The compressibility of water is `4xx10^-5` per unit atmospheric pressure. The decrease in volume of 100 cubic centimetre of water under a pressure of 100 atmosphere will be

Options

A. `0.4cm^(3)`
B. `4xx10^(-5) cm^(3)`
C. `0.25cm^(3)`
D. `0.4cm^(3)`

Explanation

To solve the problem of finding the decrease in volume of 100 cubic centimeters of water under a pressure of 100 atmospheres, given the compressibility of water, we can follow these steps:

### Step 1: Understand the given data
- Compressibility of water, \( \beta = 4 \times 10^{-5} \) per unit atmospheric pressure.
- Initial volume of water, \( V = 100 \) cm³.
- Pressure applied, \( P = 100 \) atm.

### Step 2: Convert compressibility to appropriate units
Compressibility is often expressed in terms of volume change per unit pressure. The formula for compressibility is given by:
\[ \beta = -\frac{\Delta V}{P \cdot V} \]
Where:
- \( \Delta V \) is the change in volume,
- \( P \) is the pressure applied,
- \( V \) is the original volume.

### Step 3: Convert atmospheric pressure to SI units
1 atm = \( 10^5 \) N/m². Therefore, 100 atm can be converted to:
\[ P = 100 \times 10^5 \, \text{N/m}^2 = 10^7 \, \text{N/m}^2 \]

### Step 4: Convert the volume from cm³ to m³
Since we need to work in SI units, we convert the volume:
\[ V = 100 \, \text{cm}^3 = 100 \times 10^{-6} \, \text{m}^3 = 10^{-4} \, \text{m}^3 \]

### Step 5: Rearranging the compressibility formula
From the compressibility formula, we can rearrange it to find \( \Delta V \):
\[ \Delta V = -\beta \cdot P \cdot V \]

### Step 6: Substitute the values into the equation
Substituting the values we have:
\[ \Delta V = - (4 \times 10^{-5}) \cdot (10^7) \cdot (10^{-4}) \]

### Step 7: Calculate \( \Delta V \)
Calculating the above expression:
\[ \Delta V = - (4 \times 10^{-5}) \cdot (10^{3}) = -4 \times 10^{-2} \, \text{m}^3 \]

### Step 8: Convert \( \Delta V \) back to cm³
To convert \( \Delta V \) to cm³:
\[ \Delta V = -4 \times 10^{-2} \, \text{m}^3 = -4 \times 10^{4} \, \text{cm}^3 = -40 \, \text{cm}^3 \]

### Step 9: Present the final answer
The decrease in volume of 100 cubic centimeters of water under a pressure of 100 atmospheres is:
\[ \Delta V = -0.4 \, \text{cm}^3 \]

### Final Answer
The decrease in volume is \( 0.4 \, \text{cm}^3 \).

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