The pressure of a medium is changed from 1.01 x 10^5 Pa to 1.165 x 10^5 Pa and change in volume is 1

Q: Why is there a negative sign in the formula for Bulk Modulus?

The negative sign is a physical convention. When pressure increases ($\Delta P > 0$), the volume of the material decreases ($\Delta V < 0$). The negative sign ensures that the Bulk Modulus ($B$) remains a positive physical quantity.

Q: What is the difference between Isothermal and Adiabatic Bulk Modulus?

Isothermal Bulk Modulus is measured when temperature is kept constant during compression ($B = P$ for an ideal gas). Adiabatic Bulk Modulus is measured when no heat is exchanged with the surroundings ($B = \gamma P$ for an ideal gas). Adiabatic Bulk Modulus is always larger because the temperature rise during rapid compression increases the pressure resistance.

Q: Why does the fractional change in volume not have any units?

Volumetric strain is defined as the ratio of change in volume ($\Delta V$) to the original volume ($V$). Since both quantities are measured in the same units (e.g., cubic meters), the units cancel out, making strain a dimensionless quantity.

Q: Can Bulk Modulus be negative?

No, for any stable material, Bulk Modulus must be positive. A negative bulk modulus would mean that squeezing an object causes it to expand, which violates thermodynamic stability.

Q: How does Bulk Modulus relate to the speed of sound?

The speed of sound in a medium depends directly on its Bulk Modulus ($B$) and density ($\rho$) through the formula $v = \sqrt{B/\rho}$. A stiffer medium (higher $B$) allows sound waves to propagate much faster.

Question

The pressure of a medium is changed from $1.01 	imes 10^5	ext{ Pa}$ to $1.165 	imes 10^5	ext{ Pa}$ and change in volume is $10\%$ keeping temperature constant. The bulk modulus of the medium is

Options:

(A) $204.8 	imes 10^5	ext{ Pa}$

(B) $102.4 	imes 10^5	ext{ Pa}$

(C) $51.2 	imes 10^5	ext{ Pa}$

(D) $1.55 	imes 10^5	ext{ Pa}$

Sigma Percentile · Accepted Answer

### The Magic of Squeezing: Understanding Bulk Modulus

Imagine holding a sponge ball in your hand and squeezing it from all directions. What happens? The ball shrinks uniformly, its volume decreases, and you feel a resistive force pushing back against your fingers. This simple, everyday experience is the gateway to one of the most fundamental concepts in the physics of materials: **Bulk Modulus**.

In this problem, we are asked to find the Bulk Modulus of a medium whose pressure is increased from $1.01 	imes 10^5	ext{ Pa}$ to $1.165 	imes 10^5	ext{ Pa}$, causing its volume to contract by $10\%$. Let us embark on a journey to understand how materials respond to being squeezed from all sides, and how we can mathematically quantify this resistance.

### The Physics of Volumetric Stress and Strain

When we apply a force uniformly over the entire surface of an object, we are applying pressure. This uniform pressure is called **volumetric stress** or **bulk stress**. Unlike tensile stress, which pulls an object along a single axis, or shear stress, which slides layers of an object past each other, volumetric stress acts in three dimensions simultaneously.

How does the material respond? It undergoes **volumetric strain**, which is defined as the fractional change in volume:

$$	ext{Volumetric Strain} = \frac{\Delta V}{V}$$

Here, $\Delta V$ is the change in volume, and $V$ is the original volume. Because a positive increase in pressure always leads to a decrease in volume, the change in volume $\Delta V$ is negative. To keep our physical constant positive, we define the **Bulk Modulus ($B$)** with a negative sign:

$$B = -\frac{\Delta P}{\frac{\Delta V}{V}}$$

This elegant formula tells us that Bulk Modulus is simply the ratio of the change in pressure to the fractional change in volume. It is a direct measure of how incompressible a substance is. A massive Bulk Modulus, like that of steel or diamond, means it takes an enormous amount of pressure to cause even a tiny change in volume. A small Bulk Modulus, like that of air or water, means the substance is much easier to compress.

### Step-by-Step Execution: Breaking Down the Math

Let us look at the numbers given to us. The initial pressure of our medium is:

$$P_1 = 1.01 	imes 10^5	ext{ Pa}$$

And the final pressure is:

$$P_2 = 1.165 	imes 10^5	ext{ Pa}$$

To find the volumetric stress, we calculate the change in pressure, $\Delta P$:

$$\Delta P = P_2 - P_1 = 1.165 	imes 10^5 - 1.01 	imes 10^5$$

$$\Delta P = (1.165 - 1.01) 	imes 10^5 = 0.155 	imes 10^5	ext{ Pa}$$

This is the extra pressure squeezing our medium.

Now, let us address the volumetric strain. The problem states that the volume changes by $10\%$. This means the fractional change in volume is:

$$\frac{\Delta V}{V} = -10\% = -0.10$$

Substituting these values into our Bulk Modulus formula, we get:

$$B = -\frac{0.155 	imes 10^5}{-0.10}$$

Notice how the two negative signs cancel out beautifully, leaving us with a positive value for the Bulk Modulus:

$$B = \frac{0.155}{0.10} 	imes 10^5 = 1.55 	imes 10^5	ext{ Pa}$$

And there we have it! The Bulk Modulus of the medium is **$1.55 	imes 10^5	ext{ Pa}$**, which corresponds perfectly to **Option (d)**.

### Deep Physical Insights: Compressibility and Beyond

Why is Bulk Modulus so important? It is the key to understanding sound waves traveling through fluids, the behavior of deep-sea submersibles, and even the stability of stars!

There is another closely related concept called **Compressibility ($K$)**, which is simply the reciprocal of the Bulk Modulus:

$$K = \frac{1}{B}$$

While Bulk Modulus measures a material's *resistance* to compression, compressibility measures how *easy* it is to compress. Gases have extremely high compressibility (and low bulk modulus), while solids and liquids have very low compressibility (and high bulk modulus).

In our problem, the temperature was kept constant, which means we calculated the **isothermal bulk modulus**. If the compression had happened so quickly that no heat could escape, we would have calculated the **adiabatic bulk modulus**, which is always larger because the rising temperature of the compressed material adds an extra thermal pressure resisting the squeeze. Physics is full of these beautiful, interconnected layers!

Animated Solution for Physics - Properties of Solids and Liquids: The pressure of a medium is changed from $1.01 \times 1 0^{5} Pa$ to $1.165 \times 1 0^{5} Pa$ and change in volume is $10%$ keeping temperature constant. The bulk modulus of the medium is

Select Answer:

Visualized Solution

Visualizing Volumetric Compression

Defining Bulk Modulus

Extracting the Given Pressures

Calculating Change in Pressure ( $Δ P$ )

Determining Volumetric Strain

Substituting into the Bulk Modulus Formula

Computing the Final Value

Matching the Option & Physical Insight

The Sigma Insight: Young's Modulus, Bulk Modulus and Modulus of Rigidity

The Magic of Squeezing

The Physics of Volumetric Stress and Strain

Step-by-Step Execution

Deep Physical Insights

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