The second law of thermodynamics states that there is never a decrease in the entropy of an isolated system. In other words, the entropy change of an isolated system can never hold a negative value. However, the entropy can remain constant if the system is in a state of thermodynamic equilibrium.

In an ideal reversible process, the overall entropy remains the same. An increase in the entropy results in an irreversible change, resulting in an asymmetry between the previous and the future states of the body.

## Statements of the Second Law of Thermodynamics

Many different statements of the second law of thermodynamics have been formulated. One of the earliest statements of this law was formulated by Nicolas Leonard Sadi Carnot, a French scientist who is often referred to as the â€˜father of thermodynamicsâ€™.

### Carnotâ€™s Principle and Theorem

- Carnotâ€™s principle states that the efficiency of a cycle of a Carnot engine is solely dependent on the temperatures of the heat reservoirs.
- It also states that this Carnot engine has the highest possible efficiency a heat engine can have using these two temperatures.
- Carnotâ€™s Theorem states that the efficiency of all irreversible heat engines that operate between two heat reservoirs is less than that of a Carnot engine.
- It goes on to state that the efficiency of a reversible heat engine that operates between two heat reservoirs is equal to that of a Carnot engine.

### The Clausius Statement

- The relationship between heat and work was examined by the German physicist Rudolf Clausius in his statement of the second law of thermodynamics.
- He stated that heat can never pass from a cold body to a hotter body unless some work is done by an external source of energy.
- For example, in a refrigerator, the heat flows from a cold region to a hot region due to the work performed by the refrigerating system.

### The Kelvin-Planck Statement

- This statement was derived from the independent statements of the second law put forth by Lord Kelvin and Max Planck.
- It states that it is impossible to construct a device that operates via a thermodynamic cycle and converts all the thermal energy from a heat reservoir into work.
- This statement of the second law of thermodynamics dictates that all of the heat absorbed from heat source cannot be converted into work and that some of the heat must be passed on to a colder body.

Another important principle related to the second law is Planckâ€™s principle, which states that an adiabatic process (a process that occurs without a transfer of heat or mass between the system and its surroundings) causes an increase in the internal energy of a closed system.

## Derivation of the Second Law of Thermodynamics

### What is Entropy?

Entropy is the degree of randomness or disorder in a system. In thermodynamics, this quantity is a measure of the thermal energy in a system that cannot be used to perform work. The entropy of a system can be expressed in terms of the changes undergone by the system when its initial and final states are compared.

In order to measure this quantity, it is expressed in terms of entropy change (denoted by Î”S). Some important points detailing the relationship between entropy and the second law of thermodynamics are listed below.

- An isentropic process is a process in which the overall entropy of the system remains constant, i.e. Î”S = 0. This implies the entropy of the initial and final states of the system holds a constant value.
- The mass of a closed system generally remains constant. However, an exchange of heat between the system and the surrounding can still occur. This implies that a change in the entropy of a system arises when the heat content of a closed system is disturbed by its surroundings.
- The movement of molecules in a closed system leads to a transfer of energy. This can, in turn, lead to an increase in the entropy of the system.

The Clausius concept of entropy is used to measure the direction of a spontaneous change, stating that the spontaneous changes caused by an irreversible process must proceed in the direction of increasing entropy.

### Derivation

The change in the entropy of the universe is given by the sum of the entropy change of the system and its surroundings. This can be equated as follows.

\(\Delta S_{universe} = \Delta S_{system} + \Delta S_{surrounding} = \frac{q_{system}}{T} + \frac{q_{surrounding}}{T}\)

Where Î”S is the entropy change, q is the heat absorbed, and T is the temperature. The equation for the heat energy absorbed by the system in a reversible and isothermal (no change in temperature) process is given by:

\(q = nRTln\frac{V_{2}}{V_{1}}\)

In an isothermal process, the total amount of heat absorbed by the system must be equal to the total amount of heat lost by the surrounding. This implies that **q _{system} = – q_{surrounding} **

Therefore, the change in the entropy of the universe can be expressed as:

\(\Delta S_{universe} = \frac{nRTln\frac{V_{2}}{V_{1}}}{T} + \frac{-nRTln\frac{V_{2}}{V_{1}}}{T} = 0\)

This implies that the entropy change in a reversible process is zero. Now, if the process is irreversible, the entropy change must be greater than that of a reversible process, i.e.

\(\Delta S_{universe} = \frac{nRTln\frac{V_{2}}{V_{1}}}{T} > 0\)

Combining the equation for entropy for a reversible and irreversible process, the following equation can be obtained.

\(\Delta S_{universe} = \Delta S_{system} + \Delta S_{surroundings} \geq 0\)

Which is the equation for the second law of thermodynamics, i.e. the change in the entropy of an isolated system cannot be negative.

### Relationship Between Entropy and Gibbs Free Energy

The change in the entropy of the surrounding can be expressed via the formula:

\(\Delta S_{surrounding} = \frac{\Delta H_{system}}{T}\)

The entropy change of the universe is the sum of the entropy change of the system and the surrounding (Î”S_{total} = Î”S_{system} + Î”S_{surrounding}). Substituting the value of Î”S_{surrounding}, the following equation can be obtained.

\(-T \Delta S_{total} = \Delta H_{system} – T\Delta S_{system}\)

Replacing the left-hand side of the equation with Gibbs free energy (G), the equation can be written as:

**Î”G = Î”H – TÎ”S**

Where Î”G is the Gibbs energy, Î”H is the enthalpy change, and Î”S is the entropy change. It is important to note that a spontaneous process must have a negative value of Î”G. Positive values of Î”H imply that the process is endothermic. Similarly, a negative enthalpy change suggests that the process is exothermic.

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