Question

Two identical thermally insulated cylindrical calorimeters of height h= 75 cm are filled to one-third. The first calorimeter is filled with ice formed as a result of freezing water poured into it, and the second is filled with water at T, = 10 °C. Water from the second calorimeter is poured into the first one, and as a result it becomes to be filled to two-thirds. After the temperature has been stabilized in the first calorimeter, its level of water increases by Δh = 0.5 cm. The density of ice is ρice = 0.9ρw, the latent heat of fusion of ice is λ = 340 kJ/kg, the specific heat of ice is cice =2 .1 kJ/(kg . K), and the specific heat of water is cw = 4.2 kJ/(kg . K).

Determine the initial temperature Tice of ice in the first calorimeter.

Answer 1

If the level of water in a calorimeter has become higher, it means that a part of water has been frozen (the volume of water increases during freezing). On the other hand, we can state that some amount of water has not been frozen since otherwise its volume would have increased by a factor of ρw/ρice ≈ 1.1, and the level of water in the calorimeter would have increased by (h/3) (1.1 - 1) ≈ 2.5 cm, while by hypothesis Δh = 0.5 cm. Thus, the temperature established in the calorimeter is 0 °C.

Using this condition, we can write

where Δm is the mass of frozen water, and Tice is the initial temperature of ice. As was mentioned above, the volume of water increases as a result of freezing by a factor of ρw/ρice, and hence

where * is the cross-sectional area of the calorimeter. Substituting Δm from Eq. (2) into Eq. (1) and using the relations mw (h/3)ρwS and mice = (h/3) ρiceS, we obtain