Question

Choose the correct option for free expansion of an ideal gas under adiabatic condition from the following :

• A. \( q = 0, \, \Delta T \neq 0, \, w = 0 \)
• B. \( q = 0, \, \Delta T = 0, \, w = 0 \)
• C. \( q \neq 0, \, \Delta T = 0, \, w = 0 \)
• D. \( q = 0, \, \Delta T<0, \, w \neq 0 \)

Answer 1

The Correct Option is B

This inquiry concerns the definitive thermodynamic parameters (heat exchanged \(q\), temperature variation \( \Delta T \), and work performed \(w\)) associated with the adiabatic free expansion of an ideal gas.

Core Principles:

The resolution relies upon the First Law of Thermodynamics, the fundamental definitions of adiabatic processes and free expansion, and the characteristic behavior of ideal gases.

1. First Law of Thermodynamics: The internal energy change of a system is the sum of heat input to the system and work done on the system.
2. Adiabatic Process: Characterized by the absence of heat transfer between the system and its environment. Thus, \( q = 0 \).
3. Free Expansion: The expansion of a gas into a vacuum. This scenario involves zero external pressure (\( P_{ext} = 0 \)), and the work done on the system is computed as \( w = -P_{ext}\Delta V \).
4. Internal Energy of an Ideal Gas: The internal energy (\(U\)) of an ideal gas is solely dependent on its temperature. Consequently, any alteration in internal energy corresponds directly to a change in temperature: \( \Delta U = nC_v\Delta T \), where \(n\) denotes the number of moles and \(C_v\) represents the molar heat capacity at constant volume.

Detailed Analysis:

Step 1: Determination of heat (\(q\)) for the process.

The process is defined as adiabatic, meaning no heat is exchanged with the surroundings. Therefore, \(q\) is zero.

\[ q = 0 \]

Step 2: Determination of work done (\(w\)) for the process.

The expansion is free, implying the external pressure is zero (\( P_{ext} = 0 \)). The work done on the system is calculated as:

\[ w = -P_{ext}\Delta V \]

Substituting \( P_{ext} = 0 \):

\[ w = -(0) \times \Delta V = 0 \]

Hence, the work done is zero.

Step 3: Calculation of internal energy change (\(\Delta U\)) using the First Law of Thermodynamics.

Applying the First Law of Thermodynamics, \( \Delta U = q + w \), with the values from Steps 1 and 2:

\[ \Delta U = 0 + 0 = 0 \]

The internal energy of the gas does not change.

Step 4: Determination of temperature change (\(\Delta T\)) for the ideal gas.

For an ideal gas, internal energy is exclusively a function of temperature, described by \( \Delta U = nC_v\Delta T \). As \( \Delta U = 0 \) from the First Law, we have:

\[ nC_v\Delta T = 0 \]

Given that \(n\) and \(C_v\) are non-zero constants, the change in temperature \( \Delta T \) must be zero.

\[ \Delta T = 0 \]

This indicates that the initial and final temperatures of the ideal gas are identical.

Conclusion:

Following the systematic analysis, for an ideal gas undergoing adiabatic free expansion, the thermodynamic parameters are:

• Heat exchanged, \(q = 0\)
• Work done, \(w = 0\)
• Change in temperature, \(\Delta T = 0\)

The correct option that aligns with these findings is:

The correct option is \( q = 0, \, \Delta T = 0, \, w = 0 \).