One litre glass bottle is completely filled with water at `25^@C`. If the bottle and water both are heated to `75^@C`, then the volume of water that overflows is

Options

A. `30 cm^3` , if the expansion of bottle is neglected
B. `32 cm^3` , if the expansion of bottle is neglected
C. `27 cm^3` , if the expansion of bottle is not neglected
D. `29 cm^3` , if the expansion of bottle is neglected

Explanation

To solve the problem of how much water overflows from a 1-litre glass bottle when both the bottle and the water are heated from 25°C to 75°C, we will consider the thermal expansion of both the water and the glass bottle.

### Step-by-Step Solution:

1. **Identify Initial Conditions**:
- Initial volume of water \( V_i = 1 \, \text{litre} = 1000 \, \text{cm}^3 \).
- Initial temperature \( T_i = 25^\circ C \).
- Final temperature \( T_f = 75^\circ C \).

2. **Calculate the Change in Temperature**:
\[ \Delta T = T_f - T_i = 75^\circ C - 25^\circ C = 50^\circ C \]

3. **Volumetric Expansion of Water**:
- The coefficient of volumetric expansion of water \( \gamma_w \approx 6 \times 10^{-4} \, \text{°C}^{-1} \).
- The change in volume of water due to heating can be calculated using the formula:
\[ \Delta V_w = V_i \cdot \gamma_w \cdot \Delta T = 1000 \, \text{cm}^3 \cdot 6 \times 10^{-4} \, \text{°C}^{-1} \cdot 50 \, \text{°C} \]
\[ \Delta V_w = 1000 \cdot 6 \times 10^{-4} \cdot 50 = 30 \, \text{cm}^3 \]

4. **Volumetric Expansion of the Glass Bottle**:
- The coefficient of volumetric expansion of glass \( \gamma_g \approx 1.8 \times 10^{-5} \, \text{°C}^{-1} \).
- The change in volume of the glass bottle can be calculated similarly:
\[ \Delta V_g = V_i \cdot \gamma_g \cdot \Delta T = 1000 \, \text{cm}^3 \cdot 1.8 \times 10^{-5} \, \text{°C}^{-1} \cdot 50 \, \text{°C} \]
\[ \Delta V_g = 1000 \cdot 1.8 \times 10^{-5} \cdot 50 = 0.9 \, \text{cm}^3 \]

5. **Calculate the Net Overflow Volume**:
- The net overflow volume of water can be calculated by subtracting the increase in volume of the bottle from the increase in volume of the water:
\[ V_{overflow} = \Delta V_w - \Delta V_g = 30 \, \text{cm}^3 - 0.9 \, \text{cm}^3 = 29.1 \, \text{cm}^3 \]

6. **Final Answer**:
- The volume of water that overflows when both the bottle and water are heated from 25°C to 75°C is approximately \( 29.1 \, \text{cm}^3 \).