Two versions of the Debye-Hückel limiting law

The Debye-Hückel limiting law is usually written for the mean activity coefficient $$\log(\gamma_\pm)=-|z_+z_-|A\sqrt{I}$$ However, occasionally you will also see it written for the (unmeasurable) activity coefficient of each individual ion $$\log(\gamma_+)=-z_+^2A\sqrt{I}$$ For the compound $M_pX_q$ $$\gamma_\pm=(\gamma_+^p\gamma_-^q)^{1/(p+q)}$$ But that implies $$\log(\gamma_\pm)=-\frac{(pz_+^2+qz_-^2)}{p+q}A\sqrt{I}$$ I couldn't find the connection between these seemingly different definitions of $\gamma_\pm$ anywhere on the web so here it comes.  It's actually rather simple in hindsight.  Charge neutrality dictates $$pz_++qz_-=0$$ and this means $$pz_+^2+qz_-z_+=0\\ pz_+z_-+qz_-^2=0\\pz_+^2+(p+q)z_-z_++qz_-^2=0$$ which is what we need.  For some reason $z_-z_+$ is usually written $-|z_-z_+|$.

One things that's usually not mentioned is that the activities based on the mean and ionic activity coefficients will be different, i.e. $\gamma_\pm b_+ \ne \gamma_+ b_+$.  But of course things like the equilibrium constant will of course.



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