# Q12  Compute the bulk modulus of water from the following data: Initial volume = 100.0 litre, Pressure increase = 100.0 atm ($1 atm = 1.013 \times 10 ^ 5 Pa$ ), Final volume = 100.5 litre. Compare the bulk modulus of water with that of air (at constant temperature). Explain in simple terms why the ratio is so large.

S Sayak

Pressure Increase, P = 100.0 atm

Initial Volume = 100.0 l

Final volume = 100.5 l

Change in Volume = 0.5 l

Let the Bulk Modulus of water be B

$\\B=\frac{Stress}{Volumetric\ Strain}\\ B=\frac{P}{\frac{\Delta V}{V}}\\ B=\frac{100\times 1.013\times 10^{5}\times 100\times 10^{-3}}{0.5\times 10^{-3}}\\ B=2.026\times 10^{9}Nm^{-2}$

The bulk modulus of air is $B_{a}=1.0\times 10^{5}Nm^{-2}$

The Ratio of the Bulk Modulus of water to that of air is

$\\\frac{B}{B_{a}}=\frac{2.026\times 10^{9}}{1.0\times 10^{5}}\\ \frac{B}{B_{a}}=20260$

This ratio is large as for the same pressure difference the strain will be much larger in the air than that in water.

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