Editing Second law of thermodynamics (section)

== Corollaries ==

=== Perpetual motion of the second kind ===
{{main|Perpetual motion}}
Before the establishment of the second law, many people who were interested in inventing a perpetual motion machine had tried to circumvent the restrictions of [[first law of thermodynamics]] by extracting the massive internal energy of the environment as the power of the machine. Such a machine is called a "perpetual motion machine of the second kind". The second law declared the impossibility of such machines.

=== Carnot's theorem ===
[[Carnot's theorem (thermodynamics)|Carnot's theorem]] (1824) is a principle that limits the maximum efficiency for any possible engine. The efficiency solely depends on the temperature difference between the hot and cold thermal reservoirs. Carnot's theorem states:
* All irreversible heat engines between two heat reservoirs are less efficient than a [[Carnot engine]] operating between the same reservoirs.
* All reversible heat engines between two heat reservoirs are equally efficient with a Carnot engine operating between the same reservoirs.

In his ideal model, the heat of caloric converted into work could be reinstated by reversing the motion of the cycle, a concept subsequently known as [[thermodynamic reversibility]]. Carnot, however, further postulated that some caloric is lost, not being converted to mechanical work. Hence, no real heat engine could realize the [[Carnot cycle]]'s reversibility and was condemned to be less efficient.

Though formulated in terms of caloric (see the obsolete [[caloric theory]]), rather than [[entropy]], this was an early insight into the second law.

=== Clausius inequality ===
The [[Clausius theorem]] (1854) states that in a cyclic process
: <math>\oint \frac{\delta Q}{T_\text{surr}} \leq 0.</math>

The equality holds in the reversible case<ref>[http://scienceworld.wolfram.com/physics/ClausiusTheorem.html ''Clausius theorem''] at [[Wolfram Research]]</ref> and the strict inequality holds in the irreversible case, with ''T''<sub>surr</sub> as the temperature of the heat bath (surroundings) here. The reversible case is used to introduce the state function [[entropy]]. This is because in cyclic processes the variation of a state function is zero from state functionality.

=== Thermodynamic temperature ===
{{main|Thermodynamic temperature}}
For an arbitrary heat engine, the efficiency is:
{{NumBlk|: |<math>\eta = \frac {|W_n|}{q_\text{H}} = \frac{q_H+q_\text{C}}{q_\text{H}} = 1 - \frac{|q_\text{C}|}{|q_\text{H}|}</math>|{{EquationRef|1}}}}
where ''W''<sub>''n''</sub> is the net work done by the engine per cycle, ''q''<sub>H</sub> > 0 is the heat added to the engine from a hot reservoir, and ''q''<sub>C</sub> = −{{abs|''q''<sub>''C''</sub>}} < 0<ref name="PlanckBook">{{cite book |last=Planck |first=M. |title=Treatise on Thermodynamics |page=§90 |quote=eq.(39) & (40) |publisher=Dover Publications |year=1945}}.</ref> is waste [[Heat|heat given off]] to a cold reservoir from the engine. Thus the efficiency depends only on the ratio {{abs|''q''<sub>C</sub>}} / {{abs|''q''<sub>H</sub>}}.

[[Carnot theorem (thermodynamics)|Carnot's theorem]] states that all reversible engines operating between the same heat reservoirs are equally efficient. Thus, any reversible heat engine operating between temperatures ''T''<sub>H</sub> and ''T''<sub>C</sub> must have the same efficiency, that is to say, the efficiency is a function of temperatures only: 
{{NumBlk|:|<math>\frac{|q_\text{C}|}{|q_\text{H}|} = f(T_\text{H},T_\text{C}).</math>|{{EquationRef|2}}}}

In addition, a reversible heat engine operating between temperatures ''T''<sub>1</sub> and ''T''<sub>3</sub> must have the same efficiency as one consisting of two cycles, one between ''T''<sub>1</sub> and another (intermediate) temperature ''T''<sub>2</sub>, and the second between ''T''<sub>2</sub> and ''T''<sub>3</sub>, where ''T''<sub>1</sub> > ''T''<sub>2</sub> > ''T''<sub>3</sub>. This is because, if a part of the two cycle engine is hidden such that it is recognized as an engine between the reservoirs at the temperatures ''T''<sub>1</sub> and ''T''<sub>3</sub>, then the efficiency of this engine must be same to the other engine at the same reservoirs. If we choose engines such that work done by the one cycle engine and the two cycle engine are same, then the efficiency of each heat engine is written as the below.
: <math>\eta _1 = 1 - \frac{|q_3|}{|q_1|} = 1 - f(T_1, T_3)</math>,
: <math>\eta _2 = 1 - \frac{|q_2|}{|q_1|} = 1 - f(T_1, T_2)</math>,
: <math>\eta _3 = 1 - \frac{|q_3|}{|q_2|} = 1 - f(T_2, T_3)</math>.

Here, the engine 1 is the one cycle engine, and the engines 2 and 3 make the two cycle engine where there is the intermediate reservoir at ''T''<sub>2</sub>. We also have used the fact that the heat <math>q_2</math> passes through the intermediate thermal reservoir at <math>T_2</math> without losing its energy. (I.e., <math>q_2</math> is not lost during its passage through the reservoir at <math>T_2</math>.) This fact can be proved by the following.
: <math>\begin{align}
  & {{\eta }_{2}}=1-\frac{|{{q}_{2}}|}{|{{q}_{1}}|}\to |{{w}_{2}}|=|{{q}_{1}}|-|{{q}_{2}}|,\\ 
 & {{\eta }_{3}}=1-\frac{|{{q}_{3}}|}{|{{q}_{2}}^{*}|}\to |{{w}_{3}}|=|{{q}_{2}}^{*}|-|{{q}_{3}}|,\\ 
 & |{{w}_{2}}|+|{{w}_{3}}|=(|{{q}_{1}}|-|{{q}_{2}}|)+(|{{q}_{2}}^{*}|-|{{q}_{3}}|),\\ 
 & {{\eta}_{1}}=1-\frac{|{{q}_{3}}|}{|{{q}_{1}}|}=\frac{(|{{w}_{2}}|+|{{w}_{3}}|)}{|{{q}_{1}}|}=\frac{(|{{q}_{1}}|-|{{q}_{2}}|)+(|{{q}_{2}}^{*}|-|{{q}_{3}}|)}{|{{q}_{1}}|}.\\ 
\end{align}</math>

In order to have the consistency in the last equation, the heat <math>q_2</math> flown from the engine 2 to the intermediate reservoir must be equal to the heat <math>q_2^*</math> flown out from the reservoir to the engine 3.

Then
: <math>f(T_1,T_3) = \frac{|q_3|}{|q_1|} = \frac{|q_2| |q_3|} {|q_1| |q_2|} = f(T_1,T_2)f(T_2,T_3).</math>

Now consider the case where <math>T_1</math> is a fixed reference temperature: the temperature of the [[triple point]] of water as 273.16&nbsp;K; <math>T_1 = \mathrm{273.16~K}</math>. Then for any ''T''<sub>2</sub> and ''T''<sub>3</sub>,
: <math>f(T_2,T_3) = \frac{f(T_1,T_3)}{f(T_1,T_2)} = \frac{273.16 \text{ K} \cdot f(T_1,T_3)}{273.16 \text{ K} \cdot f(T_1,T_2)}.</math>

Therefore, if thermodynamic temperature ''T''* is defined by
: <math>T^* = 273.16 \text{ K} \cdot f(T_1,T)</math>
then the function ''f'', viewed as a function of thermodynamic temperatures, is simply
: <math>f(T_2,T_3) = f(T_2^*,T_3^*) = \frac{T_3^*}{T_2^*},</math>
and the reference temperature ''T''<sub>1</sub>* = 273.16 K × ''f''(''T''<sub>1</sub>,''T''<sub>1</sub>) = 273.16 K. (Any reference temperature and any positive numerical value could be used{{snd}}the choice here corresponds to the [[Kelvin]] scale.)

=== Entropy ===
{{main|Entropy (classical thermodynamics)}}
According to the [[Clausius theorem|Clausius equality]], for a ''reversible process''
: <math>\oint \frac{\delta Q}{T}=0</math>

That means the line integral <math>\int_L \frac{\delta Q}{T}</math> is path independent for reversible processes.

So we can define a state function ''S'' called entropy, which for a reversible process or for pure heat transfer satisfies
: <math>dS = \frac{\delta Q}{T} </math>

With this we can only obtain the difference of entropy by integrating the above formula. To obtain the absolute value, we need the [[third law of thermodynamics]], which states that ''S'' = 0 at [[absolute zero]] for perfect crystals.

For any irreversible process, since entropy is a state function, we can always connect the initial and terminal states with an imaginary reversible process and integrating on that path to calculate the difference in entropy.

Now reverse the reversible process and combine it with the said irreversible process. Applying the [[Clausius inequality]] on this loop, with ''T''<sub>surr</sub> as the temperature of the surroundings,
: <math>-\Delta S+\int\frac{\delta Q}{T_\text{surr}}=\oint\frac{\delta Q}{T_\text{surr}} \leq 0</math>

Thus,
: <math>\Delta S \ge \int \frac{\delta Q}{T_\text{surr}}</math>

where the equality holds if the transformation is reversible. If the process is an [[adiabatic process]], then <math>\delta Q=0</math>, so <math>\Delta S \ge 0</math>.

=== Energy, available useful work ===
{{See also|Exergy}}
An important and revealing idealized special case is to consider applying the second law to the scenario of an isolated system (called the total system or universe), made up of two parts: a sub-system of interest, and the sub-system's surroundings. These surroundings are imagined to be so large that they can be considered as an ''unlimited'' heat reservoir at temperature ''T''<sub>R</sub> and pressure ''P''<sub>R</sub> {{snd}}so that no matter how much heat is transferred to (or from) the sub-system, the temperature of the surroundings will remain ''T''<sub>R</sub>; and no matter how much the volume of the sub-system expands (or contracts), the pressure of the surroundings will remain ''P''<sub>R</sub>.

Whatever changes to ''dS'' and ''dS''<sub>R</sub> occur in the entropies of the sub-system and the surroundings individually, the entropy ''S''<sub>tot</sub> of the isolated total system must not decrease according to the second law of thermodynamics:
: <math> dS_{\mathrm{tot}}= dS + dS_\text{R} \ge 0 </math>

According to the [[first law of thermodynamics]], the change ''dU'' in the internal energy of the sub-system is the sum of the heat ''δq'' added to the sub-system, ''minus'' any work ''δw'' done ''by'' the sub-system, ''plus'' any net chemical energy entering the sub-system ''d'' Σ''μ<sub>iR</sub>N<sub>i</sub>'', so that:
: <math> dU = \delta q - \delta w + d\left(\sum \mu_{iR}N_i\right)</math>
where ''μ''<sub>''iR''</sub> are the [[chemical potential]]s of chemical species in the external surroundings.

Now the heat leaving the reservoir and entering the sub-system is
: <math> \delta q = T_\text{R} (-dS_\text{R}) \le T_\text{R} dS </math>
where we have first used the definition of entropy in classical thermodynamics (alternatively, in statistical thermodynamics, the relation between entropy change, temperature and absorbed heat can be derived); and then the second law inequality from above.

It therefore follows that any net work ''δw'' done by the sub-system must obey
: <math> \delta w \le - dU + T_\text{R} dS + \sum \mu_{iR} dN_i </math>

It is useful to separate the work ''δw'' done by the subsystem into the ''useful'' work ''δw<sub>u</sub>'' that can be done ''by'' the sub-system, over and beyond the work ''p<sub>R</sub> dV'' done merely by the sub-system expanding against the surrounding external pressure, giving the following relation for the useful work (exergy) that can be done:
: <math> \delta w_u \le -d \left(U - T_\text{R} S + p_\text{R} V - \sum \mu_{iR} N_i \right)</math>

It is convenient to define the right-hand-side as the exact derivative of a thermodynamic potential, called the ''availability'' or ''[[exergy]]'' ''E'' of the subsystem,
: <math> E = U - T_\text{R} S + p_\text{R} V - \sum \mu_{iR} N_i </math>

The second law therefore implies that for any process which can be considered as divided simply into a subsystem, and an unlimited temperature and pressure reservoir with which it is in contact,
: <math> dE + \delta w_u \le 0 </math>
i.e. the change in the subsystem's exergy plus the useful work done ''by'' the subsystem (or, the change in the subsystem's exergy less any work, additional to that done by the pressure reservoir, done ''on'' the system) must be less than or equal to zero.

In sum, if a proper ''infinite-reservoir-like'' reference state is chosen as the system surroundings in the real world, then the second law predicts a decrease in ''E'' for an irreversible process and no change for a reversible process.
: <math>dS_\text{tot} \ge 0 </math> is equivalent to <math> dE + \delta w_u \le 0 </math>

This expression together with the associated reference state permits a [[design engineer]] working at the macroscopic scale (above the [[thermodynamic limit]]) to utilize the second law without directly measuring or considering entropy change in a total isolated system (see also ''[[Process engineer]]''). Those changes have already been considered by the assumption that the system under consideration can reach equilibrium with the reference state without altering the reference state. An efficiency for a process or collection of processes that compares it to the reversible ideal may also be found (see ''[[Exergy efficiency]]'').

This approach to the second law is widely utilized in [[engineering]] practice, [[environmental accounting]], [[systems ecology]], and other disciplines.