A 100-g aluminum calorimeter contains a mixture of 40 g of ice and 200 g of water at equilibrium. A copper cylinder of mass 300 g is heated to 350 C and then dropped into the calorimeter. What is the final temperature of the calorimeter and its contents if no heat is lost to the surroundings?

specific heat of water = 1.00 cal/g·C specific heat of aluminum = 0.22 cal/g·C
specific heat of copper = 0.093 cal/g·C heat of fusion of water = 79.7 cal/g

Assume all the heat lost by the copper is gained by the water, ice and aluminum.

In general, one can either assume all the ice melts and solve for final T, or that some ice remains (at 0 C final T) and solve for the amount that melts. Only one assumption will give a meaningful result. There is more than enough mass of Cu to melt all the ice in this case.

20 C

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To find the final temperature of the calorimeter and its contents, we need to consider the heat gained and lost by the substances involved.

Step 1: Calculate the heat gained by the water to reach its final temperature.
The heat gained by the water can be calculated using:

Qwater = (mass of water) * (specific heat of water) * (change in temperature)

Here, the mass of the water is 200 g, the specific heat of water is 1.00 cal/g·C, and the change in temperature is the final temperature of the calorimeter minus the initial temperature (which is 0 C since the ice is at equilibrium). Let's call the final temperature of the calorimeter and its contents "T".

Qwater = (200 g) * (1.00 cal/g·C) * (T - 0 C)

Step 2: Calculate the heat gained by the aluminum calorimeter to reach its final temperature.
The heat gained by the aluminum calorimeter can be calculated using:

Qcalorimeter = (mass of calorimeter) * (specific heat of aluminum) * (change in temperature)

Here, the mass of the calorimeter is 100 g, the specific heat of aluminum is 0.22 cal/g·C, and the change in temperature is T - 0 C.

Qcalorimeter = (100 g) * (0.22 cal/g·C) * (T - 0 C)

Step 3: Calculate the heat gained by the copper cylinder.
The heat gained by the copper cylinder can be calculated using:

Qcopper = (mass of copper) * (specific heat of copper) * (change in temperature)

Here, the mass of the copper cylinder is 300 g, the specific heat of copper is 0.093 cal/g·C, and the change in temperature is the final temperature of the copper minus the initial temperature of 350 C.

Qcopper = (300 g) * (0.093 cal/g·C) * (T - 350 C)

Step 4: Calculate the heat lost by the ice to reach its melting point.
The heat lost by the ice can be calculated using:

Qice = (mass of ice) * (heat of fusion of water)

Here, the mass of the ice is 40 g, and the heat of fusion of water is 79.7 cal/g.

Qice = (40 g) * (79.7 cal/g)

Step 5: Set up the heat transfer equation assuming no heat is lost to the surroundings.
Since no heat is lost to the surroundings, the total heat gained by the water, the aluminum calorimeter, and the copper cylinder should be equal to the heat lost by the ice.

Qwater + Qcalorimeter + Qcopper = Qice

Substituting the respective equations from steps 1 to 4:

(200 g) * (1.00 cal/g·C) * (T - 0 C) + (100 g) * (0.22 cal/g·C) * (T - 0 C) + (300 g) * (0.093 cal/g·C) * (T - 350 C) = (40 g) * (79.7 cal/g)

Simplifying the equation, we can solve for T, the final temperature of the calorimeter and its contents.