Charge Mobility and Conductivity of Protons

The charge mobility of protons is given via the Nernst-Einstein equation, as follows  

The conductivity 7, of any species i is given by its charge z,c, volume concentration of charge carrier particles c and mobility m,  

If the concentration is, instead, expressed as volume concentration of moles of charge carriers c. i, the conductivity becomes  

For protons, z = I, and it is convenient to obtain the conductivity in terms of the fraction of protons over oxide ion sitesx/f+ and the molar concentration of oxide ion sites qi-  

In regions where the proton concentration is given by the acceptor dopants, Xm,H- = Xm,Acc = Constant and the activation energy of proton conduction is given by that of the proton mobility. In regions where the protons are minor defects, and oxygen vacancies instead compensate the acceptors, the concentration term has a temperature dependency given by AH /2 so that the overall activation enthalpy of proton conduction is = AHm,m + AH /2. As 

---