A boundary reaction minimum with a one-sided extent

Use species $A,B$, one reaction column $\nu =(-1,1{)}^{\mathsf{T}}$, and starting amounts $n_A^0=0$ mol, $n_B^0=1$ mol. The admissible extents are $\xi \in [-1,0]$ mol because $n_A(\xi )=-\xi$ and $n_B(\xi )=1\mathrm{m}\mathrm{o}\mathrm{l}+\xi$ must both be nonnegative. Let ${\mathrm{\Phi }}_0$ have units joules and define an ambient affine potential on an open neighborhood of the nonnegative amount region by

$$\widehat{\mathrm{\Phi }}(n_A,n_B)={\mathrm{\Phi }}_0+1\mathrm{J}-(1\mathrm{J}\mathrm{m}\mathrm{o}{\mathrm{l}}^{-1})n_B\mathrm{.}$$

Along $n(\xi )=(-\xi ,1\mathrm{m}\mathrm{o}\mathrm{l}+\xi )$, set $\mathrm{\Phi }(\xi ):=\widehat{\mathrm{\Phi }}(n(\xi ))={\mathrm{\Phi }}_0-(1\mathrm{J}\mathrm{m}\mathrm{o}{\mathrm{l}}^{-1})\xi$. Its derivative-defined amount coefficients are ${\mu }_A:={\mathrm{\partial }}_{n_A}\widehat{\mathrm{\Phi }}=0$ and ${\mu }_B:={\mathrm{\partial }}_{n_B}\widehat{\mathrm{\Phi }}=-1\mathrm{J}\mathrm{m}\mathrm{o}{\mathrm{l}}^{-1}$, including at ${\xi }_{\ast }=0$ mol. This ambient extension is constructed optimization data only; no positive-mole thermodynamic chart at $n_A=0$ or physical realization is asserted.