The question asks about the conditions during the free expansion of an ideal gas under adiabatic conditions. Let's analyze each of the given options and identify the correct one through reasoning based on the principles of thermodynamics.
Concept Explained: In thermodynamics, the free expansion of a gas is a process where the gas expands into a vacuum or expands without being hindered by external forces. When this process happens in an ideal gas, particularly under adiabatic conditions, it implies the following:
Adiabatic Process: There is no heat exchange between the system and the surroundings. Mathematically, this is represented as \(q = 0\).
Work Done (w): In free expansion, no work is done because the expansion occurs against zero external pressure. Hence, \(w = 0\).
Change in Temperature (\(\Delta T\)): For an ideal gas undergoing free expansion, there is no temperature change because there is no exchange of heat or work done on/by the system. Hence, \(\Delta T = 0\).
Analysis of Options:
Option A: \(q = 0\), \(\Delta T = 0\), and \(w = 0\). This matches all the characteristics described above for an adiabatic free expansion of an ideal gas.
Option B: \(q = 0\), \(\Delta T < 0\), and \(w > 0\). This suggests an incorrect change in temperature and work done, which does not agree with free expansion.
Option C: \(q < 0\), \(\Delta T = 0\), and \(w = 0\). The condition \(q < 0\) is incorrect for an adiabatic process.
Option D: \(q > 0\), \(\Delta T > 0\), and \(w > 0\). This contradicts all free expansion conditions.
Conclusion: The correct option is Option A: \(q = 0\), \(\Delta T = 0\), and \(w = 0\), which perfectly aligns with the principles of free expansion of an ideal gas under adiabatic conditions.
