To find the final temperature of the water, we can use the principle of energy conservation.
First, let's calculate the energy required to melt the ice. The formula for the energy required to change a substance from solid to liquid is given by:
Q = m * L
Where:
Q is the heat energy required
m is the mass of the substance
L is the latent specific heat of fusion
In this case, the mass of ice is 1 kilogram, and the latent specific heat of fusion is 336,000 joules/kilogram.
Q = 1 kg * 336,000 J/kg = 336,000 J
So, 336,000 joules of energy is required to melt 1 kilogram of ice.
Now, let's calculate the energy released by the water when it cools down from its initial temperature (50°C) to the final temperature.
The formula to calculate the heat energy released by a substance is given by:
Q = m * C * ΔT
Where:
Q is the heat energy released
m is the mass of the substance
C is the specific heat capacity of the substance
ΔT is the change in temperature
In this case, mass of water is 9 kilograms, the initial temperature is 50°C, and the final temperature is what we want to find.
Q = 9 kg * C * (final temperature - 50°C)
We know that the heat energy released by the water must be equal to the energy required to melt the ice for the system to reach equilibrium:
336,000 J = 9 kg * C * (final temperature - 50°C)
From this equation, we can solve for the final temperature.
336,000 J = 9 kg * C * final temperature - 450 kg * C
336,000 J + 450 kg * C = 9 kg * C * final temperature
final temperature = (336,000 J + 450 kg * C) / (9 kg * C)
Now, we need the specific heat capacity of water. The specific heat capacity of water is approximately 4,186 J/kg°C.
Plugging this value into the equation and solving for the final temperature:
final temperature = (336,000 J + 450 kg * (4,186 J/kg°C)) / (9 kg * (4,186 J/kg°C))
final temperature ≈ (336,000 J + 1,881,900 J) / 37,674 J
final temperature ≈ 2,217,900 J / 37,674 J
final temperature ≈ 58.89°C
Therefore, the final temperature of the water is approximately 58.89°C.