The heat capacity of liquid water is 75.6 J/mol K, while the enthalpy of fusion of ice is 6.0 kJ/mol. What is the smallest number of ice cubes at `0^(@)C`, each containing 9.0g of water, needed to cool 500g of liquid water from `20^(@)C " to " 0^(@0C`?

Options

A. 1
B. 7
C. 14
D. None of these

Answer

To solve the problem, we need to determine how many ice cubes are required to cool 500 g of liquid water from 20°C to 0°C. We will use the heat capacity of water and the enthalpy of fusion of ice in our calculations.

### Step-by-Step Solution:

1. **Calculate the moles of water in 500 g:**
\[ \text{Molar mass of water (H}_2\text{O)} = 18 \text{ g/mol} \]
\[ \text{Moles of water} = \frac{500 \text{ g}}{18 \text{ g/mol}} \approx 27.78 \text{ moles} \]

**Hint:** Remember to convert grams to moles using the molar mass.

2. **Calculate the heat lost by the liquid water when cooled from 20°C to 0°C:**
\[ \Delta T = 20 - 0 = 20 \text{ °C} \]
\[ \text{Heat lost (q)} = n \cdot C \cdot \Delta T \]
Where \( C = 75.6 \text{ J/mol K} \).
\[ q = 27.78 \text{ moles} \times 75.6 \text{ J/mol K} \times 20 \text{ K} \]
\[ q = 27.78 \times 75.6 \times 20 \approx 42,000 \text{ J} = 42 \text{ kJ} \]

**Hint:** Use the formula for heat transfer, \( q = n \cdot C \cdot \Delta T \), to find the total heat lost.

3. **Calculate the heat absorbed by one ice cube:**
Each ice cube contains 9 g of water.
\[ \text{Moles of water in one ice cube} = \frac{9 \text{ g}}{18 \text{ g/mol}} = 0.5 \text{ moles} \]
The heat absorbed by one ice cube when it melts (using the enthalpy of fusion):
\[ \text{Enthalpy of fusion} = 6.0 \text{ kJ/mol} \]
\[ \text{Heat absorbed by one ice cube} = 0.5 \text{ moles} \times 6.0 \text{ kJ/mol} = 3.0 \text{ kJ} \]

**Hint:** Remember to convert grams to moles for the ice cube and use the enthalpy of fusion to find the heat absorbed.

4. **Calculate the number of ice cubes needed:**
To find the number of ice cubes required to absorb the heat lost by the water:
\[ \text{Number of ice cubes} = \frac{\text{Total heat lost}}{\text{Heat absorbed per ice cube}} = \frac{42 \text{ kJ}}{3 \text{ kJ/ice cube}} = 14 \]

**Hint:** Divide the total heat lost by the heat absorbed by one ice cube to find the total number of ice cubes needed.

### Final Answer:
The smallest number of ice cubes needed is **14**.