The bulk moduli of ethanol, mercury and water are given as 0.9, 25 and 2.2 respectively in units of $$10^9$$ Nm$$^{-2}$$. For a given value of pressure, the fractional compression in volume is $$\frac{\Delta V}{V}$$. Which of the following statements about $$\frac{\Delta V}{V}$$ for these three liquids is correct?

The bulk modulus $$K$$ is defined as the ratio of the applied pressure change $$\Delta P$$ to the fractional decrease in volume $$-\frac{\Delta V}{V}$$. The formula is:

$$ K = - \frac{\Delta P}{\frac{\Delta V}{V}} $$

Rearranging for the fractional compression $$\frac{\Delta V}{V}$$, we get:

$$ \frac{\Delta V}{V} = - \frac{\Delta P}{K} $$

Since the fractional compression is typically considered as a magnitude for comparison, we take the absolute value:

$$ \left| \frac{\Delta V}{V} \right| = \frac{\Delta P}{K} $$

For a given pressure change $$\Delta P$$ (same for all liquids), the fractional compression $$\left| \frac{\Delta V}{V} \right|$$ is inversely proportional to the bulk modulus $$K$$. Therefore, a smaller bulk modulus results in a larger fractional compression.

Given the bulk moduli in units of $$10^9$$ Nm$$^{-2}$$:

• Ethanol: $$K_e = 0.9$$
• Mercury: $$K_m = 25$$
• Water: $$K_w = 2.2$$

Comparing the values:

$$ K_e = 0.9, \quad K_w = 2.2, \quad K_m = 25 $$

So, $$K_e < K_w < K_m$$.

Since fractional compression is inversely proportional to $$K$$:

$$ \left| \frac{\Delta V}{V} \right| \propto \frac{1}{K} $$

The order of fractional compression from largest to smallest is:

Ethanol (smallest $$K$$) > Water (next smallest $$K$$) > Mercury (largest $$K$$)

Thus, $$\frac{\Delta V}{V}$$ is largest for ethanol, followed by water, and smallest for mercury.

Now, comparing with the options:

A. Ethanol > Water > Mercury

B. Water > Ethanol > Mercury

C. Mercury > Ethanol > Water

D. Ethanol > Mercury > Water

Option A matches our result.

Hence, the correct answer is Option A.