Deriving the Formula of Gibbs Free Energy: Exploring Reversible and Irreversible Processes

Introduction

In the realm of thermodynamics, the Gibbs Free Energy formula plays a crucial role in understanding the spontaneity of reactions and processes. This article delves into the derivation of the Gibbs Free Energy formula, shedding light on the concepts of reversible and irreversible processes. We'll use a theoretical framework, involving entropy, enthalpy, and specific conditions that define these processes.

Understanding the Basics of Gibbs Free Energy

The Gibbs Free Energy (G) is defined as:

[ G = H - TS ]

where H is enthalpy, T is the temperature, and S is the entropy of the system. This equation helps identify whether a process is spontaneous or not. A negative change in Gibbs Free Energy (( \Delta G < 0 \)) indicates a spontaneous process.

Key Terms:

Gibbs Free Energy (G): A thermodynamic potential that measures the usable energy in a system.
Enthalpy (H): A measurement of heat content in a system at constant pressure.
Entropy (S): A measure of the disorder or randomness in a system.
Spontaneous Process: A process that occurs without external intervention or additional energy input.

Exploring Reversible and Irreversible Processes

To derive the Gibbs Free Energy formula, we need to distinguish between reversible and irreversible processes.

Reversible Processes

A reversible process is an ideal process that can be reversed without any net change in the system and the surroundings.

Characteristics:
- Occurs infinitely slow.
- Always in equilibrium at each state.
- No energy losses due to friction or other irreversible factors.

As an example, imagine a piston-cylindrical system where a gas expands slowly. The system allows for energy transfer with a heat reservoir, keeping the process at a constant temperature (T).

Irreversible Processes

In contrast, an irreversible process cannot return to its original state without changes in the surroundings.

Characteristics:
- Occurs rapidly and creates a gradient in the system.
- Leads to friction and energy loss, which contributes to heat generation.

For example, if we were to remove large blocks from the pistons instead of pebbles, the system would quickly transition to the new state without ever maintaining equilibrium.

Relation of Entropy in Both Processes

Both processes can lead to a change in entropy (ΔS).

For the reversible process, the change in entropy is given by:

[ \Delta S_{rev} = \frac{Q_{rev}}{T} ]

where ( Q_{rev} ) is the heat absorbed by the reversible process.

For the irreversible process, even though less heat is absorbed, the change in entropy is still expressed similarly:

[ \Delta S_{irrev} = \frac{Q_{irrev}}{T} ]

As established before, while both systems have the same change in entropy, the heat absorbed in the irreversible process (( Q_{irrev} )) is less than that in the reversible process (( Q_{rev} )), leading to an increase in entropy overall.

Total Change in Entropy of the Universe

For a reversible process, the total change in entropy of the universe (system + surroundings) is zero:

[ \Delta S_{universe, rev} = 0 ].

For irreversible processes, however, since there is an increase in entropy due to the irreversible nature and heat generation, we can show: [ \Delta S_{universe, irr} > 0 ].

Derivation of the Gibbs Free Energy Formula

Now, let’s see how we derive the Gibbs Free Energy equation from these discussions.

Start with the relationship: [ Q_{irrev} - T * \Delta S_{irrev} < 0 ]
Multiply through by -1: [ -Q_{irrev} + T * \Delta S_{irrev} > 0 ]
Rearranging gives: [ Q_{irrev} < T * \Delta S_{irrev} ]
By acknowledging that ( Q ) is equivalent to the enthalpy (under constant pressure):
- Thus, we receive: [ \Delta G = \Delta H - T * \Delta S < 0 ].

At this stage, we observe that a negative Gibbs Free Energy indicates a spontaneous reaction.

Conclusion

The exploration of Gibbs Free Energy through the lens of reversible and irreversible processes provides insight into the spontaneous nature of physical and chemical reactions. By dissecting these concepts, we discover the interconnectedness of entropy, heat transfer, and thermodyamical states that lead to the derivation of the Gibbs Free Energy formula. This rigorous approach not only clarifies how spontaneous reactions occur but also deepens our understanding of fundamental principles in thermodynamics.