Definition 0.0.1  The chemical work $W_{chem}$ is the work done on the system by the generalized chemical potentials $\{\mu_\alpha\}$ and the generalized numbers of particles $\{N_\alpha\}$ which is the work done by a change in particle number.
\[\newcommand{\dj}{\rlap{d}{\bar{\phantom{w}}}}\dj W_{chem} = \sum_\alpha{\mu_\alpha dN_\alpha} = \mathbf{\mu}\cdot d\mathbf{N}\]