A given quantity of an ideal gas is at pressure p and absolute temperature T. The isothermal bulk mo

Q: Why is there a negative sign in the definition of Bulk Modulus?

The negative sign is introduced because an increase in pressure ($dP > 0$) always causes a decrease in volume ($dV < 0$). To ensure that the Bulk Modulus $B$ is a positive physical quantity, we multiply by $-1$.

Q: Does the bulk modulus of a gas remain constant during an isothermal process?

No, because the isothermal bulk modulus is equal to the pressure ($B = P$). As the gas is compressed isothermally, its pressure increases, which means its bulk modulus also increases, making it harder to compress further.

Q: What is the difference between isothermal and adiabatic bulk modulus?

The isothermal bulk modulus is $B_{\text{iso}} = P$, while the adiabatic bulk modulus is $B_{\text{ad}} = \gamma P$. Since $\gamma > 1$, the adiabatic bulk modulus is always greater, meaning a gas resists compression more when heat is trapped.

Q: Can we apply this formula to real gases?

This derivation assumes an ideal gas ($PV = nRT$). For real gases, intermolecular forces and molecular volume cause deviations, so the bulk modulus will differ slightly, especially at very high pressures or low temperatures.

Q: What are the units of Bulk Modulus?

Since volumetric strain ($\frac{dV}{V}$) is dimensionless, the Bulk Modulus has the same units as pressure, which is Pascals ($\text{Pa}$) or Newtons per square meter ($\text{N/m}^2$).

Question

A given quantity of an ideal gas is at pressure $p$ and absolute temperature $T$. The isothermal bulk modulus of the gas is

Options:

(A) $\frac{2}{3}p$

(B) $p$

(C) $\frac{3}{2}p$

(D) $2p$

Sigma Percentile · Accepted Answer

### Introduction to Elasticity in Gases

When we think of elasticity, our minds often jump to solid objects—stretching a rubber band, bending a steel rod, or compressing a spring.

However, **fluids (both liquids and gases) also exhibit elastic properties** when subjected to uniform compression.

For a gas, this resistance to compression is highly dependent on the thermodynamic conditions under which the compression occurs.

In this article, we will explore the concept of **Bulk Modulus** and derive its value specifically for an **isothermal process** of an ideal gas.

---

### Understanding Bulk Modulus ($B$)

The **Bulk Modulus** ($B$) is a measure of a substance's resistance to uniform compression.

It is defined as the ratio of the infinitesimal pressure increase (volumetric stress) to the resulting relative decrease of the volume (volumetric strain):

$$B = -\frac{dP}{\frac{dV}{V}} = -V \frac{dP}{dV}$$

Why is there a negative sign in the formula?

When we increase the pressure on a gas ($dP > 0$), its volume decreases ($dV < 0$).

To ensure that the Bulk Modulus $B$ remains a **positive physical quantity**, we introduce the negative sign.

---

### The Isothermal Condition

In an **isothermal process**, the temperature $T$ of the system remains constant throughout the process.

For a given quantity of an ideal gas, the equation of state is given by Boyle's Law:

$$PV = 	ext{constant}$$

Let us denote this constant as $C$:

$$PV = C$$

This simple hyperbolic relationship on a $P-V$ diagram tells us that pressure and volume are inversely proportional when temperature is held constant.

---

### Deriving the Isothermal Bulk Modulus

To find the term $\frac{dP}{dV}$ needed for our Bulk Modulus formula, we differentiate both sides of the isothermal equation of state with respect to volume $V$:

$$\frac{d}{dV}(PV) = \frac{d}{dV}(C)$$

Since $C$ is a constant, its derivative with respect to $V$ is zero:

$$\frac{d}{dV}(PV) = 0$$

Applying the **product rule** of differentiation on the left-hand side:

$$P \frac{d(V)}{dV} + V \frac{dP}{dV} = 0$$

Since $\frac{d(V)}{dV} = 1$, this simplifies to:

$$P + V \frac{dP}{dV} = 0$$

Now, we rearrange the terms to solve for the derivative $\frac{dP}{dV}$:

$$V \frac{dP}{dV} = -P$$

$$\frac{dP}{dV} = -\frac{P}{V}$$

This derivative represents the slope of the tangent to the isothermal curve on a $P-V$ diagram at any point $(V, P)$.

---

### Final Substitution and Calculation

Now, let us substitute this expression for $\frac{dP}{dV}$ back into our fundamental definition of the Bulk Modulus:

$$B = -V \frac{dP}{dV}$$

$$B = -V \left(-\frac{P}{V}ight)$$

Notice how beautifully the terms simplify!

The volume $V$ in the numerator cancels out with the $V$ in the denominator, and the two negative signs multiply to become positive:

$$B = P$$

Thus, the **isothermal bulk modulus of an ideal gas is exactly equal to its pressure $P$**.

This is an incredibly elegant result! It tells us that the resistance of a gas to compression under constant temperature is directly proportional to its current pressure.

A gas at high pressure is much harder to compress than a gas at low pressure.

---

### The Way Forward: Isothermal vs. Adiabatic Bulk Modulus

What if the compression happened so quickly that no heat could escape? This would be an **adiabatic process**, described by:

$$PV^{\gamma} = 	ext{constant}$$

If we perform a similar derivation for the adiabatic bulk modulus ($B_{	ext{ad}}$), we get:

$$B_{	ext{ad}} = \gamma P$$

Since the adiabatic index $\gamma = \frac{C_p}{C_v}$ is always greater than $1$ for any gas, we have:

$$B_{	ext{ad}} > B_{	ext{iso}}$$

This means that **a gas is always harder to compress adiabatically than isothermally**.

In an adiabatic compression, the temperature rises because heat cannot escape, which further increases the pressure and makes the gas resist compression even more strongly!

Animated Solution for Physics - Thermodynamics: A given quantity of an ideal gas is at pressure $p$ and absolute temperature $T$ . The isothermal bulk modulus of the gas is

Select Answer:

Visualized Solution

Visualizing the Isothermal Process

Defining Bulk Modulus ( $B$ )

The Isothermal Condition

Differentiating the Equation of State

Applying the Product Rule

Simplifying the Derivative

Solving for $\frac{d P}{d V}$

Substituting into the Bulk Modulus Formula

Final Calculation of Isothermal Bulk Modulus

The Way Forward: Adiabatic Bulk Modulus

The Sigma Insight: Thermodynamic Processes

Introduction to Elasticity in Gases

Understanding Bulk Modulus ( $B$ )

The Isothermal Condition

Deriving the Isothermal Bulk Modulus

Final Substitution and Calculation

The Way Forward

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Animated Solution for Physics - Thermodynamics: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A given quantity of an ideal gas is at pressure p and absolute temperature T. The isothermal bulk modulus of the gas is

Select Answer:

Visualized Solution

Visualizing the Isothermal Process

Defining Bulk Modulus ( .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } B)

The Isothermal Condition

Differentiating the Equation of State

Applying the Product Rule

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