Final answer:
To find the initial temperature of the hot water, we apply the conservation of energy, which accounts for the heat needed to melt the ice and the heat lost by the hot water. The calculations show that the initial temperature of the hot water was 57.14°C.
Step-by-step explanation:
The question relates to the thermal equilibrium between ice and water, which involves concepts of heat transfer, specific heat, and latent heat of fusion. To find the initial temperature of the hot water before mixing with ice, we can use the principle of conservation of energy, which states that the heat lost by the hot water will equal the heat gained by the ice to melt and reach 0°C.
First, the ice needs to be brought to the melting point and then melted. The heat required to melt 0.50 kg of ice is:
Q1 = mass of ice × latent heat of fusion
= 500 g × 80 cal/g
= 40000 cal
= 40 kcal
When the hot water gives off heat to melt the ice, its temperature drops. The amount of heat Q2 lost by the hot water is proportional to the temperature change (denoted as ΔT), the mass of the water, and the specific heat capacity of water:
Q2 = mass of water × specific heat of water × ΔT
= 700 g × 1 cal/g°C × ΔT
Since Q1 = Q2 and the final temperature of the water is 0°C:
700 g × 1 cal/g°C × ΔT = 40000 cal
ΔT = 40000 cal / (700 g × 1 cal/g°C) = 57.14°C
Therefore, the initial temperature (T_initial) of the hot water is:
T_initial = final temperature + ΔT = 0°C + 57.14°C
T_initial = 57.14°C