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 problem asks for the correct thermodynamic conditions (heat \(q\), change in temperature \( \Delta T \), and work done \(w\)) for the free expansion of an ideal gas under adiabatic conditions.

Concept Used:

The solution is based on the First Law of Thermodynamics, the definitions of adiabatic and free expansion processes, and the properties of an ideal gas.

1. First Law of Thermodynamics: The change in internal energy of a system is equal to the heat supplied to the system plus the work done on the system.
2. Adiabatic Process: A process that occurs without any heat transfer between the system and its surroundings. By definition, for an adiabatic process, \( q = 0 \).
3. Free Expansion: The expansion of a gas into a vacuum. This means the gas expands against zero external pressure (\( P_{ext} = 0 \)). The work done on the system is given by the formula \( w = -P_{ext}\Delta V \).
4. Internal Energy of an Ideal Gas: For an ideal gas, the internal energy (\(U\)) is a function of temperature only. Therefore, any change in internal energy is directly proportional to the change in temperature: \( \Delta U = nC_v\Delta T \), where \(n\) is the number of moles and \(C_v\) is the molar heat capacity at constant volume.

Step-by-Step Solution:

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

The problem states that the condition is adiabatic. By definition, in an adiabatic process, there is no exchange of heat between the system and the surroundings. Therefore, the heat \(q\) is zero.

\[ q = 0 \]

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

The process is a free expansion. This means the gas is expanding against an external pressure of zero (\( P_{ext} = 0 \)). The work done on the system is calculated as:

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

Substituting \( P_{ext} = 0 \) into the equation:

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

So, the work done is also zero.

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

The First Law of Thermodynamics is \( \Delta U = q + w \). Using the values we found in Step 1 and Step 2:

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

The change in the internal energy of the gas is zero.

Step 4: Determine the change in temperature (\(\Delta T\)) for the ideal gas.

For an ideal gas, the internal energy is solely a function of its temperature. The relationship is given by \( \Delta U = nC_v\Delta T \). Since we have determined from the First Law that \( \Delta U = 0 \), we can write:

\[ nC_v\Delta T = 0 \]

Since \(n\) (number of moles) and \(C_v\) (molar heat capacity) are not zero, the change in temperature \( \Delta T \) must be zero.

\[ \Delta T = 0 \]

This implies that the initial and final temperatures of the ideal gas are the same.

Final Result:

Based on our step-by-step analysis, for the free expansion of an ideal gas under adiabatic conditions, we have:

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

Comparing these results to the given choices, the correct option is the one that matches all three conditions.

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