Question

The change in volume of a certain mass of liquid is observed to be \(1/500\) of original volume, when pressure on it is increased by \(50 \text{ kgf/cm}^2\) (or \(5000\text{ kN/m}^2\)). Find the bulk modulus of liquid

• A. \(2.5\times10^4 \text{ kgf/cm}^2\)
• B. \(5.5\times10^4 \text{ kgf/cm}^2\)
• C. \(9.5\times10^4 \text{ kgf/cm}^2\)
• D. \(15.6\times10^4 \text{ kgf/cm}^2\)

Hint:

Bulk modulus is simply the reciprocal of compressibility.
Multiply the change in pressure directly by the inverse of the fractional change:
\[ 50 \cdot 500 = 25000 = 2.5 \times 10^4 \]

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The Bulk Modulus (\(K\)) measures a fluid's resistance to uniform compression.
It is defined as the ratio of the increase in pressure to the resulting relative decrease in volume.
Key Formula or Approach:
The formula for the bulk modulus is:
\[ K = \frac{\Delta P}{-\left(\frac{\Delta V}{V}\right)} \]
Where:
- \(\Delta P\) is the change in pressure.
- \(\Delta V / V\) is the volumetric strain (the fractional change in volume).
The negative sign represents the decrease in volume that occurs as pressure increases, ensuring \(K\) remains positive.

Step 2: Detailed Explanation:
Let us substitute the given values into the formula:
- Change in pressure (\(\Delta P\)) = \(50 \text{ kgf/cm}^2\).
- Volumetric strain (\(\Delta V / V\)) = \(-1/500\) (since the volume decreases).
Now, calculate the bulk modulus:
\[ K = \frac{50}{-\left(-\frac{1}{500}\right)} \]
\[ K = 50 \cdot 500 \]
\[ K = 25000 \text{ kgf/cm}^2 \]
Express this value in scientific notation:
\[ K = 2.5 \times 10^4 \text{ kgf/cm}^2 \]
This matches Option A.

Step 3: Final Answer:
The bulk modulus of the liquid is \(2.5\times10^4 \text{ kgf/cm}^2\).