Editing Second law of thermodynamics (section)

== History ==
{{See also|History of entropy}}
[[File:Sadi Carnot.jpeg|thumb|upright|Nicolas Léonard Sadi Carnot in the traditional uniform of a student of the [[École Polytechnique]]]]
The first theory of the conversion of heat into mechanical work is due to [[Nicolas Léonard Sadi Carnot]] in 1824. He was the first to realize correctly that the efficiency of this conversion depends on the difference of temperature between an engine and its surroundings.

Recognizing the significance of [[James Prescott Joule]]'s work on the conservation of energy, [[Rudolf Clausius]] was the first to formulate the second law during 1850, in this form: heat does not flow ''spontaneously'' from cold to hot bodies. While common knowledge now, this was contrary to the [[caloric theory]] of heat popular at the time, which considered heat as a fluid. From there he was able to infer the principle of Sadi Carnot and the definition of entropy (1865).

Established during the 19th century, the [[Kelvin-Planck statement|Kelvin-Planck statement of the second law]] says, "It is impossible for any device that operates on a [[cyclic process|cycle]] to receive heat from a single [[heat reservoir|reservoir]] and produce a net amount of work." This statement was shown to be equivalent to the statement of Clausius.

The [[ergodic hypothesis]] is also important for the [[Boltzmann]] approach. It says that, over long periods of time, the time spent in some region of the phase space of microstates with the same energy is proportional to the volume of this region, i.e. that all accessible microstates are equally probable over a long period of time. Equivalently, it says that time average and average over the statistical ensemble are the same.

There is a traditional doctrine, starting with Clausius, that entropy can be understood in terms of molecular 'disorder' within a [[Macroscopic bodies|macroscopic system]]. This doctrine is obsolescent.<ref>Denbigh, K.G., Denbigh, J.S. (1985). ''Entropy in Relation to Incomplete Knowledge'', Cambridge University Press, Cambridge UK, {{ISBN|0-521-25677-1}}, pp. 43–44.</ref><ref>Grandy, W.T., Jr (2008). ''Entropy and the Time Evolution of Macroscopic Systems'', Oxford University Press, Oxford, {{ISBN|978-0-19-954617-6}}, pp. 55–58.</ref><ref name=Lambert>[http://franklambert.net/entropysite.com/ Entropy Sites — A Guide] Content selected by [[Frank L. Lambert]]</ref>

=== Account given by Clausius ===
[[File:Clausius-1.jpg|thumb|upright|Rudolf Clausius]]
In 1865, the German physicist [[Rudolf Clausius]] stated what he called the "second fundamental theorem in the [[mechanical theory of heat]]" in the following form:{{sfnp|Clausius|1867}}
: <math>\int \frac{\delta Q}{T} = -N</math>
where ''Q'' is heat, ''T'' is temperature and ''N'' is the "equivalence-value" of all uncompensated transformations involved in a cyclical process. Later, in 1865, Clausius would come to define "equivalence-value" as entropy. On the heels of this definition, that same year, the most famous version of the second law was read in a presentation at the Philosophical Society of Zurich on April 24, in which, in the end of his presentation, Clausius concludes:
<blockquote>The entropy of the universe tends to a maximum.</blockquote>

This statement is the best-known phrasing of the second law. Because of the looseness of its language, e.g. [[universe]], as well as lack of specific conditions, e.g. open, closed, or isolated, many people take this simple statement to mean that the second law of thermodynamics applies virtually to every subject imaginable. This is not true; this statement is only a simplified version of a more extended and precise description.

In terms of time variation, the mathematical statement of the second law for an [[isolated system]] undergoing an arbitrary transformation is:
: <math>\frac{dS}{dt} \ge 0</math>
where
: ''S'' is the entropy of the system and
: ''t'' is [[time]].

<!-- A reversible process requires equilibrium with the surroundings. This is not possible for an isolated system. Therefore, the discussion about the reversible process has been shifted to the analysis of closed systems --> 
The equality sign applies after equilibration. An alternative way of formulating of the second law for isolated systems is:
: <math>\frac{dS}{dt} = \dot S_\text{i}</math> with <math> \dot S_\text{i} \ge 0</math>
with <math> \dot S_\text{i}</math> the sum of the rate of [[entropy production]] by all processes inside the system. The advantage of this formulation is that it shows the effect of the entropy production. The rate of entropy production is a very important concept since it determines (limits) the efficiency of thermal machines. Multiplied with ambient temperature <math>T_\text{a}</math> it gives the so-called dissipated energy <math> P_\text{diss}=T_\text{a}\dot S_\text{i}</math>.

The expression of the second law for closed systems (so, allowing heat exchange and moving boundaries, but not exchange of matter) is:
: <math>\frac{dS}{dt} = \frac{\dot Q}{T}+\dot S_\text{i}</math> with <math> \dot S_\text{i} \ge 0</math>
Here,
: <math>\dot Q</math> is the heat flow into the system
: <math>T</math> is the temperature at the point where the heat enters the system.

The equality sign holds in the case that only reversible processes take place inside the system. If irreversible processes take place (which is the case in real systems in operation) the >-sign holds. If heat is supplied to the system at several places we have to take the algebraic sum of the corresponding terms.

For open systems (also allowing exchange of matter):
: <math>\frac{dS}{dt} = \frac{\dot Q}{T}+\dot S+\dot S_\text{i}</math> with <math> \dot S_\text{i} \ge 0</math>
Here, <math>\dot S</math> is the flow of entropy into the system associated with the flow of matter entering the system. It should not be confused with the time derivative of the entropy. If matter is supplied at several places we have to take the algebraic sum of these contributions.