Determine the fractional change in volume as the pressure of the atmosphere `(1.0xx10^(5)`Pa) around a metal block is reduced to zero by placing the block in vacuum. The bulk modulus for the block is `1.25xx10^(11) Nm^(-2)`.

Answer

To determine the fractional change in volume of a metal block when the atmospheric pressure is reduced to zero (i.e., placed in a vacuum), we can use the concept of bulk modulus. Here’s a step-by-step solution:

### Step 1: Understand the Given Values
- Initial atmospheric pressure, \( P = 1.0 \times 10^5 \, \text{Pa} \)
- Bulk modulus of the block, \( K = 1.25 \times 10^{11} \, \text{N/m}^2 \)

### Step 2: Recall the Formula for Bulk Modulus
The bulk modulus \( K \) is defined as:
\[ K = \frac{\text{Stress}}{\text{Volumetric Strain}} \]
Where:
- Stress is the change in pressure, \( \Delta P \)
- Volumetric strain is the fractional change in volume, \( \frac{\Delta V}{V} \)

### Step 3: Relate the Change in Pressure to the Bulk Modulus
Since the block is placed in a vacuum, the change in pressure \( \Delta P \) is equal to the initial atmospheric pressure:
\[ \Delta P = P = 1.0 \times 10^5 \, \text{Pa} \]

### Step 4: Express the Volumetric Strain
From the definition of bulk modulus, we can rearrange the formula to express the volumetric strain:
\[ \frac{\Delta V}{V} = \frac{\Delta P}{K} \]

### Step 5: Substitute the Values into the Formula
Now, we can substitute the values of \( \Delta P \) and \( K \) into the equation:
\[ \frac{\Delta V}{V} = \frac{1.0 \times 10^5 \, \text{Pa}}{1.25 \times 10^{11} \, \text{N/m}^2} \]

### Step 6: Calculate the Fractional Change in Volume
Perform the calculation:
\[ \frac{\Delta V}{V} = \frac{1.0 \times 10^5}{1.25 \times 10^{11}} = 8.0 \times 10^{-7} \]

### Final Answer
The fractional change in volume of the metal block when the atmospheric pressure is reduced to zero is:
\[ \frac{\Delta V}{V} = 8.0 \times 10^{-7} \]

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