What is the Second Law of Thermodynamics?

The first law of thermodynamics is the special case of conservation of energy principle. It states that the net heat input (or in mathematical words, the cyclic integral of net heat supplied) to the system is equal to the network output (or in mathematical terms the cyclic integral of net work done) to the surroundings.

But, is it really sufficient to conclude here? Does it answer all the expert questions that arise in a scientific mind? No! Consider a system as shown in the figure below. It illustrates an object maintained at elevated temperature Ti in a room at atmospheric air temperature To.

With the passage of time, the hotter object would cool off and the surrounding air would heat up. This is a spontaneous process that has a certain direction of heat flow (from a hotter to a colder object) and obeys definitely the principle of conservation of energy which explains the cause of such a heat transfer: the decrease in the internal energy of the hot object and consequently the rise in that of the surrounding air.

Heat transfer in diagram

Let’s raise a thoughtful question here: Is the inverse process spontaneous in nature? Would the principle of energy conservation hold or transgress if the direction of the heat flow is reversed?

The answer to the first question is a yes, that is, the transfer of heat from the surrounding air back to the object would not be a natural process because it would require the mediation of external devices or engineering systems to reverse the direction of heat flow.

The answer to the second question is that, even during the reverse process, the principle of conservation of energy would hold: the heat received by the surrounding air would be lost and subsequently regained by the hot object.

Here comes the limitation of the first law of thermodynamics: even when the energy conservation principle is in place, the first law does not fully explain the spontaneity of the process.

In the case cited above, our experiential observation can easily decide the direction of heat flow (from a hotter to a colder environment). But, it is certainly not so obvious in many complex situations at hand. In such scenarios, we need another law which is the second law of thermodynamics.

Moreover, we know that when a system-in-process with its surroundings is left to its own, it reaches an equilibrium state sooner or later. Some processes take a longer time to establish an equilibrium state; for example, the iron bar may take years to rust away than a chemical reaction, which takes a few seconds to get stable at times.

The point is that although the energy conservation principle explains the conservation of energy and thus the equilibrium state, it alone cannot fully decide about the final equilibrium state- an explanation that is only provided by the second law of thermodynamics.

Lastly, what if instead of losing the heat from the hot object to its surroundings, it is provided to a system undergoing a power cycle in order to perform useful work? Such a process would be exploitable only until the state of thermal equilibrium is achieved.

So, the answers to the questions: What would be the maximum value of obtainable useful work? What factors would decide upon the maximum value of work output? The second law of thermodynamics would explain it well.

The Second Law of Thermodynamics: Two Statements

There are two statements of the second law of thermodynamics which are called:

  1. The Clausius Statement
  2. The Kelvin-Planck Statement

As a point of departure, the Clausius statement is easy to accept as it is experientially accorded. In contrast, the second statement is based on the useful deductions of the first statement, which is imperative for the study of systems undergoing thermodynamic cycles.

The Clausius (C) Statement

It states that:

It is impossible to construct a device or a system that operates in such a way that its sole purpose would be an energy transfer by heat from a cooler body to a hotter one.

The Clausius statement cited above does not make impossible the construction of devices or systems that would cause the transfer of heat from a cooler to a hotter body, because it is what the refrigerators are made up of their purpose is to extract a certain amount of energy from the refrigerated space and reject it to the hotter environment.

But, to serve that purpose, a refrigerator is connected to an air compressor which is driven by an electric motor (work input). Calusius’s statement is clear and concise in stating that no such device system or cycle can be constructed without input work (in our case, an electric motor for running compressors in the refrigeration cycle).

Again, because thermodynamic processes occur in a system in which at the end of each cycle, the system restores to its initial state, there must be some changes external to the system, in its surroundings, which must be taken into account.

The Kelvin-Planck (K-P) Statement

It states that:

It is impossible to construct a system undergoing a thermodynamic cycle and operate to give a net amount of work while exchanging heat with a single thermal reservoir.

By reservoir, it means a system that is approximated to maintain a constant temperature such as the earth’s atmosphere, lakes, ocean, large copper, etc. Its operation is shown schematically below.

Kelvin Planck Statement

Coming back to the K-P statement, which of course does not state that it is impossible to devise a system that receives heat from a single reservoir and produces an equivalent amount of work.

However, it imposes constraints on a system operating in cycles in which at the end of each cycle, the system is restored to its initial state. For such a cyclic process, the K-P statement asserts that without the mediation of a third reservoir (maintained at lower temperatures), the construction of devices such as heat engines is not possible.

It is shown schematically in the diagram below:

Thermodynamic Cyclic process

So, one can write the K-P statement in analytical form as follows:

W_{cycle}\leq0

This shows clearly that the net amount of work done by the system is less than or equal to zero in the account of the single thermal reservoir. The less than and equal to of the statements are associated respectively with the presence and absence of the system’s internal irreversibilities.

Difference Between Clausius and Kelvin-Planck Statement

Both statements have a privileged place in thermodynamic studies. Calusius’s statement deals more with the direction of energy flow and under what conditions. Meanwhile, the Kelvin-Planck statement has to do with the limitations of heat-work transformation processes that cyclically take place. Clausius’s statement is more useful in the development of refrigeration cycles, whereas Kelvin-Planck’s statement is concerned with heat engines and their thermal efficiency.

Applications of the Kelvin-Planck Statement

The Kelvin-Planck statement is helpful in understanding the limitations on the performance of heat engines operating between two thermal reservoirs as shown in the figure below.

kelvin planck applications

As thermal efficiency is network divided by heat supplied, one can write,

\eta=\frac{Output}{Input}=\frac{W_{cycle}}{Q_H}=\frac{Q_H-Q_C}{Q_H}=1-\frac{Q_C}{Q_H}

If QC = 0, then the thermal efficiency would be:

\eta\;=\;100\%

Which is practically impossible.

So, the K-P statement accounts for the fact that no heat engine can have 100% thermal efficiency on account of thermal irreversibilities. Thus, the K-P statement results in the maximum theoretical performance efficiency of an ideal thermal cycle (which is called a Carnot cycle proper to heat engine)  against which the thermal efficiencies of other practical cycles are compared.

Irreversibilities and The Second Law of Thermodynamics

The second law is intimately linked with the concept of irreversibilities of the system, which renders the natural process impossible in the inverse direction. There may be external as well as internal irreversibilities which are both associated with the natural processes. A few of them are mentioned below:

  1. Heat loss due to the transfer of energy at a finite temperature difference between the system and surroundings
  2. Unconstrained expansion of a gas to a lower pressure
  3. Static/dynamic friction as well as fluid internal friction
  4. Electric current passing through an electric resistance
  5. A chemical reaction occurring in a spontaneous manner
  6. Plastic or inelastic deformation

Entropy and the Second Law

Apart from the two statements cited above, it’s worthwhile to mention what the second law predicts in the attainment of the final state of thermal equilibrium of a system undergoing a thermal cycle.

To answer this question, one needs to introduce oneself to the concept of Entropy which is the measure of the system’s randomness, disorder, chaos, missing information freedom, and so on.

The second law predicts that a system reaches a final equilibrium state unless its entropy ceases to increase any further. That is, at the system’s equilibrium state, its entropy is maximum, which means, the entropy change of the system plus surrounding would be highest at the final equilibrium state of the system and the heat flow would be in the spontaneous direction.

Implications

There are certain implications of the second law of thermodynamics (and of the deductions of the second law)  which are mentioned below one by one:

  1. It predicts the direction of the process undergoing the thermodynamic cycle (Gibbs free energy equation.)
  2. It establishes the conditions necessary for the attainment of the final equilibrium state.
  3. It is helpful in the optimization (optimum theoretical performance) of power cycles, steam engines, etc.
  4. It defines a temperature scale (K).
  5. It helps in defining certain properties such as internal energy and enthalpy more succinctly that can be obtained quite readily on the experimental level (Gibbs Free Energy).