Depth of sea is maximum at Mariana Trench in West Pacific Ocean. Trench has a maximum depth of about `11km`. At bottom of trench water column above it exerts `1000 atm `pressure. Percentage change in density of sea water at such depth will be around
(Given , B `= 2xx10^(9) Nm^(-2) and P_(atm) = 1xx10^(5 Nm^(-2)))`

Options

A. about `5%`
B. about `10%`
C. about `3 %`
D. about `7 %`

Explanation

To find the percentage change in density of seawater at the maximum depth of the Mariana Trench, we can follow these steps:

### Step 1: Identify Given Values
- Maximum depth of the Mariana Trench, \( h = 11 \, \text{km} = 11000 \, \text{m} \)
- Pressure at this depth, \( P = 1000 \, \text{atm} \)
- Bulk modulus of seawater, \( B = 2 \times 10^9 \, \text{N/m}^2 \)
- Atmospheric pressure, \( P_{\text{atm}} = 1 \times 10^5 \, \text{N/m}^2 \)

### Step 2: Convert Pressure to Pascals
Convert the pressure from atm to pascals:
\[ P = 1000 \, \text{atm} \times P_{\text{atm}} = 1000 \times 1 \times 10^5 \, \text{N/m}^2 = 1 \times 10^8 \, \text{N/m}^2 \]

### Step 3: Use the Bulk Modulus Formula
The bulk modulus \( B \) is defined as:
\[ B = -\frac{\Delta P}{\frac{\Delta V}{V}} \]
Where:
- \( \Delta P \) is the change in pressure,
- \( \Delta V \) is the change in volume,
- \( V \) is the original volume.

Rearranging gives:
\[ \frac{\Delta V}{V} = -\frac{\Delta P}{B} \]

### Step 4: Calculate the Change in Volume
Substituting the values:
\[ \Delta V/V = -\frac{1 \times 10^8}{2 \times 10^9} = -\frac{1}{20} \]

### Step 5: Relate Volume Change to Density Change
Let the initial volume be \( V_1 \) and the final volume at depth be \( V_2 \):
\[ V_2 = V_1 \left(1 - \frac{1}{20}\right) = V_1 \left(\frac{19}{20}\right) \]

### Step 6: Calculate the Density at Depth
Density is defined as:
\[ \rho = \frac{m}{V} \]
Since mass \( m \) remains constant, we can express the density at depth \( \rho_2 \) as:
\[ \rho_2 = \frac{\rho_1 V_1}{V_2} = \frac{\rho_1 V_1}{V_1 \left(\frac{19}{20}\right)} = \frac{\rho_1}{\frac{19}{20}} = \frac{20 \rho_1}{19} \]

### Step 7: Calculate the Percentage Change in Density
The percentage change in density is given by:
\[ \text{Percentage Change} = \frac{\rho_2 - \rho_1}{\rho_1} \times 100 \]
Substituting for \( \rho_2 \):
\[ \text{Percentage Change} = \frac{\frac{20 \rho_1}{19} - \rho_1}{\rho_1} \times 100 = \frac{20 - 19}{19} \times 100 = \frac{1}{19} \times 100 \approx 5.26\% \]

### Final Answer
Thus, the percentage change in density of seawater at the depth of the Mariana Trench is approximately **5%**.

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