The volume of a block of a metal changes by `0.12%` when it is heated through `20^(@)C` . The coefficient of linear expansion of the metal is

Options

A. `4 xx 10^(-5)"^(@) C^(-1)`
B. `2 xx 10^(-5)"^(@) C^(-1)`
C. `0.5 xx 10^(-5)"^(@) C^(-1)`
D. `4 xx 10^(-4)"^(@) C^(-1)`

Explanation

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

### Step 1: Understand the relationship between volume change and linear expansion
The change in volume (ΔV) of a material when it is heated can be expressed in terms of the coefficient of volume expansion (γ) and the change in temperature (Δθ):
\[ \frac{\Delta V}{V} = \gamma \Delta \theta \]
Where:
- \(\Delta V\) is the change in volume,
- \(V\) is the original volume,
- \(\gamma\) is the coefficient of volume expansion,
- \(\Delta \theta\) is the change in temperature.

### Step 2: Relate the coefficient of volume expansion to the coefficient of linear expansion
The coefficient of volume expansion (γ) is related to the coefficient of linear expansion (α) by the formula:
\[ \gamma = 3\alpha \]

### Step 3: Substitute the known values into the equation
We know that the volume change is 0.12%, which can be expressed as:
\[ \frac{\Delta V}{V} = \frac{0.12}{100} = 0.0012 \]
The change in temperature is given as:
\[ \Delta \theta = 20°C \]

Now, substituting these values into the volume change equation:
\[ 0.0012 = \gamma \times 20 \]

### Step 4: Substitute γ in terms of α
Using the relationship between γ and α:
\[ 0.0012 = (3\alpha) \times 20 \]

### Step 5: Solve for α
Rearranging the equation gives:
\[ 0.0012 = 60\alpha \]
Now, solving for α:
\[ \alpha = \frac{0.0012}{60} \]
\[ \alpha = 0.00002 \]

### Step 6: Convert to scientific notation
Converting 0.00002 to scientific notation gives:
\[ \alpha = 2 \times 10^{-5} \text{ per degree Celsius} \]

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

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