The approximate depth of an ocean is 2700 m. The compressibility of water is `45.4 xx 10^(-11) Pa^(-1)` and density of water is `10^(3) kg//m^(3)`. What fractional compression of water will be obtained at the bottom of the ocean ?

Options

A. `1.0 xx 10^(-2)`
B. `1.2 xx 10^(-2)`
C. `1.4 xx 10^(-2)`
D. `0.8 xx 10^(-2)`

Explanation

To find the fractional compression of water at the bottom of the ocean, we can follow these steps:

### Step 1: Understand the relationship between compressibility, pressure, and fractional change in volume.
The compressibility (β) of a substance is defined as:
\[ \beta = -\frac{1}{V} \frac{dV}{dP} \]
where \(dV\) is the change in volume, \(V\) is the original volume, and \(dP\) is the change in pressure. This can be rearranged to express the fractional change in volume (\(\frac{dV}{V}\)):
\[ \frac{dV}{V} = -\beta \cdot dP \]

### Step 2: Calculate the change in pressure (\(dP\)) at the depth of the ocean.
The change in pressure due to the depth of the ocean can be calculated using the formula:
\[ dP = \rho g h \]
where:
- \(\rho = 1000 \, \text{kg/m}^3\) (density of water)
- \(g = 9.8 \, \text{m/s}^2\) (acceleration due to gravity)
- \(h = 2700 \, \text{m}\) (depth of the ocean)

Substituting the values:
\[ dP = 1000 \times 9.8 \times 2700 \]
Calculating this gives:
\[ dP = 26,460,000 \, \text{Pa} \, \text{(or N/m}^2\text{)} \]

### Step 3: Substitute \(dP\) into the fractional change in volume formula.
Now that we have \(dP\), we can substitute it into the formula for fractional change in volume:
\[ \frac{dV}{V} = -\beta \cdot dP \]
Given that \(\beta = 45.4 \times 10^{-11} \, \text{Pa}^{-1}\), we substitute:
\[ \frac{dV}{V} = -45.4 \times 10^{-11} \cdot 26,460,000 \]

### Step 4: Calculate the fractional change in volume.
Calculating this gives:
\[ \frac{dV}{V} = -45.4 \times 10^{-11} \times 26,460,000 \approx -0.012012 \]

### Step 5: Express the result as a positive value for fractional compression.
Since we are interested in the magnitude of fractional compression:
\[ \text{Fractional Compression} = 0.012012 \approx 1.2 \times 10^{-2} \]

### Final Answer:
The fractional compression of water at the bottom of the ocean is approximately:
\[ 1.2 \times 10^{-2} \]

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