The fraction of the volume of a glass flask must be filled with mercury so that the volume of the empty space may be the same at all temperature is
`(alpha_("glass") = 9 xx 10^(-6)//^(0)C, gamma_(Hg) = 18.9 xx 10^(-5) // ^(0)C)`

Options

A. `(1)/(2)`
B. `(1)/(7)`
C. `(1)/(4)`
D. `(1)/(5)`

Explanation

To solve the problem, we need to determine the fraction of the volume of a glass flask that must be filled with mercury so that the volume of the empty space remains constant at all temperatures. We are given the coefficients of linear expansion for glass and volumetric expansion for mercury.

### Step-by-Step Solution:

1. **Understand the Coefficients of Expansion**:
- The coefficient of linear expansion of glass is given as \( \alpha_{glass} = 9 \times 10^{-6} \, ^\circ C^{-1} \).
- The volumetric expansion coefficient of mercury is given as \( \gamma_{Hg} = 18.9 \times 10^{-5} \, ^\circ C^{-1} \).

2. **Relate the Volumes and Expansions**:
- Let \( V_g \) be the volume of glass and \( V_m \) be the volume of mercury.
- The change in volume due to temperature change \( \Delta T \) can be expressed as:
\[ \Delta V_g = V_g \cdot 3\alpha_{glass} \cdot \Delta T \]
\[ \Delta V_m = V_m \cdot \gamma_{Hg} \cdot \Delta T \]
- Since we want the empty space to remain constant, we need the changes in volumes to be equal:
\[ \Delta V_g = \Delta V_m \]

3. **Set Up the Equation**:
- From the above relationships, we can set up the equation:
\[ V_g \cdot 3\alpha_{glass} = V_m \cdot \gamma_{Hg} \]

4. **Substitute the Values**:
- Substituting the values of \( \alpha_{glass} \) and \( \gamma_{Hg} \):
\[ V_g \cdot 3 \cdot (9 \times 10^{-6}) = V_m \cdot (18.9 \times 10^{-5}) \]
- Simplifying this gives:
\[ V_g \cdot 27 \times 10^{-6} = V_m \cdot 18.9 \times 10^{-5} \]

5. **Express the Volumes in Terms of Each Other**:
- Rearranging the equation to find the ratio of the volumes:
\[ \frac{V_m}{V_g} = \frac{27 \times 10^{-6}}{18.9 \times 10^{-5}} \]
- Simplifying the right side:
\[ \frac{V_m}{V_g} = \frac{27}{189} = \frac{1}{7} \]

6. **Conclusion**:
- The fraction of the volume of the glass flask that must be filled with mercury is \( \frac{1}{7} \).

### Final Answer:
The fraction of the volume of the glass flask that must be filled with mercury is \( \frac{1}{7} \).