Question

The volume of a material reduces by 2% when the pressure is increased from 1 atm to 2 atm. What is its bulk modulus?

• A. \( 10^5 \, \text{N/m}^2 \)
• B. \( 5 \times 10^5 \, \text{N/m}^2 \)
• C. \( 10^6 \, \text{N/m}^2 \)
• D. \( 5 \times 10^6 \, \text{N/m}^2 \)

Hint:

Bulk modulus is always positive. Use \( B = \frac{\Delta P}{|\Delta V/V|} \). Remember: \( 1 \, \text{atm} = 1.013 \times 10^5 \, \text{N/m}^2 \approx 10^5 \, \text{N/m}^2 \) for quick calculations.

Answer 1

The Correct Option is D

Concept: Bulk modulus \( B \) is defined as the ratio of the change in pressure to the fractional change in volume: \[ B = - \frac{\Delta P}{\Delta V / V} \] The negative sign ensures \( B \) is positive since an increase in pressure causes a decrease in volume.

Step 1: Identify given values. Initial pressure \( P_1 = 1 \, \text{atm} \) Final pressure \( P_2 = 2 \, \text{atm} \) \[ \Delta P = P_2 - P_1 = 1 \, \text{atm} \] Volume reduces by 2%, so: \[ \frac{\Delta V}{V} = -0.02 \quad (\text{negative sign indicates decrease}) \]

Step 2: Convert pressure to SI units. \[ 1 \, \text{atm} = 1.013 \times 10^5 \, \text{N/m}^2 \approx 10^5 \, \text{N/m}^2 \] Thus: \[ \Delta P = 1 \times 10^5 \, \text{N/m}^2 \]

Step 3: Apply bulk modulus formula. \[ B = - \frac{\Delta P}{\Delta V / V} = - \frac{10^5}{-0.02} = \frac{10^5}{0.02} \] \[ 0.02 = 2 \times 10^{-2} \quad \Rightarrow \quad B = \frac{10^5}{2 \times 10^{-2}} = \frac{10^5 \times 10^2}{2} = \frac{10^7}{2} \] \[ B = 5 \times 10^6 \, \text{N/m}^2 \]