When the pressure on a sphere is increased by 80 atmospheres then its volume decreases by `0.01%`. Find the bulk modulus of elasticity of the material of sphere. (in N/m^2)

Options

A. 5.4 x 10^10
B. 11.4 x 10^10
C. 6.4 x 10^10
D. 8.1 x 10^10

Explanation

To find the bulk modulus of elasticity of the material of the sphere, we can follow these steps:

### Step 1: Identify the given data
- Increase in pressure (ΔP) = 80 atm
- Decrease in volume percentage (ΔV/V) = -0.01%

### Step 2: Convert the pressure from atmospheres to Pascals
1 atm = 1.013 × 10^5 N/m²
So,
\[ \Delta P = 80 \, \text{atm} \times 1.013 \times 10^5 \, \text{N/m}^2/\text{atm} = 80 \times 1.013 \times 10^5 \, \text{N/m}^2 \]
Calculating this gives:
\[ \Delta P = 8.104 \times 10^6 \, \text{N/m}^2 \]

### Step 3: Convert the volume change percentage to a decimal
The percentage decrease in volume is given as -0.01%. To convert this to a decimal:
\[ \frac{\Delta V}{V} = -0.01\% = -\frac{0.01}{100} = -0.0001 \]

### Step 4: Use the formula for bulk modulus
The bulk modulus (K) is defined as:
\[ K = -\frac{\Delta P}{\Delta V/V} \]
Substituting the values we have:
\[ K = -\frac{8.104 \times 10^6 \, \text{N/m}^2}{-0.0001} \]

### Step 5: Calculate the bulk modulus
Calculating this gives:
\[ K = \frac{8.104 \times 10^6}{0.0001} = 8.104 \times 10^{10} \, \text{N/m}^2 \]

### Final Answer
The bulk modulus of elasticity of the material of the sphere is:
\[ K = 8.104 \times 10^{10} \, \text{N/m}^2 \]

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