A 28.0-g sample of ice at -18.0°C is mixed with 124.0 g of water at 85.0°C. Calculate the final temperature of the mixture assuming no heat loss to the surroundings. The heat capacities of H2O(s) and H2O(l) are 2.08 and 4.18 J/g · °C, respectively, and the enthalpy of fusion for ice is 6.02 kJ/mol.
To calculate the final temperature of the mixture, we need to determine the amount of heat gained or lost by each component and then use that information to calculate the final temperature.
First, let's find the amount of heat gained or lost by the ice. The heat gained or lost by a substance is given by the formula:
Q = m × C × ΔT
where Q is the heat gained or lost, m is the mass of the substance, C is the specific heat capacity, and ΔT is the change in temperature.
For the ice, the initial temperature is -18.0°C, and the final temperature is the same as the final temperature of the mixture. The mass of the ice is 28.0 g, and the specific heat capacity of ice (H2O(s)) is 2.08 J/g · °C. Thus, the heat gained by the ice is:
Q_ice = m × C_ice × ΔT_ice
= 28.0 g × 2.08 J/g · °C × (T - (-18.0°C))
= 58.24 J/g · °C × (T + 18.0°C)
Next, let's find the amount of heat gained or lost by the water. The initial temperature of the water is 85.0°C, and the final temperature is the same as the final temperature of the mixture. The mass of the water is 124.0 g, and the specific heat capacity of liquid water (H2O(l)) is 4.18 J/g · °C. Thus, the heat lost by the water is:
Q_water = m × C_water × ΔT_water
= 124.0 g × 4.18 J/g · °C × (T - 85.0°C)
= 518.32 J/g · °C × (T - 85.0°C)
Since there is no heat loss to the surroundings, the heat gained by the ice must be equal to the heat lost by the water. Therefore, we can set the two equations equal to each other:
Q_ice = Q_water
58.24 J/g · °C × (T + 18.0°C) = 518.32 J/g · °C × (T - 85.0°C)
Now, solve this equation for T, the final temperature of the mixture:
58.24 T + 58.24 × 18.0°C = 518.32 T - 518.32 × 85.0°C
Distribute the terms:
58.24 T + 1048.32°C = 518.32 T - 44057.2°C
Combine like terms:
-460.08 T = -45105.52°C
Divide both sides by -460.08:
T ≈ 98.0°C
Therefore, the final temperature of the mixture, assuming no heat loss to the surroundings, is approximately 98.0°C.