Question

The pressure of a medium is changed from \[1.01 \times {10^5}\,{\text{Pa}}\] to \[1.165 \times {10^5}\,{\text{Pa}}\] and change in volume is 10% keeping the temperature constant. The bulk modulus of the medium is

Answer 1

Hint: Use the formula for bulk modulus of a material. This formula gives the relation between the bulk modulus of the material, change in pressure on the material, original volume of the material and change in volume of the material. Substitute all the values in this equation and determine the value of bulk modulus of the medium.

Formula used:
The formula for bulk modulus \[K\] of a material is given by
\[K = \dfrac{{V\Delta P}}{{\Delta V}}\] …… (1)
Here, \[\Delta P\] is the change in pressure on the material, \[V\] is the original volume of the material and \[\Delta V\] is the change in volume of the material.

Complete step by step solution:
We have given that the pressure of a medium is changed from \[1.01 \times {10^5}\,{\text{Pa}}\] to \[1.165 \times {10^5}\,{\text{Pa}}\].

The initial pressure of the medium is \[1.01 \times {10^5}\,{\text{Pa}}\] and the final pressure of the medium is \[1.165 \times {10^5}\,{\text{Pa}}\].
\[{P_i} = 1.01 \times {10^5}\,{\text{Pa}}\]
\[{P_f} = 1.165 \times {10^5}\,{\text{Pa}}\]

We can determine the change in pressure of the medium using the following equation.
\[\Delta P = {P_f} - {P_i}\]

Substitute \[1.165 \times {10^5}\,{\text{Pa}}\] for \[{P_f}\] and \[1.01 \times {10^5}\,{\text{Pa}}\] for \[{P_i}\] in the above equation.
\[\Delta P = \left( {1.165 \times {{10}^5}\,{\text{Pa}}} \right) - \left( {1.01 \times {{10}^5}\,{\text{Pa}}} \right)\]
\[ \Rightarrow \Delta P = 0.155 \times {10^5}\,{\text{Pa}}\]
\[ \Rightarrow \Delta P = 1.55 \times {10^4}\,{\text{Pa}}\]

Hence, the change in pressure of the medium is \[1.55 \times {10^4}\,{\text{Pa}}\].

Also the change in volume of the medium 10%. Hence, the fraction of the change in volume of the medium with the original volume of the medium is 0.1.
\[\dfrac{{\Delta V}}{V} = 0.1\]

We can determine the bulk modulus of the medium using equation (1).

Substitute \[1.55 \times {10^4}\,{\text{Pa}}\] for \[\Delta P\] and 0.1 for \[\dfrac{{\Delta V}}{V}\] in equation (1).
\[K = \dfrac{{1.55 \times {{10}^4}\,{\text{Pa}}}}{{0.1}}\]
\[ \Rightarrow K = 1.55 \times {10^5}\,{\text{Pa}}\]

Hence, the bulk modulus of the medium is \[1.55 \times {10^5}\,{\text{Pa}}\].

Note:
One can also solve the same question in another way. The bulk modulus of a material is the ratio of bulk stress and bulk strain.
One can determine first bulk stress and bulk strain on the medium and then substitute these values in the formula to determine the bulk modulus of the medium.