The biological membrane potential: A thermodynamic approach

Summary

The emf of the following galvanic cell

$$\begin{gathered} Ag|AgCl\left| {_{3M}^{KCl} } \right||C_K (1)C_{Na} (1)C_{Cl} (1)C_R (1)|Membrane| \hfill \\ C_K (2)C_{Na} (2)C_{Cl} (2)C_R (2)\left| {_{3M}^{KCl} } \right|AgCl|Ag \hfill \\ \end{gathered} $$

is calculated on the basis of classical irreversible thermodynamics. The membrane is permeable to water, K⁺, Na⁺ and Cl⁻ ions, but impermeable to the anion R⁻. If the membrane consists of separate channels for transport of cations and anions, these channels being charged or neutral, the emf calculated for the present cell can be approximated by the equation

$$E = \frac{{RT}}{F}\left\{ {(1 - t'_{_{Cl} } )1n\frac{{C_{Na} (1) + \frac{{u'_K }}{{u'_{Na} }}KC_K (1)}}{{C_{Na} (2) + \frac{{u'_K }}{{u'_{Na} }}KC_K (2)}} - t'_{Cl} 1n\frac{{C_{Cl} (1)}}{{C_{Cl} (2)}}} \right\}$$

which for zero transport number of Cl⁻ ions in the membrane,t′_Cl=0, gives

$$E = \frac{{RT}}{F}1n\frac{{C_{Na} (1) + \frac{{u'_K }}{{u'_{Na} }}KC_K (1)}}{{C_{Na} (2) + \frac{{u'_K }}{{u'_{Na} }}KC_K (2)}}.$$

In these equationsu′ is the mobility of an ion in the membrane andC is the concentration of an ion in solution.K is the equilibrium constant for the equilibrium KCl(aq)+NaM(membrane)=NaCl(aq)+KM(Membrane). The second equation is recognized as the Goldman-Hodgkin-Katz equation for “the membrane potential” when the membrane is cation conducting. The present equations, however, give the total emf for the galvanic cell includingnot negligible contributions to the cell potential from the KCl salt bridges.