Editing Second law of thermodynamics (section)

=== 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.