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 ρ_ice = 0.9ρ_w, the latent heat of fusion of ice is λ = 340 kJ/kg, the specific heat of ice is c_ice =2 .1 kJ/(kg . K), and the specific heat of water is c_w = 4.2 kJ/(kg . K).

Determine the initial temperature T_ice 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 ρ_w/ρ_ice ≈ 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 T_ice is the initial temperature of ice. As was mentioned above, the volume of water increases as a result of freezing by a factor of ρ_w/ρ_ice, and hence

where S is the cross-sectional area of the calorimeter. Substituting Δm from Eq. (2) into Eq. (1) and using the relations m_w (h/3)ρ_wS and m_ice = (h/3) ρ_iceS, we obtain