# Second law of thermodynamics

The second law of thermodynamics makes statements about the direction of processes and the principle of irreversibility . The definition of the thermodynamic temperature and the state variable entropy can be derived from the second law. Also follows from the second law of thermodynamics the distinction between exergy and anergy and the fact that the efficiency of a heat engine cannot exceed the Carnot efficiency .

## Formulations of the statements of the second law

For the Second Law of Thermodynamics, many different, sometimes equivalent formulations have been established, some of which are reproduced below:

• Heat cannot by itself pass from a body with a lower temperature to a body with a higher temperature.
• Heat can not be completely converted into work by a machine that operates intermittently . This would be a realization of a perpetual motion machine of the second kind.
• The efficiency of the Carnot process cannot be exceeded.
• All spontaneous (in one direction) processes are irreversible .
• All processes in which friction occurs are irreversible.
• Equalization and mixing processes are irreversible (apart from reversible mixtures of ideal gases).
• In a closed adiabatic system, the entropy cannot decrease.
• The equilibrium of isolated thermodynamic systems is characterized by a maximum principle of entropy.

## Statements of the Second Law

### Preferred direction of processes

The second law of thermodynamics in the formulation of Clausius reads:

" There is no change of state, the only result of which is the transfer of heat from a body at a lower temperature to a body at a higher temperature."

To put it more simply: heat cannot by itself pass from a body with a lower temperature to a body with a higher temperature. There is a preferred direction for the process of heat transfer. This statement seems superfluous at first, because it corresponds to everyday experience. However, it is synonymous with all other, less “obvious” statements, because all contradictions to the other statements can be traced back to a contradiction to this one.

The second law of thermodynamics in the formulation of Kelvin and Planck reads:

It is impossible to construct a periodically operating machine that does nothing more than lift a load and cool a heat reservoir. "

The first law would not contradict the assumption that it is possible to supply a constant flow of heat to an engine - of whatever type - which it emits completely as mechanical or electrical power. Such a machine is called a perpetual motion machine of the second type. A corresponding formulation of the second main clause reads:

A perpetual motion machine of the second kind is impossible.

The heat engine (a) describes a perpetual motion machine of the second kind. If such a machine existed, it could convert heat into work without loss (green arrow). If a heat pump (b) were driven with this work, heat could be pumped from the cold to the warm reservoir without any external influence .

If one assumes that there is this engine that is independent of a heat sink for heat dissipation, then heat could be extracted from the surroundings and converted into mechanical work. For example, a ship could be propelled by gaining propulsion power by cooling the seawater.

It could also be used to extract heat from a reservoir or container, as shown in the picture on the right, and use the converted energy to drive a heat pump that uses a reversible Carnot process to convey heat from another container with a lower temperature to the former with a higher temperature. The amount of heat fed into the warmer container would then be greater than that absorbed by the engine, because the energy released by the heat pump consists of the sum of absorbed heat and drive work. If the system boundary is drawn around both machines, including the two heat containers, then within this closed system  - i.e. without energy and material exchange with the environment - heat would ultimately have flowed from a colder to a warmer body. This contradicts the first statement.

### Efficiency of heat engines

In principle, the same contradiction also arises with the assumption that a prime mover could be built that is more efficient than a machine operating with a Carnot process. This machine would also take less heat from the warmer container than the Carnot heat pump it drives feeds in there. The corresponding form of the second main clause is:

There is no heat engine that has a higher efficiency than the Carnot efficiency formed from these temperatures at a given average temperature of the heat supply and heat removal .

${\ displaystyle \ eta _ {c} = 1 - {\ frac {T _ {\ mathrm {cold}}} {T _ {\ mathrm {hot}}}}}$

Mentioning the mean temperatures is important because, as a rule, a heat reservoir changes its temperature when heat is added or removed.

Here, T is not any temperature (eg not. Celsius - or Fahrenheit temperature) of the system but by the equation of state of the "ideal gas" better defined here, or by just indicated efficiency of the Carnot cycle " absolute Temperature "(Kelvin).

Immediately in this context, the following can be further formulated:

All reversible heat and power processes with the same mean temperatures of heat supply and heat removal have the same efficiency as the corresponding Carnot process.

and:

All irreversible heat and power processes are less efficient.

### entropy

The first law of thermodynamics can be summarized into a quantitative formula using the state variable energy and the associated energy balance. The quantitative implementation of the Second Law of Thermodynamics is analogous to this through the state variable entropy introduced by Clausius and the establishment of an entropy balance. Entropy balances can be drawn up for closed, closed and open systems. In flow processes, the balance relates to a fluid particle that moves through the system and can be viewed as a closed moving system.

For a closed system the entropy balance is:

${\ displaystyle \ mathrm {d} S = {\ frac {\ delta Q} {T}} + {\ frac {\ delta W _ {\ mathrm {diss}}} {T}}}$

The source term is the work dissipated within the system : work that does not reach the outside, but increases the internal energy as a result of friction, throttling or impact processes. She is always positive. The corresponding term in the equation is called the entropy produced in the system. indicates an incomplete differential , while indicates a complete differential . ${\ displaystyle \ delta W _ {\ mathrm {diss}}}$${\ displaystyle \ delta}$${\ displaystyle \ mathrm {d}}$

If one looks at an open system, a further term is added to the entropy balance given above, which takes into account the increase or decrease in entropy as a result of material transport across the system boundary. If, on the other hand, one considers a closed system, i.e. a closed system that is also adiabatic and in which the following applies: ${\ displaystyle \ delta Q = 0}$

${\ displaystyle \ mathrm {d} S = {\ frac {\ delta W _ {\ mathrm {diss}}} {T}}}$

and the following statement on entropy production :

In a closed adiabatic system, entropy cannot decrease; it usually increases. It only remains constant in reversible processes.

Here, too, the equivalence with the first statement by Clausius is easy to see. An automatic heat flow from the colder to the warmer container in the arrangement outlined above would mean that the entropy of the colder container (lower temperature T in the denominator) decreases more than that of the warmer container increases, i.e. H. the total entropy in the system decreases, which is not possible.

All spontaneous processes are irreversible. There is always an increase in entropy. Examples are the mixing of two different gases and the flow of heat from a hot to a cold body without generating work. The restoration of the (often called “ordered”) initial state then requires the use of energy or information (see Maxwell's demon ). Reversible processes are not associated with an increase in total entropy and therefore do not take place spontaneously. Through the theoretical description of spontaneously occurring processes, the Second Law of Thermodynamics distinguishes a direction of time that corresponds to our intuitive world of experience (see the example below).

### Thermodynamic temperature

For a reversible process, the following applies

${\ displaystyle {\ dot {S}} = {\ frac {\ dot {Q}} {T}}}$.

Based on this relationship, the definition of the thermodynamic temperature can be obtained by solving for : ${\ displaystyle T}$

${\ displaystyle T = {\ frac {\ dot {Q}} {\ dot {S}}}}$.

From this formula it can be seen that the thermodynamic temperature has a zero point , but cannot become negative and that heat flows from higher to lower temperatures. An empirical temperature, on the other hand, could also assume negative temperatures, as is the case with the Celsius scale , or be defined with the opposite sign.

### Exergy and anergy

With the contexts described, the following sentence is also a statement of the second main clause:

The thermal energy of a system consists of an exergy part and an anergy part , whereby the exergetic part disappears when the system is transferred into the surrounding state.

Exergy and anergy of heat (thermal energy = anergy + exergy)

Exergy is the portion of thermal energy that can be converted into other forms of energy . If a body or system with a state that deviates from that of its surroundings is reversibly brought into the surrounding state, its exergy is given off as work. The heat that a body (e.g. hot flue gas in the boiler of a power plant) gives off when it cools down to ambient temperature can theoretically be used to convert it into work via a sequence of differential Carnot processes, as shown in the picture on the right become. The exergetic portion results from adding up the differential (pink) surface portions above the ambient temperature . ${\ displaystyle T_ {U}}$

${\ displaystyle E _ {\ mathrm {ex}} = \ int _ {S_ {2}} ^ {S_ {1}} \ left ({T (S) -T_ {U}} \ right) \ mathrm {d} S}$

The heat sink for these processes to absorb the anergy (blue area below ) is the environment. If a gas in its initial state has not only a higher temperature but also a higher pressure than the ambient state, the total exergy consists not only of the exergetic part of the heat, but also of a part of volume work . ${\ displaystyle T_ {U}}$

The thermal efficiency of the real heat engine is therefore always less than 1 and - due to the process control specified by the machines and the inevitable dissipative effects - also always smaller than that of the ideal heat engine:

${\ displaystyle \ eta _ {\ mathrm {th}} = 1 - {\ frac {T_ {U}} {T_ {m _ {\ mathrm {zu}}}}} = {\ frac {\ text {Exergy}} {\ text {thermal energy}}}}$

where is the ambient temperature and the mean temperature of the heat input. It results when the yellow area of ​​the exergy is replaced by a rectangle of the same area above the line of the ambient temperature. ${\ displaystyle T_ {U}}$${\ displaystyle T_ {m _ {\ mathrm {zu}}}}$

The second law therefore has considerable technical implications. Since many machines that supply mechanical energy generate it indirectly from thermal energy (e.g. diesel engine : chemical energy, thermal energy, mechanical energy), the limitations of the 2nd law always apply to their efficiency. In comparison, hydropower plants, which do not require an intermediate stage via thermal energy for the conversion, offer significantly higher levels of efficiency. ${\ displaystyle \ rightarrow}$${\ displaystyle \ rightarrow}$

### Entropy sink

An entropy sink is a system that imports entropy from another system. This increases the entropy of the sink. Since entropy is a broad term, there are a multitude of different entropy sinks. In a classic Carnot thermodynamic heat engine , work is generated by manipulating the flow of energy between hot energy sources, a cold entropy sink and a work reservoir. In reality, the atmosphere acts as an entropy sink in most heat engines.

## Example for the second law

A force-free gas is always distributed in such a way that it completely and evenly fills the available volume. You can understand why this is so if you consider the opposite case. Imagine an airtight box in weightlessness in which a single particle moves. The probability of finding this in the left half of the box during a measurement is then exact . If, on the other hand, there are two particles in the box, then the probability of finding both in the left half is only , and correspondingly with N particles . The number of atoms in a volume of one cubic meter at normal pressure is on the order of around particles. The resulting probability that the gas in the crate will spontaneously concentrate in one half is so small that such an event will probably never occur. ${\ displaystyle {\ tfrac {1} {2}}}$${\ displaystyle {\ tfrac {1} {2}} \ cdot {\ tfrac {1} {2}} = {\ tfrac {1} {4}}}$${\ displaystyle \ left ({\ tfrac {1} {2}} \ right) ^ {N}}$${\ displaystyle 3 \ cdot 10 ^ {25}}$

How the symmetry breaking macroscopic equation follows from the time-reversible microscopic equations of classical mechanics (without friction) is clarified in statistical mechanics . In addition, entropy has a clear meaning there: it is a measure of the disorder of a system or of the information contained in the system. However, in statistical mechanics, the Second Law loses its status as a “strictly valid” law and is regarded there as a law in which exceptions on a macroscopic level are in principle possible, but at the same time so improbable that they do not occur in practice. Viewed at the microscopic level, e.g. For example, small statistical fluctuations around the equilibrium state, even in closed systems, mean that the entropy also fluctuates somewhat around the maximum value and can also decrease for a short time.

## validity

The second law of thermodynamics is a fact of experience. It has not yet been possible to prove this fundamental law of classical physics in its general validity for arbitrary macroscopic systems based on the basic equation of quantum theory, the many-body Schrödinger equation .

Of course, this also applies the other way round: The Schrödinger equation is a fact of experience. It has not yet been possible to prove the general validity of this fundamental law of quantum mechanical systems for arbitrary macroscopic systems, starting from the main principles of physics (and not just thermodynamics).

With regard to the validity of the Second Law, a distinction must be made between the microscopic or submicroscopic and the macroscopic area. In Brownian molecular motion, particles can not only come to rest from motion, but also start moving again from rest. The latter process corresponds to the conversion of thermal energy into higher-value kinetic energy and must be accompanied by the cooling of the environment.

## The second law of force as a law of force

The second main clause indicates a preferred direction for spontaneous, real processes. They run from a starting state A in such a way that the entropy increases until an equilibrium state B is reached. On the basis of classical statistical mechanics, the following statement can be derived:

A process runs spontaneously from A to B when the mean force component in the direction of the process path is positive.

What is meant is double averaging over (a) ensemble or time and (b) the path length. This is not a proof of the Second Law, but just says that it is equivalent to a law of force that determines the direction. The force law applies to systems in which thermodynamic states such as A and B are given by the spatial arrangement of masses and forces are defined. It applies to machines, but also to chemical reactions, as long as they take place in the electronic ground state.

## literature

• Karl Stephan , Franz Mayinger: Thermodynamics. Basics and technical applications. 2 volumes, Springer Verlag
• Hans D. Baehr, S. Kabelac: Thermodynamics, Fundamentals and Technical Applications 13., revised. u. exp. Ed., Springer Verlag, 2006, ISBN 3-540-32513-1 .
• Hans D. Baehr, Karl Stephan: Heat and mass transfer 5., revised. Ed., 2006, Springer Verlag, ISBN 3-540-32334-1 .
• Klaus Langeheinecke, Peter Jany, Eugen Sapper: Thermodynamics for Engineers . 5th edition. Vieweg Verlag, Wiesbaden 2004, ISBN 3-528-44785-0 .
• Hannelore Bernhardt : On the history of the statistical interpretation of the 2nd law of thermodynamics. Rostock physical manuscripts 3/1 (1978), pp. 95-104.

## Individual evidence

1. ^ A b c d Hans Dieter Baehr, Stephan Kabelac: Thermodynamics: Basics and technical applications . 15th edition. Springer Verlag, Berlin / Heidelberg 2012, ISBN 978-3-642-24160-4 , p. 93-176 .
2. ^ Quote from Max Planck , after entry on main clauses. In: Römpp Online . Georg Thieme Verlag, accessed on August 24, 2011.
3. ^ Klaus Langeheinecke: Thermodynamics for engineers. Vieweg, 1993.
4. Selcuk Cakmak, Ferdi Altintas, Ozgur E. Mustecaplioglu: Irreversible Work and Internal Friction in a Quantum Otto Cycle of a Single Arbitrary Spin. arxiv : 1605.02522 (2016).
5. Pankaj Mehta, Anatoly Polkovnikov: Efficiency bounds for Nonequilibrium heat engines. Annals of Physics 332 (2012): 110-126.
6. Lottermoser, Alfred: Brief Introduction to Colloid Chemistry. 1st edition, Dresden and Leipzig, 1944, pp. 70/71.
7. ^ Schlitter, Jürgen: The Second Law of Thermodynamics as a Force Law , Entropy 2018, 20, 234; doi: 10.3390 / e20040234