The bulk modulus of water is `2.0xx10^(9) N//m^(2)`. The pressure required to increase the density of water by `0.1%` is

Options

A. `2xx10^(9)N//m^(2)`
B. `2xx10^(8)N//m^(2)`
C. `2xx10^(6)//m^(2)`
D. `2xx10^(4)N//m^(2)`

Explanation

To find the pressure required to increase the density of water by 0.1%, we can use the relationship between bulk modulus, pressure, and density change. Here’s a step-by-step solution:

### Step 1: Understand the Bulk Modulus
The bulk modulus (B) is defined as the ratio of the change in pressure (ΔP) to the relative change in volume (ΔV/V). Mathematically, it is expressed as:
\[ B = -\frac{\Delta P}{\frac{\Delta V}{V}} \]

### Step 2: Relate Density Change to Volume Change
Since mass (m) is constant, we can express the relationship between density (ρ) and volume (V) as:
\[ m = \rho \cdot V \]
Differentiating this gives:
\[ \rho \cdot dV + V \cdot d\rho = 0 \]
From this, we can derive:
\[ -\frac{dV}{V} = \frac{d\rho}{\rho} \]

### Step 3: Substitute into Bulk Modulus Equation
From the bulk modulus equation, we can express the relative change in volume in terms of pressure:
\[ \frac{\Delta V}{V} = -\frac{\Delta P}{B} \]
Substituting this into the equation derived from density gives:
\[ \frac{d\rho}{\rho} = \frac{\Delta P}{B} \]

### Step 4: Rearrange for Pressure
From the above relationship, we can rearrange to find the pressure required to achieve a certain change in density:
\[ \Delta P = B \cdot \frac{d\rho}{\rho} \]

### Step 5: Calculate the Change in Density
Given that the density of water is to be increased by 0.1%, we can express this as:
\[ \frac{d\rho}{\rho} = \frac{0.1}{100} = 0.001 \]

### Step 6: Substitute Values
Now we can substitute the values into the equation:
- Bulk modulus, \( B = 2.0 \times 10^9 \, \text{N/m}^2 \)
- Change in density ratio, \( \frac{d\rho}{\rho} = 0.001 \)

Thus,
\[ \Delta P = 2.0 \times 10^9 \cdot 0.001 \]

### Step 7: Calculate the Pressure
Calculating this gives:
\[ \Delta P = 2.0 \times 10^6 \, \text{N/m}^2 \]

### Final Answer
The pressure required to increase the density of water by 0.1% is:
\[ \Delta P = 2.0 \times 10^6 \, \text{N/m}^2 \]

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