At time t = 0, a vessel contains a mixture of 11 kg of water and an unknown mass of ice in equilibrium at 0°C. The temperature of the mixture is measured over a period of an hour, with the following results: During the first 45 min, the mixture remains at 0°C; from 45 min to 60 min, the temperature increases steadily from 0°C to 2.0°C. Neglecting the heat capacity of the vessel, determine the mass of ice that was initially placed in the vessel. Assume a constant power input to the container. kg
Question
At time t = 0, a vessel contains a mixture of 11 kg of water and an unknown mass of ice in equilibrium at 0°C. The temperature of the mixture is measured over a period of an hour, with the following results: During the first 45 min, the mixture remains at 0°C; from 45 min to 60 min, the temperature increases steadily from 0°C to 2.0°C. Neglecting the heat capacity of the vessel, determine the mass of ice that was initially placed in the vessel. Assume a constant power input to the container. kg
Solution
To solve this problem, we need to use the concept of heat transfer and the specific heat capacities of water and ice.
Step 1: Calculate the heat absorbed by the water to increase its temperature from 0°C to 2.0°C.
The formula for heat transfer is Q = mcΔT, where Q is the heat transferred, m is the mass, c is the specific heat capacity, and ΔT is the change in temperature.
For water, the specific heat capacity (c) is 4.18 kJ/kg°C. The mass (m) is 11 kg and the change in temperature (ΔT) is 2°C.
So, Q = 11 kg * 4.18 kJ/kg°C * 2°C = 92.0 kJ.
Step 2: Calculate the heat absorbed by the ice to melt it completely.
The heat of fusion of ice is the amount of energy needed to change 1 kg of ice at 0°C to 1 kg of water at 0°C without changing temperature. This value is 334 kJ/kg.
Since the total heat absorbed by the system is the same as the heat provided, the heat absorbed by the ice to melt it is also 92.0 kJ.
So, using the formula Q = mL, where m is the mass and L is the heat of fusion, we can find the mass of the ice.
92.0 kJ = m * 334 kJ/kg
Solving for m gives m = 92.0 kJ / 334 kJ/kg = 0.275 kg.
So, the mass of ice that was initially placed in the vessel is 0.275 kg.
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