the enthalpy of fusion of ice is 6.02 kJ/mol. the heat capacity of liquid water is 75.4 J/molC. What is the smallest number of ice cubes at 0 degrees C each containing one mole of water, necessary to cool 500g of liquid water initially at 20.0 degree C to 0.0 degree C

(mass ice*heat fusion) + [mass water x specific heat water x (Tfinal-Tinitial)] = 0

Solve for mass ice then convert to number of cubes. Remember 1 mole water = 18 grams. The way I read the problem, they don't want any fractional ice cubes. If my interpretation is correct you will calculate the number of ice cubes, then round to the next higher whole number.

egw

To solve this problem, we need to calculate the amount of heat required to cool the liquid water from 20.0 °C to 0.0 °C and compare it to the amount of heat released by the ice cubes when they warm up from 0.0 °C to 0.0 °C.

Step 1: Calculate the heat required to cool the liquid water from 20.0 °C to 0.0 °C.
The heat (q) required to cool a substance can be calculated using the equation:
q = mass × specific heat capacity × change in temperature

Given:
- Mass of liquid water = 500 g
- Specific heat capacity of liquid water = 75.4 J/mol°C (This means it takes 75.4 J of heat to raise the temperature of 1 mole of water by 1°C)
- Change in temperature = 20.0 °C - 0.0 °C = 20.0 °C

Convert the mass of liquid water to moles:
The molar mass of water (H2O) is approximately 18 g/mol (2 g/mol for 2 hydrogen atoms + 16 g/mol for 1 oxygen atom).
Therefore, 500 g of water is equal to 500 g / 18 g/mol ≈ 27.78 moles of water.

Now, calculate the heat required to cool the water:
q = 27.78 mol × 75.4 J/mol°C × 20.0 °C
≈ 33372 J

Step 2: Calculate the heat released by the ice cubes warming from 0.0 °C to 0.0 °C.
The heat released (q) by the ice cubes can be calculated using the equation:
q = enthalpy of fusion × number of moles

Given:
- Enthalpy of fusion of ice = 6.02 kJ/mol = 6020 J/mol
- Number of moles of water in each ice cube = 1 mole

Calculate the heat released:
q = 6020 J/mol × 1 mol
= 6020 J

Step 3: Determine the number of ice cubes needed.
To cool the liquid water, we need the amount of heat removed to be equal to or greater than the heat required to cool the water.

Therefore, the number of ice cubes needed can be calculated by:
Number of ice cubes = (heat required to cool the water) / (heat released by each ice cube)
= 33372 J / 6020 J
≈ 5.53

Since we cannot have a fraction of an ice cube, we need to round up to the nearest whole number.

Answer:
The smallest number of ice cubes needed to cool 500 g of liquid water from 20.0 °C to 0.0 °C is 6 ice cubes.