The volume of a solid decreases by 0.6% when it is cooled through `50^(@)C`. Its coefficient of linear expansion is

Options

A. `4xx10^(-6) K`
B. `5xx10^(-5) K`
C. `6xx10^(4)K`
D. `4xx10^(-5) K`

Explanation

To find the coefficient of linear expansion (α) of a solid that decreases in volume by 0.6% when cooled through 50°C, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between volume change and temperature change**:
The volumetric expansion (γ) is related to the change in volume (ΔV) and the original volume (V) by the formula:
\[ \frac{\Delta V}{V} = \gamma \Delta T \]
where ΔT is the change in temperature.

2. **Convert the percentage change in volume to a decimal**:
Given that the volume decreases by 0.6%, we can express this as:
\[ \frac{\Delta V}{V} = -0.006 \]
(the negative sign indicates a decrease).

3. **Substitute the values into the formula**:
We know that ΔT = -50°C (since we are cooling), so we can rearrange the formula to find γ:
\[ \gamma = \frac{\Delta V/V}{\Delta T} \]
Substituting the known values:
\[ \gamma = \frac{-0.006}{-50} = \frac{0.006}{50} = 0.00012 \]

4. **Convert γ to scientific notation**:
\[ \gamma = 1.2 \times 10^{-4} \, \text{per °C} \]

5. **Relate volumetric expansion to linear expansion**:
The relationship between the coefficient of volumetric expansion (γ) and the coefficient of linear expansion (α) is given by:
\[ \gamma = 3\alpha \]
Therefore, we can find α:
\[ \alpha = \frac{\gamma}{3} = \frac{1.2 \times 10^{-4}}{3} = 0.4 \times 10^{-4} \]

6. **Convert α to scientific notation**:
\[ \alpha = 4 \times 10^{-5} \, \text{per °C} \]

### Final Answer:
The coefficient of linear expansion (α) is:
\[ \alpha = 4 \times 10^{-5} \, \text{per °C} \]