Question

How much should the pressure on a litre of water be changed to compress it by 0.1%? (Bulk modulus of water $B=2.2\times {{10}^{9}}N{{m}^{-2}}$).

Answer 1

Hint:The change in volume of a substance with respect to a change in its pressure is understood by a term called bulk modulus of the substance. Bulk modulus for a material is defined as the ratio of the pressure to the fraction of the change in volume of the material.

Formula used:
$B=\dfrac{P}{\left( \dfrac{\Delta V}{V} \right)}$
$B$ is bulk modulus, $P$ is applied pressure and $\dfrac{\Delta V}{V}$ is fractional change in volume.

Complete step by step answer:
When we apply force on a body, the force has the ability to change the shape and size of the body. If we apply the force on the whole surface of the body, then we talk about the applied pressure in the pace of force since pressure is the force exerted per unit area. Therefore, when some extra (excessive) pressure is exerted on a liquid or a gas, the volume of the liquid or the gas changes. The change in volume of a substance with respect to a change in its pressure is understood by a term called bulk modulus of the substance.

Bulk modulus for a material is defined as the ratio of the pressure to the fraction of the change in volume of the material.
i.e. $B=\dfrac{P}{\left( \dfrac{\Delta V}{V} \right)}$ …… (i),
where B is the bulk modulus, P is the pressure applied and $\dfrac{\Delta V}{V}$ is the fraction by which the volume of the substance changes.
It is given that the initial volume of water is 1 litre and we have to compress it by 0.1%. This means that the fractional change in the volume of the water is $\dfrac{\Delta V}{V}=\dfrac{0.1}{100}=0.001$.
The bulk modulus of water is given to be $B=2.2\times {{10}^{9}}N{{m}^{-2}}$.
Substitute the values in (i).
$2.2\times {{10}^{9}}=\dfrac{P}{0.001}$
$\Rightarrow P=2.2\times {{10}^{9}}\times 0.001\\
\therefore P =2.2\times {{10}^{6}}N{{m}^{-2}}$.

This means that the pressure of the water must be increased by $2.2\times {{10}^{6}}N{{m}^{-2}}$ to compress it by 0.1%.

Note:In liquids and gas always have some pressure exerted by the molecules within the liquid or the gas. Therefore, we apply an external force to compress the liquid, there is a change in pressure of the liquid. Fractional change in volume is the ratio of the change in volume to the original volume. Note that the unit of bulk modulus is the same as that of pressure.