Question 23

A certain pressure '$$P$$' is applied to 1 litre of water and 2 litre of a liquid separately. Water gets compressed to 0.01% whereas the liquid gets compressed to 0.03%. The ratio of Bulk modulus of water to that of the liquid is $$\frac{3}{x}$$. The value of $$x$$ is ______.


Correct Answer: 1

Solution

Bulk modulus $$B$$ is defined as the ratio of applied pressure to fractional decrease in volume.
Formula: $$B = \frac{P}{\Delta V / V}$$ $$-(1)$$

For water, initial volume does not affect the modulus. The fractional compression is 0.01%.
Thus $$\frac{\Delta V}{V} = 0.01\% = 0.0001$$ $$-(2)$$ Substituting into (1): $$B_{\text{water}} = \frac{P}{0.0001} = 10000\,P$$

For the liquid, the fractional compression is 0.03%.
Thus $$\frac{\Delta V}{V} = 0.03\% = 0.0003$$ $$-(3)$$ Substituting into (1): $$B_{\text{liquid}} = \frac{P}{0.0003} = 3333.33\,P$$

Ratio of bulk moduli:
$$\frac{B_{\text{water}}}{B_{\text{liquid}}} = \frac{10000\,P}{3333.33\,P} = 3$$ $$-(4)$$ Given that this ratio equals $$\frac{3}{x}$$, we set $$\frac{3}{x} = 3$$ Therefore, $$x = 1$$

Final Answer: $$x = 1$$

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