Chemical work convention

Definition and physical meaning

For an open thermodynamic system , the particle number $N$ can change as matter crosses the boundary . The energetic effect of this exchange is governed by the chemical potential $\mu$.

In the differential form of the fundamental energy relation for a single-component simple system, the reversible change in internal energy includes the term $\mu\,dN$:

Physically, $\mu$ is the energy change associated with adding particles to the system at fixed entropy and volume (for the reversible/quasistatic setting).

Convention used in this blog

We keep the same work sign convention as for mechanical work: $\delta W>0$ means work done by the system. With that choice, it is often convenient to treat particle exchange as a kind of generalized work and define the chemical work (by the system) as

• Single species: $\delta W_{\mathrm{chem}} := -\mu\, dN$,
• Multiple species: $\delta W_{\mathrm{chem}} := -\sum_i \mu_i\, dN_i$.

This definition is chosen so that when particles enter the system ($dN>0$), the system receives energy and the work done by the system is negative: $\delta W_{\mathrm{chem}}<0$.

With pressure–volume work included via the $P\,dV$ convention , the first law can be written in the “work by the system” form as

Key relations and common consequences

• Grand potential sign check: with grand potential $\Omega := U - TS - \sum_i \mu_i N_i$, this convention leads to

  $$d\Omega = -S\, dT - P\, dV - \sum_i N_i\, d\mu_i$$

  for reversible variations, matching the natural variables $(T,V,\mu_i)$ used in the grand-canonical setting .

• Translation to “work on the system” convention: if instead one writes $dU=\delta Q+\delta W^{(\mathrm{on})}$, then the chemical contribution is typically taken as $\delta W^{(\mathrm{on})}_{\mathrm{chem}}=+\sum_i \mu_i\, dN_i$.

• Interpretation as reservoir exchange: the $\mu\,dN$ term corresponds to energy transferred with matter from a particle reservoir (sometimes informally called a “chemical reservoir”), just as $T\,dS$ corresponds to exchange with a thermal reservoir .