It is to cool down `(3)/(4) kg` of water from `80^(circ) C` to `40^(circ) C` by putting `0^(circ) C` ice into it. What minimum volume (in litre) should the vessel have to prevent the water from overflow?

Answer

To solve the problem of cooling down \( \frac{3}{4} \) kg of water from \( 80^{\circ} C \) to \( 40^{\circ} C \) by adding \( 0^{\circ} C \) ice, we need to find the minimum volume of the vessel to prevent overflow. We will use the principle of conservation of energy, where the heat lost by the water equals the heat gained by the ice.

### Step-by-Step Solution:

1. **Identify the Variables:**
- Mass of water, \( m_w = \frac{3}{4} \) kg
- Initial temperature of water, \( T_{i} = 80^{\circ} C \)
- Final temperature of water, \( T_{f} = 40^{\circ} C \)
- Mass of ice, \( m_i \) (unknown, to be determined)
- Initial temperature of ice, \( T_{ice} = 0^{\circ} C \)
- Latent heat of fusion of ice, \( L = 335 \) kJ/kg
- Specific heat capacity of water, \( c = 4.2 \) kJ/kg°C

2. **Calculate Heat Lost by Water:**
The heat lost by the water as it cools down can be calculated using the formula:
\[ Q_{lost} = m_w \cdot c \cdot (T_{i} - T_{f}) = \frac{3}{4} \cdot 4.2 \cdot (80 - 40) \]
\[ Q_{lost} = \frac{3}{4} \cdot 4.2 \cdot 40 = \frac{3 \cdot 4.2 \cdot 40}{4} = 126 \text{ kJ} \]

3. **Calculate Heat Gained by Ice:**
The heat gained by the ice consists of two parts: the heat required to melt the ice and the heat required to raise the temperature of the resulting water to \( 40^{\circ} C \).
\[ Q_{gained} = m_i \cdot L + m_i \cdot c \cdot (T_{f} - T_{ice}) = m_i \cdot 335 + m_i \cdot 4.2 \cdot (40 - 0) \]
\[ Q_{gained} = m_i \cdot 335 + m_i \cdot 4.2 \cdot 40 = m_i \cdot (335 + 168) = m_i \cdot 503 \text{ kJ} \]

4. **Set Heat Lost Equal to Heat Gained:**
Since the heat lost by the water equals the heat gained by the ice, we can set the two equations equal to each other:
\[ 126 = m_i \cdot 503 \]
Solving for \( m_i \):
\[ m_i = \frac{126}{503} \approx 0.25 \text{ kg} \]

5. **Calculate Total Mass of Water After Melting Ice:**
The total mass of water in the vessel after the ice melts and reaches \( 40^{\circ} C \) is:
\[ m_{total} = m_w + m_i = \frac{3}{4} + 0.25 = 1 \text{ kg} \]

6. **Determine the Minimum Volume of the Vessel:**
Since the density of water is approximately \( 1 \text{ kg/L} \), the minimum volume of the vessel required to hold \( 1 \text{ kg} \) of water is:
\[ V = m_{total} = 1 \text{ L} \]

### Final Answer:
The minimum volume of the vessel should be **1 litre**.