A gaseous system changes from state \(A \)\((P_1, V_1, T_1)\) to \(B\) \((P_2, V_2, T_2)\) to \(C\) \((P_3, V_3, T_3)\) and finally from \(C\) to \(A\). The whole process may be called

Cyclic process

The correct answer is option 2. Cyclic process. 

Let us delve into each option and explain why the correct answer is a cyclic process.

1. Reversible Process:

A reversible process is one where the system undergoes changes in such a way that it can be exactly reversed by infinitesimally modifying the external conditions (such as pressure, volume, temperature) without leaving any trace on the surroundings. While a cyclic process can involve reversible processes at each stage \(A \rightarrow B \rightarrow C \rightarrow A\), it doesn't exclusively define a cyclic process. Cyclic processes can include both reversible and irreversible stages.

2. Cyclic Process:

A cyclic process is a thermodynamic process where the system returns to its initial state after undergoing a series of changes in state. In the context of the question, the gaseous system starts at state \(A\) with conditions \(P_1, V_1, T_1\), moves to state \(B\) with conditions \(P_2, V_2, T_2\), then to state \(C\) with conditions \(P_3, V_3, T_3\), and finally returns back to state \(A\) with conditions \(P_1, V_1, T_1\). The process completes a closed loop or cycle in the \(P-V\) (pressure-volume) or \(T-S\) (temperature-entropy) diagram. A key characteristic of a cyclic process is that the net change in any state function (such as internal energy or enthalpy) is zero because the system returns to its initial state. Therefore, the process described \(A \rightarrow B \rightarrow C \rightarrow A\) is classified as a cyclic process because it completes a closed loop in state space.

3. Isobaric Process:

An isobaric process is one where the pressure of the system remains constant throughout the process. The question does not specify that the process \(A \rightarrow B \rightarrow C \rightarrow A\) occurs at constant pressure \(P\) at all stages, so it cannot be definitively classified as an isobaric process.

4. Spontaneous Process:

A spontaneous process is one that occurs without requiring external intervention. The term "spontaneous" does not describe the cyclic nature of the process (returning to the initial state after a series of changes).

Conclusion:

The process described in the question, where a gaseous system undergoes changes from state \(A\) to \(B\) to \(C\) and then returns to \(A\), is best categorized as a cyclic process. This is because it completes a closed loop in the state space, returning to its initial state after undergoing a series of changes in pressure, volume, and temperature. Therefore, option (2) Cyclic process is the correct classification for the described thermodynamic process.