a glass tube of length `133cm` and of uniform cross-section is to be filled with mercury so that the volume of the unoccupied by mercury remains the same at all temperatures. If cubical coefficient for glass and mercury are respectively `0.000026//^(@)C` and `0.000182//^(@)C` , calculate the length of mercury column.

Options

A. 19 cm
B. 133 cm
C. 7 cm
D. None of these

Explanation

To solve the problem of finding the length of the mercury column in a glass tube, we will follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**: We have a glass tube of length \( L = 133 \, \text{cm} \) filled with mercury. The volume unoccupied by mercury must remain constant with temperature changes. This means that the change in volume of the mercury must equal the change in volume of the glass.

2. **Define the Coefficients**: The cubical expansion coefficients are given as:
- For glass, \( \alpha_G = 0.000026 \, \text{°C}^{-1} \)
- For mercury, \( \alpha_M = 0.000182 \, \text{°C}^{-1} \)

3. **Volume Change Relations**: The change in volume of the glass and mercury can be expressed as:
- Change in volume of glass: \( \Delta V_G = V_G \cdot \alpha_G \cdot \Delta T \)
- Change in volume of mercury: \( \Delta V_M = V_M \cdot \alpha_M \cdot \Delta T \)

Since the unoccupied volume remains constant, we set these two changes equal:
\[ V_G \cdot \alpha_G \cdot \Delta T = V_M \cdot \alpha_M \cdot \Delta T \]

4. **Initial Volumes**: The initial volume of the mercury, \( V_M \), is given by:
\[ V_M = L_M \cdot A \]
where \( L_M \) is the length of the mercury column and \( A \) is the cross-sectional area of the tube. The initial volume of the glass, \( V_G \), is given by:
\[ V_G = (L - L_M) \cdot A \]

5. **Substituting Volumes**: Substituting the expressions for \( V_G \) and \( V_M \) into the volume change equation gives:
\[ (L - L_M) \cdot A \cdot \alpha_G = L_M \cdot A \cdot \alpha_M \]

6. **Canceling Area**: The cross-sectional area \( A \) cancels out from both sides:
\[ (L - L_M) \cdot \alpha_G = L_M \cdot \alpha_M \]

7. **Rearranging the Equation**: Rearranging the equation gives:
\[ L \cdot \alpha_G - L_M \cdot \alpha_G = L_M \cdot \alpha_M \]
\[ L \cdot \alpha_G = L_M \cdot (\alpha_M + \alpha_G) \]

8. **Solving for \( L_M \)**: Now, we can solve for \( L_M \):
\[ L_M = \frac{L \cdot \alpha_G}{\alpha_M + \alpha_G} \]

9. **Substituting Values**: Now we substitute the known values:
\[ L_M = \frac{133 \, \text{cm} \cdot 0.000026}{0.000182 + 0.000026} \]

10. **Calculating the Denominator**:
\[ \alpha_M + \alpha_G = 0.000182 + 0.000026 = 0.000208 \]

11. **Final Calculation**:
\[ L_M = \frac{133 \cdot 0.000026}{0.000208} = \frac{0.003458}{0.000208} \approx 16.6 \, \text{cm} \]

12. **Final Result**: The length of the mercury column \( L_M \) is approximately \( 16.6 \, \text{cm} \).