Problem 17 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

When one mole of monoatomic ideal gas at \(T \mathrm{~K}\) undergoes adiabatic change under a constant external pressure ofl atm, volume changes from 1 litre to 2 litre. The final temperature in Kelvin would be [2005S] (a) \(\frac{T}{2^{(2 / 3)}}\) (b) \(T+\frac{2}{3} \times 0.0821\) (c) \(T\) (d) \(T-\frac{2}{3} \times 0.0821\)

Step-by-Step Solution

Answer

The final temperature is \( \frac{T}{2^{(2 / 3)}} \), which corresponds to option (a).

Step 1: Understand Adiabatic Process

An adiabatic process is one that occurs without the transfer of heat or mass between a thermodynamic system and its surroundings. For an ideal gas, such processes involve changes in temperature and volume following specific relationships.

Step 2: Use the Adiabatic Condition for Monoatomic Gas

For a monoatomic ideal gas undergoing adiabatic change, the relationship \( PV^{\gamma} = \text{constant} \) holds, where \( \gamma \) (gamma) is the adiabatic index, which for monoatomic gases is \( \frac{5}{3} \). We rearrange it to get \( V^{\gamma-1}T = \text{constant} \).

Step 3: Apply the Condition to the Initial and Final States

The condition \( V_1^{\gamma-1}T_1 = V_2^{\gamma-1}T_2 \) can be used, given initial conditions (\( V_1 = 1\, L, T_1 = T \)) and final conditions (\( V_2 = 2\, L, T_2 = T_f \)).

Step 4: Solve for Final Temperature

Substitute \( \gamma = \frac{5}{3} \) into the equation: \( 1^{\frac{2}{3}} \times T = 2^{\frac{2}{3}} \times T_f \). This simplifies to \( T_f = \frac{T}{2^{(2 / 3)}} \).

Step 5: Evaluate the Options

The derived final temperature \( T_f = \frac{T}{2^{(2/3)}} \) matches option (a) exactly.

Key Concepts

Monoatomic Ideal Gas

A monoatomic ideal gas is a theoretical gas composed of single atoms, which do not interact except when they collide elastically. Common examples of monoatomic gases include the noble gases like helium, neon, and argon.

In physics, the idea of an ideal gas is useful because it simplifies the behavior of a real gas to predictable patterns described by simple equations. An ideal gas follows the general gas law, often written as:

\[ PV = nRT \]where:

• \( P \) is the pressure,
• \( V \) is the volume,
• \( n \) is the number of moles,
• \( R \) is the universal gas constant, approximately 8.314 J/mol·K, and
• \( T \) is the temperature in Kelvin.

This equation helps to understand changes in pressure, volume, or temperature of the gas during various processes. In cases of adiabatic processes, another important equation comes into play.

Monoatomic gases are critical in studying adiabatic processes because their simplicity allows the focus to remain on the principles rather than the complexities encountered when molecules have internal structure.

Adiabatic Index

The adiabatic index, often denoted as \( \gamma \), is a critical factor in thermodynamics, particularly for adiabatic processes. This index characterizes how a gas responds to compression or expansion when no heat is exchanged with the surroundings.

For monoatomic ideal gases, the adiabatic index \( \gamma \) is typically \( \frac{5}{3} \). This value arises from the degrees of freedom in the motion of the atoms, which involve only translational movements with no rotational or vibrational components.

The adiabatic condition can be represented as:

\[ PV^{\gamma} = \text{constant} \]This relation implies that as the volume of a gas changes in an adiabatic process, the pressure and temperature also change in a predictable manner. By knowing \( \gamma \), one can precisely determine how temperature and volume will relate during such changes.

Understanding the adiabatic index enables predictions in numerous physics and engineering applications, ranging from thermodynamic cycles in engines to atmospheric phenomena.

Thermodynamic Systems

A thermodynamic system is any defined collection of matter, with a boundary separating it from the surroundings, within which a thermodynamic process can occur. Such systems can exchange energy or mass with their surroundings or remain isolated.

There are three types of thermodynamic systems:

• Open System: Both energy and matter can be exchanged with the surroundings.
• Closed System: Only energy, not matter, is exchanged across the boundary.
• Isolated System: Neither energy nor matter is exchanged. An adiabatic process is one example, as it involves no heat exchange.

In our context of adiabatic processes, an isolated system is particularly relevant. The boundary ensures that no heat enters or leaves, and the entire process must satisfy the adiabatic condition:

\[ V^{\gamma-1}T = \text{constant} \]This means the internal energy change is due solely to work done by or on the system, like compressing or expanding the gas.

Understanding these systems is essential for any study of thermodynamics, as each type of system behaves differently under various processes and conditions, including adiabatic processes.