Question

The density of the water at the surface of the ocean is $\rho$. If the bulk modulus of the water is $B$, what is the density of ocean water at a depth where the pressure is $n{P}_0$, where $P_0$ is the atmospheric pressure?

• A. $\frac{\rho B}{B(n-1)P_0}$
• B. ${\rho }_1\left(\frac{B}{B-P_0(n-1)}\right)$
• C. $\frac{\rho B}{B-n{P}_0}$
• D. $\frac{\rho B}{B+n{P}_0}$

Answer 1

The correct option is B ${\rho }_1\left(\frac{B}{B-P_0(n-1)}\right)$

$\begin{array}{cc}\text{At surface ,}& \text{density}={\rho }_1\\ \text{volume}& ={\rho }_2\end{array}$

$\text{As pressure increases volume decreases and density}$ $\text{increases.Also, mass remains conserved.}$

$$\begin{array}{cc}{\rho }_1{V}_1& ={\rho }_2{V}_2\\ {\rho }_2& =\frac{{\rho }_1{V}_1}{V_2}\\ B& =\frac{-dP}{dV/V}=\frac{V_1\Delta P}{\Delta V}=\frac{V_1}{\Delta V}(n-1)P_0\end{array}$$

$\begin{array}{cc}\therefore \Delta V& =\frac{V_1}{B}(n-1)P_0\\ V_2& =V_1-\Delta V\\ V_2& =V_1(1-\frac{P_0(n-1)}{B})\\ & \frac{V_2}{V_1}=\frac{B-P_0(n-1)}{B}\end{array}$

$\therefore {\rho }_2={\rho }_1\left(\frac{B}{B-P_0(n-1)}\right)$