The volume of a block of a metal changes by `012%` when it is heated through `20^(@)C` . The coefficient of linear expansion of the metal is

Options

A. `2.0xx10^(-5)per^(@)C`
B. `4.0xx10^(-5)per^(@)C`
C. `6.0xx10^(-5)per^(@)C`
D. `8.0xx10^(-5)per^(@)C`

Explanation

To find the coefficient of linear expansion (α) of the metal given that the volume of a block changes by 0.12% when heated through 20°C, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between volume change and temperature change**:
The change in volume (ΔV) of a material can be expressed in terms of its volume expansion coefficient (γ) and the change in temperature (Δθ) using the formula:
\[ \frac{\Delta V}{V} = \gamma \cdot \Delta \theta \]
where \( \Delta V \) is the change in volume, \( V \) is the original volume, \( \gamma \) is the volume expansion coefficient, and \( \Delta \theta \) is the change in temperature.

2. **Convert percentage change to a fraction**:
Given that the volume changes by 0.12%, we can express this as:
\[ \frac{\Delta V}{V} = \frac{0.12}{100} = 0.0012 \]

3. **Substitute known values into the formula**:
We know the change in temperature is:
\[ \Delta \theta = 20^\circ C \]
Thus, substituting the values into the volume expansion formula gives:
\[ 0.0012 = \gamma \cdot 20 \]

4. **Solve for the volume expansion coefficient (γ)**:
Rearranging the equation to solve for γ:
\[ \gamma = \frac{0.0012}{20} = 0.00006 \, \text{per degree Celsius} = 6 \times 10^{-5} \, \text{per degree Celsius} \]

5. **Relate the volume expansion coefficient to the linear expansion coefficient (α)**:
The relationship between the volume expansion coefficient (γ) and the linear expansion coefficient (α) is given by:
\[ \gamma = 3\alpha \]
Therefore, we can express α as:
\[ \alpha = \frac{\gamma}{3} = \frac{6 \times 10^{-5}}{3} = 2 \times 10^{-5} \, \text{per degree Celsius} \]

6. **Final answer**:
The coefficient of linear expansion (α) of the metal is:
\[ \alpha = 2 \times 10^{-5} \, \text{per degree Celsius} \]

### Final Answer:
The coefficient of linear expansion of the metal is \( 2 \times 10^{-5} \, \text{per degree Celsius} \).