A brass rod and a lead rod each 80 cm long at `0^@C` are clamped together at one end with their free ends coinciding. The separatioin of free ends of the rods if the system is placed in a steam bath is (`alpha_("brass")=18xx10^(-6)//^(@)C` and `alpha_("lead")=28xx10^(-6)//^(@)C`)

Options

A. 0.2 mm
B. 0.8 mm
C. 1.4 mm
D. 1.6 mm

Explanation

To solve the problem, we need to determine the separation between the free ends of a brass rod and a lead rod when they are heated from 0°C to the temperature of a steam bath (100°C). We will use the concept of linear expansion.

### Step-by-Step Solution:

1. **Identify the Initial Length and Temperature Change:**
- The initial length of both rods, \( L_0 = 80 \, \text{cm} = 0.8 \, \text{m} \).
- The initial temperature, \( T_0 = 0^\circ C \).
- The final temperature in the steam bath, \( T_f = 100^\circ C \).
- The temperature change, \( \Delta T = T_f - T_0 = 100^\circ C - 0^\circ C = 100^\circ C \).

2. **Coefficients of Linear Expansion:**
- Coefficient of linear expansion for brass, \( \alpha_{\text{brass}} = 18 \times 10^{-6} \, \text{°C}^{-1} \).
- Coefficient of linear expansion for lead, \( \alpha_{\text{lead}} = 28 \times 10^{-6} \, \text{°C}^{-1} \).

3. **Calculate the Final Lengths of Each Rod:**
- The final length of the brass rod:
\[ L_{\text{brass}} = L_0 \times (1 + \alpha_{\text{brass}} \Delta T) = 0.8 \, \text{m} \times \left(1 + 18 \times 10^{-6} \times 100\right) \]
\[ = 0.8 \, \text{m} \times (1 + 0.0018) = 0.8 \, \text{m} \times 1.0018 = 0.80144 \, \text{m} \]

- The final length of the lead rod:
\[ L_{\text{lead}} = L_0 \times (1 + \alpha_{\text{lead}} \Delta T) = 0.8 \, \text{m} \times \left(1 + 28 \times 10^{-6} \times 100\right) \]
\[ = 0.8 \, \text{m} \times (1 + 0.0028) = 0.8 \, \text{m} \times 1.0028 = 0.80224 \, \text{m} \]

4. **Calculate the Separation Between the Free Ends:**
- The separation \( S \) between the free ends of the rods is given by:
\[ S = L_{\text{lead}} - L_{\text{brass}} = 0.80224 \, \text{m} - 0.80144 \, \text{m} = 0.0008 \, \text{m} \]

5. **Convert the Separation to Millimeters:**
- To convert meters to millimeters:
\[ S = 0.0008 \, \text{m} \times 1000 \, \text{mm/m} = 0.8 \, \text{mm} \]

### Final Answer:
The separation of the free ends of the rods when placed in a steam bath is \( 0.8 \, \text{mm} \).