If 0.055 kg of ice at 0 ∘C is added to 0.330 kg of water at 32 ∘C in a 0.100-kg aluminum calorimeter cup, what is the final temperature of the water?
To find the final temperature of the water, we can use the principle of conservation of energy. The heat lost by the water in cooling down must be equal to the heat gained by the ice in melting and warming up, as well as the heat gained by the aluminum calorimeter cup.
The first step is to determine the initial heat content of each component. We can use the specific heat capacity (c) and the initial temperature to calculate the initial heat content. The specific heat capacity of water is approximately 4.18 J/g°C, and the specific heat capacity of aluminum is approximately 0.90 J/g°C.
For the water, we have:
Initial heat content of water = mass of water × specific heat capacity of water × initial temperature of water
Initial heat content of water = 0.330 kg × 4.18 J/g°C × 32°C
For the ice, we need to account for the energy required to raise the temperature of the ice to 0°C, and then the energy required to melt the ice. The specific heat capacity of ice is approximately 2.09 J/g°C, and the heat of fusion for ice is 334 J/g.
For the ice, we have:
Initial heat content of ice = mass of ice × (specific heat capacity of ice × temperature change from 0°C to melting point + heat of fusion)
But since the ice is initially at 0°C, the temperature change is simply the melting point of ice (0°C):
Initial heat content of ice = 0.055 kg × (2.09 J/g°C × 0°C + 334 J/g)
Finally, for the aluminum calorimeter cup, we can assume that its initial temperature is the same as the initial temperature of the water (32°C) since metal is a good conductor of heat. The specific heat capacity of aluminum is approximately 0.90 J/g°C.
For the aluminum calorimeter cup, we have:
Initial heat content of aluminum calorimeter cup = mass of aluminum calorimeter cup × specific heat capacity of aluminum × initial temperature of aluminum calorimeter cup
Initial heat content of aluminum calorimeter cup = 0.100 kg × 0.90 J/g°C × 32°C
Now, we can set up the energy conservation equation:
Heat lost by water = Heat gained by ice + Heat gained by aluminum calorimeter cup
The heat lost by the water can be calculated as:
Heat lost by water = mass of water × specific heat capacity of water × change in temperature = 0.330 kg × 4.18 J/g°C × (final temperature - initial temperature)
The heat gained by the ice can be calculated as:
Heat gained by ice = initial heat content of ice + mass of ice × specific heat capacity of ice × change in temperature = initial heat content of ice + 0.055 kg × 2.09 J/g°C × (final temperature - 0°C)
The heat gained by the aluminum calorimeter cup can be calculated as:
Heat gained by aluminum calorimeter cup = mass of aluminum calorimeter cup × specific heat capacity of aluminum × change in temperature = 0.100 kg × 0.90 J/g°C × (final temperature - 32°C)
Setting up the equation using the above expressions:
0.330 kg × 4.18 J/g°C × (final temperature - 32°C) = initial heat content of ice + 0.055 kg × 2.09 J/g°C × (final temperature - 0°C) + 0.100 kg × 0.90 J/g°C × (final temperature - 32°C)
Now, we can solve this equation for the final temperature.