Problem 3 - A Complete Resource Book in Chemistry for JEE Main

Question

For an ideal gas Joule-Thomson coefficient is (a) positive (b) negative (c) zero (d) dependent on molecular weight

Step-by-Step Solution

Answer

The Joule-Thomson coefficient for an ideal gas is zero.

Step 1: Understanding the Joule-Thomson Effect

The Joule-Thomson effect describes the temperature change in a real gas when it is allowed to expand without doing external work and without exchanging heat with the environment. It is quantified by the Joule-Thomson coefficient, \( rac{dT}{dP} \), which represents the change in temperature with respect to pressure during this process.

Step 2: Ideal Gas Characteristics

For an ideal gas, the interaction between molecules is negligible, meaning that the internal energy of the gas is independent of volume at constant temperature. In this context, the Joule-Thomson coefficient is often considered to be zero, since ideal gases do not show a temperature change when expanded under adiabatic conditions.

Step 3: Applying the Joule-Thomson Coefficient to Ideal Gases

The formula for the Joule-Thomson coefficient, \( rac{dT}{dP} = rac{{T( rac{ar{V}}{C_p}) - ar{V}}}{C_p} \), simplifies to zero for ideal gases as \( ar{V} \) (molar volume) and \( C_p \) are constants and no real interactions occur (Joule's second law). Thus, the Joule-Thomson coefficient for an ideal gas is zero.

Step 4: Conclusion: Determining the Correct Option

Given that the Joule-Thomson coefficient for an ideal gas is zero, the correct answer to the exercise is option (c): zero.

Key Concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental principle in chemistry and physics used to describe the behavior of gases. It is represented by the equation \( PV = nRT \), where:

• \( P \) is the pressure of the gas,
• \( V \) is the volume occupied by the gas,
• \( n \) is the number of moles of the gas,
• \( R \) is the ideal gas constant, and
• \( T \) is the temperature of the gas in Kelvin.

This equation assumes that gas molecules do not interact with each other and are in constant, random motion. The Ideal Gas Law is exceptionally useful for calculating the relationships between the various properties of a gas, providing insights into the behavior of real gases under a wide range of conditions.
The assumptions of the Ideal Gas Law imply that gases are composed of many small particles that are far apart compared to their size, with no attraction between them. While real gases exhibit slightly different behaviors due to molecular interactions, the Ideal Gas Law serves as a good approximation for many gases under many conditions, particularly high temperatures and low pressures.

Joule's Second Law

Joule's Second Law states that the internal energy of an ideal gas is dependent only on its temperature. This principle means that, for an ideal gas, changes in volume or pressure do not affect the internal energy as long as the temperature remains constant.
In essence, the law implies that the energy stored in gas molecules is not impacted by their interactions, because ideal gas conditions assume that molecular interactions are negligible.

• No energy is required to separate gas molecules due to lack of attraction or repulsion forces,
• Energy changes only occur with changes in temperature.

This fundamental understanding helps explain why the Joule-Thomson coefficient for an ideal gas is zero. In an adiabatic expansion, as there is no heat exchange with the surrounding, the temperature remains constant according to Joule's Second Law, leading to no change in internal energy.

Adiabatic Expansion

Adiabatic Expansion refers to the process in which a gas expands without exchanging heat with its environment. In a true adiabatic process, the system is isolated, and changes in the system occur due to its internal mechanics, not due to external influences. When dealing with ideal gases, adiabatic expansion often involves understanding how pressure and volume changes can affect other properties of the gas.
This type of expansion is particularly significant in thermodynamics as it demonstrates the conservation of energy within a system. For an ideal gas, because the temperature is not expected to change (due to Joule's Second Law), adiabatic expansion doesn’t result in a temperature change.

• Energy in the system remains constant,
• No heat is lost or gained from the outside,
• Temperature is constant in ideal gases due to lack of external heat exchange.

Understanding adiabatic expansion is crucial for comprehending why the Joule-Thomson coefficient for an ideal gas is zero. Since the expansion takes place adiabatically, the absence of heat transfer aligns with the principles of ideal gases to keep temperature unchanged.