Solution for Question 27
Physics Principle:
When the system is isolated, the total energy remains conserved. We must determine the final equilibrium state. The large mass of very cold ice ($L_{ice}=80$ cm, avg temp -20°C) versus the small mass of cool water ($L_{water}=8$ cm, avg temp 8°C) suggests the water will freeze completely.
Calculation:
Since the cooling capacity of the ice is far greater than the heat content of the small amount of water, the final state will be all ice at some temperature below $0^\circ\text{C}$.
We calculate the final length based on the density change. The total mass of the substance corresponds to the initial volume of water ($L_0 = 80$ cm).
$$Mass = \rho_{water} \times A \times 80 \text{ cm}$$
In the final state, this entire mass is ice. Let the final length be $L_{final}$.
$$Mass = \rho_{ice} \times A \times L_{final}$$
Equating mass:
$$\rho_{water} \times 80 = \rho_{ice} \times L_{final}$$
$$1000 \times 80 = 900 \times L_{final}$$
$$L_{final} = \frac{80000}{900} = \frac{800}{9} \approx 88.89 \text{ cm}$$