To solve this problem, we need to calculate the amount of heat gained by the ice and the amount of heat lost by the water. Since the final temperature is 0.0 °C, we know that the heat gained by the ice is equal to the heat lost by the water.
First, let's calculate the amount of heat lost by the water. We can use the formula:
Q = m * c * ΔT
where Q is the heat lost, m is the mass of water, c is the specific heat of water, and ΔT is the change in temperature.
Given:
m = 33.6 g (mass of water)
c = 4.18 J/(g•°C) (specific heat of water)
ΔT = (0.0 °C - 45.0 °C) = -45.0 °C (change in temperature)
Q = 33.6 g * 4.18 J/(g•°C) * (-45.0 °C)
Q = -6776.64 J
Since the heat gained by the ice is equal to the heat lost by the water, the heat gained by the ice is also -6776.64 J.
Next, let's calculate the amount of heat gained by the ice using the heat of fusion of water.
Q = n * ΔH
where Q is the heat gained, n is the number of moles of ice, and ΔH is the heat of fusion of water.
We need to convert the mass of water to moles using the molar mass of water (H2O):
molar mass of water = 2(1.008 g/mol) + 16.00 g/mol = 18.02 g/mol
moles of water = mass of water / molar mass of water
moles of water = 33.6 g / 18.02 g/mol
moles of water = 1.865 mol
Since the heat gained by the ice is equal to the heat lost by the water, we can equate the two equations:
-6776.64 J = n * 6.01 kJ/mol (since 1 kJ = 1000 J)
-6776.64 J = n * 6.01 * 10^3 J/mol
n = -6776.64 J / (6.01 * 10^3 J/mol)
n ≈ -1.13 mol
The negative sign indicates that the ice is losing heat, which means it is being heated. Since we can't have a negative number of moles, we take the absolute value of n:
|n| = 1.13 mol
Therefore, approximately 1.13 moles of ice were added.