# Is a system's work output limited when changes in entropy do not occur?

Jul 7, 2017

Not necessarily.

Much of entropy has to do with energy dispersal, which can be influenced by system volume (which can be affected by pressure-volume work). But it neither depends directly on nor necessarily influences whether the system volume can change or not.

(and this is ignoring non-PV work.)

Entropy, $S$, which describes the amount of energy dispersal, is often regarded as a function of the volume AND temperature.

From the first law of thermodynamics for reversible processes in a thermodynamically-closed system (where the total number of particles is constant),

$\mathrm{dE} = \delta {q}_{\text{rev" + deltaw_"rev}} = T \mathrm{dS} - P \mathrm{dV}$

$\implies \underline{\mathrm{dS} = \frac{\mathrm{dE} + P \mathrm{dV}}{T}}$,

where:

• $\delta {q}_{r e v} = T \mathrm{dS}$ is the differential reversible heat flow (with respect to the system).
• $\delta {w}_{r e v , P V} = - P \mathrm{dV}$ is the differential reversible PV-work (done with respect to the system).
• $\mathrm{dE}$ is the differential change in internal energy, $E$.
• $P$ and $T$ are pressure and temperature, respectively.

PV-work, i.e. expansion/compression work, evidently involves a change in volume. In principle, having no entropy change in an entirely reversible process would result in:

$\mathrm{dS} = 0 = \frac{\mathrm{dE}}{T} + \frac{P}{T} \mathrm{dV}$

For ideal gases, which is what we usually treat in a first-year thermodynamics course, the internal energy is only a function of temperature.

So, if we assume $\mathrm{dE} \ne 0$, then the work must go into at least changing the temperature a little bit, if not just the volume.

At ordinary (i.e. nonzero) temperatures:

$\textcolor{b l u e}{\mathrm{dE} = - P \mathrm{dV} = \underline{{w}_{\text{rev,PV}} \ne 0}}$

This relation, however, only true for reversible isentropic processes in thermodynamically-closed systems.

And if I stopped here, I would be slightly lying. There are other kinds of work, what we'll label non-PV work, that is also done, such as electrical work.

$\mathrm{dE} = {w}_{\text{rev,PV" + w_"elec}}$ + etc.

So, no, work output is not necessarily limited when a system has no change in entropy. Much of entropy has to do with energy dispersal, and it neither depends directly on nor necessarily influences whether the system volume can change or not.

(Entropy can also be described as the tendency for system particles to redistribute into each energy state. Even without having that, work can be done on the system to compress it, shrinking energy level gaps without changing the population of states, and thus keeping entropy constant.)