The remaining volume of a glass vessel is constant at all temperature if `(1)/(x)` of its volume is filled with mercury. The coefficient of volume expansion of mercury is 7 times that of glass. The value of x should be

Options

A. 5
B. 7
C. 6
D. 8

Explanation

To solve the problem, we need to analyze the relationship between the volume expansions of mercury and glass in the context of the given conditions.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have a glass vessel that has a constant remaining volume when filled with mercury. The volume expansion coefficient of mercury is 7 times that of glass. We need to find the value of \( x \) such that \( \frac{1}{x} \) of the vessel's volume is filled with mercury.

2. **Initial Volume Setup**:
Let the total volume of the glass vessel be \( V \). The volume filled with mercury is \( \frac{V}{x} \), and the remaining volume of the glass vessel is:
\[ V_{\text{remaining}} = V - \frac{V}{x} = V\left(1 - \frac{1}{x}\right) \]

3. **Volume Expansion Relation**:
For the remaining volume to be constant, the expansion of the glass must equal the expansion of the mercury. The volume expansion can be expressed as:
\[ \Delta V_{\text{glass}} = \beta_g V_{\text{glass}} \Delta T \]
\[ \Delta V_{\text{mercury}} = \beta_m V_{\text{mercury}} \Delta T \]
where \( \beta_g \) is the coefficient of volume expansion for glass and \( \beta_m = 7\beta_g \) for mercury.

4. **Setting Up the Equation**:
The volume of glass is the remaining volume:
\[ V_{\text{glass}} = V\left(1 - \frac{1}{x}\right) \]
The volume of mercury is:
\[ V_{\text{mercury}} = \frac{V}{x} \]
Therefore, we can write the expansions as:
\[ \beta_g V\left(1 - \frac{1}{x}\right) \Delta T = 7\beta_g \left(\frac{V}{x}\right) \Delta T \]

5. **Cancelling Common Terms**:
Since \( \Delta T \) and \( V \) are common in both sides, we can cancel them out:
\[ \left(1 - \frac{1}{x}\right) = 7 \left(\frac{1}{x}\right) \]

6. **Solving for \( x \)**:
Rearranging the equation gives:
\[ 1 - \frac{1}{x} = \frac{7}{x} \]
Multiplying through by \( x \) to eliminate the fraction:
\[ x - 1 = 7 \]
Thus,
\[ x = 8 \]

### Final Answer:
The value of \( x \) should be \( 8 \).