A metal block is experiencing an atmospheric pressure of `1 xx 10^(5)N//m^(2)`, when the same block is placed vacuum chamber, the freactional change in its volume is (the bulk modulus of metal is `1.25 xx 10^(11) N//m^(2)`)

Options

A. `4 xx 10^(-7)`
B. `2 xx 10^(-7)`
C. `8 xx 10^(-7)`
D. `1 xx 10^(-7)`

Explanation

To solve the problem, we need to find the fractional change in volume of a metal block when it is subjected to a change in pressure from atmospheric pressure to a vacuum. We will use the concept of bulk modulus for this calculation.

### Step-by-Step Solution:

1. **Identify Given Values:**
- Atmospheric pressure (P) = \(1 \times 10^{5} \, \text{N/m}^2\)
- Bulk modulus (K) = \(1.25 \times 10^{11} \, \text{N/m}^2\)

2. **Understand the Concept of Bulk Modulus:**
- The bulk modulus (K) is defined as the ratio of the change in pressure (ΔP) to the fractional change in volume (ΔV/V):
\[ K = -\frac{P}{\frac{\Delta V}{V}} \]
- Rearranging this formula gives us:
\[ \frac{\Delta V}{V} = -\frac{P}{K} \]

3. **Calculate the Change in Pressure (ΔP):**
- When the block is in a vacuum chamber, the pressure outside the block becomes 0 N/m².
- Therefore, the change in pressure (ΔP) is:
\[ \Delta P = P_{\text{initial}} - P_{\text{final}} = 1 \times 10^{5} \, \text{N/m}^2 - 0 \, \text{N/m}^2 = 1 \times 10^{5} \, \text{N/m}^2 \]

4. **Substitute Values into the Formula:**
- Now we can substitute the values of ΔP and K into the rearranged bulk modulus formula:
\[ \frac{\Delta V}{V} = -\frac{1 \times 10^{5}}{1.25 \times 10^{11}} \]

5. **Perform the Calculation:**
- Calculate the right-hand side:
\[ \frac{\Delta V}{V} = -\frac{1 \times 10^{5}}{1.25 \times 10^{11}} = -8 \times 10^{-7} \]

6. **Interpret the Result:**
- The negative sign indicates a decrease in volume, but since we are interested in the fractional change in volume, we can express it as:
\[ \frac{\Delta V}{V} = 8 \times 10^{-7} \]

### Final Answer:
The fractional change in volume of the metal block when placed in a vacuum chamber is \(8 \times 10^{-7}\).

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