When a liquid in a glass vessel is heated, its apparent expansion coefficient is `10.30 xx 10^(-4) .^(@)C^(-1)`. When the same liquid is heated in a metal vessel, its apparent expansion coeffiecient is `10.06 xx 10^(-4) .^(@)C^(-1)`. If the coefficient of linear expansion of glass = `9 xx 10^(-6) .^(@)C^(-1)`, what is the coefficient of linear expansion of metal?

Options

A. `51 xx 10^(-6).^(@)C^(-1)`
B. `17 xx 10^(-6).^(@)C^(-1)`
C. `25 xx 10^(-6).^(@)C^(-1)`
D. `43 xx 10^(-6).^(@)C^(-1)`

Answer

To solve the problem, we need to find the coefficient of linear expansion of the metal vessel (let's denote it as α_m). We will use the concept of apparent expansion and real expansion.

### Step-by-Step Solution:

1. **Understanding the Apparent Expansion Coefficient**:
The apparent expansion coefficient (γ_a) is given for two cases:
- For the glass vessel: γ_a(glass) = 10.30 × 10^(-4) °C^(-1)
- For the metal vessel: γ_a(metal) = 10.06 × 10^(-4) °C^(-1)

2. **Given Coefficient of Linear Expansion of Glass**:
The coefficient of linear expansion of glass (α_g) is given as:
- α_g = 9 × 10^(-6) °C^(-1)

3. **Using the Formula for Real Expansion**:
The relationship between real expansion (γ_r), apparent expansion (γ_a), and the coefficient of linear expansion (α) is given by:
\[ \gamma_r = \gamma_a + 3\alpha \]
This formula will be used for both cases (glass and metal).

4. **Setting Up the Equations**:
- For the glass vessel:
\[ \gamma_r = \gamma_a(glass) + 3α_g \]
Substituting the known values:
\[ \gamma_r = 10.30 × 10^{-4} + 3(9 × 10^{-6}) \]

- For the metal vessel:
\[ \gamma_r = \gamma_a(metal) + 3α_m \]
Substituting the known values:
\[ \gamma_r = 10.06 × 10^{-4} + 3α_m \]

5. **Equating the Two Expressions for Real Expansion**:
Since the real expansion (γ_r) is the same for both cases, we can set the two equations equal to each other:
\[ 10.30 × 10^{-4} + 3(9 × 10^{-6}) = 10.06 × 10^{-4} + 3α_m \]

6. **Calculating the Left Side**:
Calculate the left side:
\[ 10.30 × 10^{-4} + 27 × 10^{-6} = 10.30 × 10^{-4} + 0.027 × 10^{-4} = 10.327 × 10^{-4} \]

7. **Setting Up the Final Equation**:
Now, we have:
\[ 10.327 × 10^{-4} = 10.06 × 10^{-4} + 3α_m \]

8. **Solving for α_m**:
Rearranging the equation to solve for α_m:
\[ 3α_m = 10.327 × 10^{-4} - 10.06 × 10^{-4} \]
\[ 3α_m = 0.267 × 10^{-4} \]
\[ α_m = \frac{0.267 × 10^{-4}}{3} = 0.089 × 10^{-4} = 8.9 × 10^{-6} °C^{-1} \]

### Final Answer:
The coefficient of linear expansion of the metal is:
\[ α_m = 1.7 × 10^{-6} °C^{-1} \]