Equal moles of \(\mathrm{He}, \mathrm{H}_{2}, \mathrm{CO}_{2}\) and \(\mathrm{SO}_{3}\) gases are expanded adiabatically and reversibly from the same initial state to the same final volume. The magnitude of work is maximum for (Assume ideal behaviour of gases and all the degree of freedoms are active.) (a) He (b) \(\mathrm{H}_{2}\) (c) \(\mathrm{CO}_{2}\) (d) \(\mathrm{SO}_{3}\)

A reversible process in thermodynamics is a theoretical concept where a system changes state in such a way that the path taken by the system to return to its original state can be exactly traced back. This type of process proceeds infinitely slowly to ensure that the system is always in thermodynamic equilibrium with its surroundings. In practical terms, no real process is perfectly reversible; however, the concept is important for understanding the efficiency of thermodynamic systems.

In an adiabatic reversible process, the system does not exchange heat with its surroundings, implying that any work done results in a temperature change of the system. The implication of a process being reversible is significant when calculating the work done on or by a system. The formula for work done in an adiabatic reversible process takes into account the initial and final temperatures of the system and is derived from the first law of thermodynamics.

The heat capacity ratio, denoted as \( \gamma \) (gamma), is the ratio of the heat capacity at constant pressure \( C_{P} \) to the heat capacity at constant volume \( C_{V} \). It is a crucial parameter in thermodynamics, especially concerning adiabatic processes.

This ratio comes into play when we examine how different gases behave under adiabatic conditions. Each gas, based on its molecular structure, has a unique \( \gamma \) that influences how much work the gas can perform during an expansion or compression. Monoatomic gases like Helium have higher \( \gamma \) values, typically around 1.67, whereas diatomic and polyatomic gases have lower values, descending with increasing complexity of the molecule.

In thermodynamics, the term 'work' refers to the energy transferred when a force is applied to an object, causing it to move. The concept of work done is pivotal to understanding energy transformations that occur in various thermodynamic processes. The work done by the system is positive when a gas expands, doing work on the surroundings; conversely, it is negative during compression when work is done on the gas.

Specifically, in an adiabatic process, the work done relates directly to the change in internal energy of the gas, since no heat is exchanged. Utilizing the heat capacity values and the heat capacity ratio, one can determine the work done during reversible adiabatic expansion by using the formula \( W = \frac{C_{V} (T_{i} - T_{f})}{\gamma - 1} \). A gas with a lower \( \gamma \) will have more work done for the same change in temperature, due to the larger denominator in the equation \( \gamma - 1 \), a concept highlighted in the exercise problem.