Second law of thermodynamics. Entropy

The second Law of Thermodynamics is concerned with the direction in which natural processes take place.

The first law of Thermodynamics states that energy is conserved and provides a relationship between the internal energy of a thermodynamic system and the different ways in which this energy can vary according to the processes it undergoes. However, it does not set any limits for the direction in which these processes take place.

In the real world however, natural processes occur in a certain direction and are not reversible (unless energy is supplied so that they may be reversed): heat always flows from hotter to colder bodies; heat engines absorb heat from a hot thermal reservoir and transfer part of it to a cooler thermal reservoir; ice cubes do not form spontaneously unless a freezer is used; human beings age and do not become younger… The second law of Thermodynamics allows us to define an arrow of time and to distinguish the past from the future.

The state function that allows quantifying this arrow is entropy (S). The entropy of an isolated system always increases or remains constant, it can never decrease over time since the systems spontaneously evolve towards thermodynamic equilibrium states.

Entropy is defined as:

Where δQ_R is the heat exchanged during a reversible process and T is the temperature expressed in kelvin (K). The SI unit of entropy is J/K.

Entropy is a state function: its variation depends only on the initial and final states of the thermodynamic system and not on the way that system evolves from its initial to its final state.

In order to prove it, we will make use of the expressions for the efficiency of a heat engine and the Carnot heat engine.

The efficiency of a heat engine is given by:

Where W is the heat engine’s net work output and Q₁ is the heat absorbed by the working fluid in each cycle. This expression of efficiency is valid for any heat engine, including the Carnot heat engine.

But for the latter the efficiency is also given by:

Therefore we can equal the two together. On the other hand, the Carnot heat engine’s work output equals the net heat exchanged in each cycle. Therefore:

Where Q₂ is the heat given (negative) by the Carnot heat engine in each cycle:

Not all reversible engines describe a Carnot cycle. However we can always divide any cycle into N elementary Carnot cycles, as shown in the figure below, and the expression above holds for each one of them.

Therefore, the expression below will be valid for any reversible cyclic process:

Which proves the entropy to be a state function because its line integral over a cycle is zero.

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Throughout these pages we will see some of the applications of entropy as well as different statements of the Second Law of Thermodynamics.

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