The density of water at the surface of ocean is `rho` . If the bulk modulus of water is `B`, then the density of ocean water at depth, when the pressure is `alphap_(0)` and `p_(0)` is the atmospheric pressure is

Options

A. `(pB)/(B-(alpha-1)p_(0))`
B. `(pB)/(B+(alpha-1)p_(0))`
C. `(pB)/(B-alphap_(0))`
D. `(pB)/(B+alphap_(0))`

Explanation

To find the density of ocean water at a certain depth, we can use the relationship between pressure, bulk modulus, and density. Here’s a step-by-step solution:

### Step 1: Understand the relationship between pressure and density
The density of a fluid changes when it is subjected to pressure. The relationship can be expressed using the bulk modulus (B) of the fluid. The bulk modulus is defined as:

\[ B = -V \frac{dP}{dV} \]

Where \( V \) is the volume, \( dP \) is the change in pressure, and \( dV \) is the change in volume.

### Step 2: Calculate the change in pressure
At a depth \( h \) in the ocean, the pressure \( P \) is given by:

\[ P = P_0 + \rho g h \]

Where \( P_0 \) is the atmospheric pressure, \( \rho \) is the density of water at the surface, and \( g \) is the acceleration due to gravity. In this case, we are given that the pressure at depth is \( \alpha P_0 \).

### Step 3: Relate the change in pressure to the change in density
The change in pressure from the surface to the depth can be expressed as:

\[ \Delta P = P - P_0 = \alpha P_0 - P_0 = (\alpha - 1) P_0 \]

### Step 4: Use the bulk modulus to find the change in density
Using the bulk modulus, we can relate the change in pressure to the change in density. The change in density \( \Delta \rho \) can be expressed as:

\[ \Delta \rho = -\frac{B}{V} \Delta P \]

Assuming the initial volume is constant, we can simplify this to:

\[ \Delta \rho = -\frac{B}{\rho} (\alpha - 1) P_0 \]

### Step 5: Calculate the density at depth
The density at depth \( \rho_d \) can be found by adding the change in density to the surface density:

\[ \rho_d = \rho + \Delta \rho \]

Substituting \( \Delta \rho \):

\[ \rho_d = \rho - \frac{B}{\rho} (\alpha - 1) P_0 \]

### Final Expression
Thus, the density of ocean water at the depth where the pressure is \( \alpha P_0 \) is given by:

\[ \rho_d = \rho - \frac{B (\alpha - 1)}{\rho} P_0 \]