To solve this problem, we need to determine the amount of energy required to cool the water from its initial temperature of 0°C to the final temperature of -19.2°C. We can calculate this energy by considering the heat transfer and the change in temperature.
The first step is to find the energy required to cool the water from 0°C to the freezing point of water (0°C). We can use the specific heat formula:
Q1 = mass * specific heat of water * change in temperature
Where:
- Q1 is the energy required to cool the water from 0°C to the freezing point
- mass is the mass of water (4.77 kg)
- specific heat of water is 4.186 kJ/kg·°C
- change in temperature is the difference between the initial temperature (0°C) and the freezing point (0°C)
Q1 = 4.77 kg * 4.186 kJ/kg·°C * (0°C - 0°C)
Since the change in temperature is zero, there is no energy required to cool the water from 0°C to the freezing point.
The second step is to find the energy required to freeze the water. When water freezes, it releases a certain amount of energy called the latent heat of fusion. We can calculate this energy using the formula:
Q2 = mass * latent heat of fusion of water
Where:
- Q2 is the energy required to freeze the water
- mass is the mass of water (4.77 kg)
- latent heat of fusion of water is the heat released when 1 kg of water freezes, which is 334 kJ/kg
Q2 = 4.77 kg * 334 kJ/kg
Next, we need to find the energy required to lower the temperature of the frozen water from its freezing point to the final temperature of -19.2°C. Since the water is already in solid form, we can use the specific heat of ice in this calculation.
Q3 = mass * specific heat of ice * change in temperature
Where:
- Q3 is the energy required to lower the temperature of the frozen water
- mass is the mass of water (4.77 kg)
- specific heat of ice is 2.09 kJ/kg·°C
- change in temperature is the difference between the freezing point (0°C) and the final temperature (-19.2°C)
Q3 = 4.77 kg * 2.09 kJ/kg·°C * (-19.2°C - 0°C)
Finally, the total energy required to cool the water to -19.2°C is obtained by adding the three calculated quantities:
Total Energy = Q1 + Q2 + Q3
Now you can substitute the values into the formulas and calculate the total energy required in kJ.