Raoult and Henry's law

According to my textbook, for Raoult's law, we have:


$$μ\_i(\mathrm{vap}) = μ\_i^\* (p\_i^\*,T) + RT \ln⁡ x\_i$$


where $μ\_i^\* (p\_i^\*,T)$ is the chemical potential at $T$ and $p\_i^\*$, the pressure of pure $i$. So, when $x\_i=1$, we have $μ\_i (vap)=μ\_i^\* (p\_i^\*,T)$. That's okay and makes sense.


For Henry's law, we have:


$$μ\_i(\mathrm{vap}) = μ\_i^∞ (p\_j^\*,T) + RT \ln⁡ x\_i$$


where $μ\_i^∞ (p\_j^\*,T)$ is the chemical potential at $T$ and $p\_j^\*$, the pressure of pure $j$ (the other component of the mixture). So, when $x\_i = 1$ we have $μ\_i^∞ (p\_j^\*,T) = μ\_i^∞ (p\_j^\*,T)$ and that doesn't make any sense, since we have $i$ pure and the equation says that the chemical potential is the chemical potential of pure $j$. 


Am I interpreting something wrongly?




The first equation is for the chemical potential of species i in an ideal liquid solution, neglecting the Poynting correction. The second equation is an approximation to the chemical potential of species i in a real solution in the region of low values of the mole fraction of i, and apparently includes both the Poynting correction, hence evaluation of the chemical potential at the total pressure p (not just pj) as well as a term for RT times the natural log of the infinite dilution activity coefficient of species i. So the equation only applies at low values of the mole fraction, and the first term in the equation represents the extrapolation of the low mole fraction limiting behavior to a mole fraction of 1. 


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