The authors consider a self-diffusiophoresis problem for a Janus particle in the fast-reaction limit, where the solute concentration satisfies a mixed boundary-value problem written in a dimensionless formulation as: \(\nabla ^{2}c=0\), inside the spherical particle, where \(c\) is the solute concentration, with the kinetic condition at the particle boundary \(r=1\), \( \frac{\partial c}{\partial r}=Da\ c\), if \(0<\theta <\pi /2\), and \(\frac{ \partial c}{\partial r}=0\), if \(\pi /2<\theta <\pi \), and the approach to the equilibrium concentration at large distances \(c\rightarrow 1\) as \( r\rightarrow \infty \). Here \(Da\) is Damköhler coefficient, which is here supposed to be large, implying \(c=0\) for \(0<\theta <\pi /2\), and \(\frac{ \partial c}{\partial r}=0\) for \(\pi /2<\theta <\pi \). The author proposes the following expression for the solution to this problem: \(c(r,\mu )-1=\sum^{\infty}_{n=0}B_{n}r^{-(n+1)}P_{n}(\mu )\) and he draws computations to determine its coefficients.