Problem 931 - JEE Physics

Question

The bulk modulus of an ideal gas at constant temperature..... (A) is equal to its volume \(\mathrm{V}\) (B) is equal to \(\mathrm{P} / 2\) (C) is equal to its pressure \(\mathrm{P}\) (D) cannot be determined

Step-by-Step Solution

Answer

The bulk modulus of an ideal gas at constant temperature is equal to its pressure \(\mathrm{P}\) (option C). This conclusion was reached by differentiating the ideal gas law and substituting the resulting expression for ΔP into the bulk modulus equation, which yielded \(K = P\).

Step 1: Recall the definition of bulk modulus

The bulk modulus (K) of a substance measures its resistance to compression under pressure. For an ideal gas, the bulk modulus is defined as the ratio of applied pressure (ΔP) to the fractional change in volume (ΔV/V): \[K = -\dfrac{ΔP}{ΔV/V}\]

Step 2: Write the ideal gas law

For an ideal gas, the relationship between pressure, volume, and temperature is given by the ideal gas law: \[PV = nRT\] where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature.

Step 3: Differentiate the ideal gas law

Since bulk modulus is defined for a constant temperature, we can treat T as constant and differentiate the ideal gas law with respect to volume. Using the chain rule, we get: \[\dfrac{dP}{dV}V + P = 0\]

Step 4: Rearrange to find ΔP

Solving for the change in pressure (ΔP), we can write: \[ΔP = - \dfrac{P}{V} ΔV\]

Step 5: Substitute ΔP back into the bulk modulus equation

Now, we will substitute the value of ΔP we found into the bulk modulus equation: \[K = -\dfrac{-\displaystyle\dfrac{P}{V} ΔV}{ΔV/V}\]

Step 6: Simplify the expression

We can cancel out the ΔV terms in the numerator and denominator and simplify our expression for bulk modulus: \[K = P\] This means the bulk modulus of an ideal gas at constant temperature is equal to its pressure. So, the correct answer is: (C) is equal to its pressure \(\mathrm{P}\)

Key Concepts

Understanding the Ideal Gas

The ideal gas concept is a theoretical model used to represent a gas that perfectly follows certain physical laws. These laws describe how molecules in gases behave under varying conditions of pressure, volume, and temperature. An ideal gas is characterized by its molecules, which are point particles that have no volume and no interactions between them except during elastic collisions. The ideal gas law is expressed by the equation\[ PV = nRT \]where \( P \) stands for pressure, \( V \) for volume, \( n \) for the number of moles, \( R \) the ideal gas constant, and \( T \) the temperature. This law links the measurable properties of a gas, allowing us to predict how a gas will respond to changes in these conditions.

Exploring Pressure in Gases

Pressure is a measure of the force exerted by gas particles as they collide with the walls of their container. In terms of ideal gases, pressure depends on the number of particles, their energy (temperature), and the volume of their container. The relationship between pressure and volume is crucial in understanding how gases behave. According to the ideal gas law, if the temperature and amount of gas remain constant, an increase in pressure results in a decrease in volume and vice versa. Understanding pressure helps us in calculating other gas properties, like the bulk modulus, which measures how much pressure a gas can handle before it changes its volume.

Volume and Its Role in Gas Behavior

Volume refers to the space that a gas occupies. In the context of an ideal gas, volume is directly related to pressure and temperature through the ideal gas law \( PV = nRT \). Altering the volume of a container while keeping temperature constant directly affects the pressure exerted by the gas. Decreasing the volume increases the pressure, as there is less space for the gas particles to move around, leading to more frequent collisions with the container walls. The relationship between volume and pressure is inverse, which is critical for understanding compression resistance, the ability of a gas to resist changes in volume when pressure is applied.

Concept of Compression Resistance

Compression resistance is the ability of a material, including gases, to withstand changes in volume under the application of pressure. For an ideal gas, this resistance is quantified by the bulk modulus, \( K \), which is calculated as\[ K = -\dfrac{\Delta P}{\Delta V/V} \]At constant temperature, bulk modulus simplifies to the pressure of the ideal gas. Thus, for an ideal gas, its resistance to compression is equal to its pressure \( P \). This relationship shows that understanding how an ideal gas maintains its volume under pressure helps in designing systems that manage gas flow and storage efficiently.