If compressibility of material is
`4xx10^(-5)` per atm, pressure is 100 atm and volume is `100 cm^(3)` then find `triangleV=?`

Options

A. `0.4 cm^(3)`
B. `0.8 cm^(3)`
C. `0.6 cm^(3)`
D. `0.2 cm^(3)`

Answer

To solve the problem of finding the change in volume (ΔV) given the compressibility of a material, pressure, and initial volume, we can follow these steps:

### Step 1: Understand the Given Data
- Compressibility (β) = \(4 \times 10^{-5} \, \text{per atm}\)
- Pressure (P) = \(100 \, \text{atm}\)
- Volume (V) = \(100 \, \text{cm}^3\)

### Step 2: Convert Units
To work with standard units, we need to convert pressure from atm to Newtons per square meter (N/m²):
\[ 1 \, \text{atm} = 1.013 \times 10^5 \, \text{N/m}^2 \]
Thus,
\[ P = 100 \, \text{atm} = 100 \times 1.013 \times 10^5 \, \text{N/m}^2 = 1.013 \times 10^7 \, \text{N/m}^2 \]

### Step 3: Calculate Bulk Modulus
The bulk modulus (K) is the inverse of compressibility (β):
\[ K = \frac{1}{\beta} = \frac{1}{4 \times 10^{-5}} = 2.5 \times 10^4 \, \text{atm} \]
To convert this to N/m²:
\[ K = 2.5 \times 10^4 \, \text{atm} \times 1.013 \times 10^5 \, \text{N/m}^2/\text{atm} = 2.533 \times 10^9 \, \text{N/m}^2 \]

### Step 4: Use the Formula for Change in Volume
The change in volume (ΔV) can be calculated using the formula:
\[ \Delta V = \frac{P \cdot V}{K} \]
Substituting the values:
- P = \(1.013 \times 10^7 \, \text{N/m}^2\)
- V = \(100 \, \text{cm}^3 = 100 \times 10^{-6} \, \text{m}^3 = 1 \times 10^{-4} \, \text{m}^3\)
- K = \(2.533 \times 10^9 \, \text{N/m}^2\)

Now, substituting these values into the formula:
\[ \Delta V = \frac{(1.013 \times 10^7) \cdot (1 \times 10^{-4})}{2.533 \times 10^9} \]

### Step 5: Calculate ΔV
Calculating the above expression:
\[ \Delta V = \frac{1.013 \times 10^3}{2.533 \times 10^9} \approx 0.0004 \, \text{m}^3 \]
Converting back to cm³:
\[ \Delta V \approx 0.4 \, \text{cm}^3 \]

### Final Answer
Thus, the change in volume (ΔV) is:
\[ \Delta V = 0.4 \, \text{cm}^3 \]