Question

A sample of a liquid is kept at 1 atm. It is compressed to 5 atm which leads to change of volume of 0.8 cm$^3$. If the bulk modulus of the liquid is 2 GPa, the initial volume of the liquid was ______ litre. (Take 1 atm = $10^5$ Pa)

Hint:

- Bulk modulus relates pressure change to relative volume change: \(K = -\Delta P/(\Delta V/V_0)\) - 1 m\(^3\) = 1000 litres - Watch unit conversions (1 cm\(^3\) = 10\(^{-6}\) m\(^3\), 1 GPa = 10\(^9\) Pa)

Answer 1

Correct Answer: 4

Given:

Initial liquid pressure \( P_i = 1 \, \text{atm} \)

Final liquid pressure \( P_f = 5 \, \text{atm} \)

The pressure change is calculated as:

\( \Delta P = P_f - P_i = 4 \, \text{atm} = 4 \times 10^5 \, \text{Pa} \)

The volume change is:

\( \Delta V = -0.8 \, \text{cm}^3 \)

The bulk modulus \( B \) is provided as:

\( B = 2 \times 10^9 \, \text{Pa} \)

Using the bulk modulus formula:

\( B = - \frac{\Delta P}{\frac{\Delta V}{V}} \)

The volume \( V \) can be determined using:

\( V = -B \times \left( \frac{\Delta V}{\Delta P} \right) \)

Substituting the provided values:

\( V = -2 \times 10^9 \times \left( \frac{-0.8 \times 10^{-6}}{4 \times 10^5} \right) \)

The resulting volume \( V \) is:

\( V = 4 \times 10^{-3} \, \text{m}^3 = 4 \, \text{litre} \)