Evaluating membrane cell emf from different schemes of integration of the relevant Nernstian equations

Computing the emf. E (and the related membrane potential EM) of the general concn. cell: electrode J|IJ (m1) in S| membrane I|IJ (m2) in S| electrode J (where the membrane permselective to counterion I separates 2 solns. of molalities m1 and m2 of the same electrolyte IJ, one mole of which forms nI mol of I of valency zI and nJ moles of coion J of valency ZJ with n = nI + nJ) requires integration between limits m1 and m2 of the differential equation: dE = [nRT/FnJ|zJ|][τIzI - mτsMs]dln(mγ)IJ, where the transference nos. τ1 of I and τS of the solvent S through the membrane are customarily taken as const. The key point is the integration of the term mdln (mγ), which in the current practice is oversimplified to give [(m1 + m2)/2]ln(m2γ2/m1γ1), and may cause large overestimation of the solvent-transfer contribution, with unreliable results for E and EM. The procedure recommended is instead that of slitting into the "ideal part" and the "non-ideal part", the latter being safely integrable putting lnγ as an explicit function of m in terms of the extended Debye-Hueckel equation. Alternative and equally reliable procedures are described.