.
First, let's calculate the heat lost by the water. Given that the water is initially at 96 degrees Celsius and the final temperature is unknown, we can assume the change in temperature (ΔT1) for the water is (96 - Tf), where Tf is the final mixing temperature.
Using Equation 1:
Q1 = (15 kg) (4186 J/kg°C) (96 - Tf)
Next, let's calculate the heat gained by the ice cube. Given that the ice cube is initially at -20 degrees Celsius, and the final temperature is unknown, we can assume the change in temperature (ΔT2) for the ice is (Tf - (-20)).
Using Equation 2:
Q2 = (1 kg) (2108 J/kg°C) (Tf - (-20))
Since no heat is lost to the surroundings, we can set Q1 equal to Q2:
Q1 = Q2
(15 kg) (4186 J/kg°C) (96 - Tf) = (1 kg) (2108 J/kg°C) (Tf - (-20))
Now, we can solve this equation to find the final mixing temperature (Tf).
(15 kg) (4186 J/kg°C) (96 - Tf) = (1 kg) (2108 J/kg°C) (Tf + 20)
(15) (4186) (96 - Tf) = (2108) (Tf + 20)
625560 (96 - Tf) = 2108 (Tf + 20)
60120960 - 625560Tf = 2108Tf + 42160
(625560 + 2108)Tf = 60120960 - 42160
628668Tf = 60078800
Tf ≈ 95.63°C
Therefore, the final mixing temperature will be approximately 95.63 degrees Celsius.