How to calculate membrane potentials? I have recently been reading about membrane potentials in the cell and how they are derived. The textbook (Guyton and Hall Textbook of Medical Physiology, 13th Ed) gave this equation:
$$EMF=\pm\frac{61}{z}\log\frac{C_i}{C_o}$$
Where $i$ refers to the interior of the cell, $o$ refers to the exterior of the cell, $z$ is the relative magnitude of the charge of the cation/anion, $EMF$ is the potential in the cell, taking the external potential as 0. Additionally, the sign is + for anions and - for cations.
I understand that this comes from the Nernst Equation:
$$E=E^o-\frac{RT}{\nu_eF}\ln Q$$
Where $\nu_e$ is the number of electrons transferred per mole of reaction.
I understand that $E^o=0$ because at equilibrium, the concentrations of ions on both sides of the membrane are equal and I also understand where the magnitude comes from: $$61=\frac{1000RT}{\log eF}$$
Where $\frac{1000}{\log e}$ is used to change the base of the logarithm and change the units to mV and T is taken to be the body temperature of 37 C, converted to Kelvins.
However, what confuzzles me is

The sign convention: Why are cations - and anions +.
How do we derive the $-\frac{RT}{\nu_eF}\ln Q$ for transfer of ions, since unlike the Nernst Equation, which is derived for electrons, ions are being transferred instead?
How do we measure the membrane potential? According to the text, they put one electrode in the extracellular fluid and pierce another electrode (a typical Ag/AgCl electrode) into the cell, in contact with the intracellular fluid, as shown below. Since there is no transfer of electrons, how does this apparatus actually work?
Your textbook description is correct. Recall the principle of potentiometry, the potential difference is measured when there is no current flowing in the system. This is true under ideal conditions. Do you recall the pH meter? What is the electrode made of? It is made of glass, do you feel that current can flow across glass when you are measuring pH of your solution?
BTW, the working of the glass electrode puzzled electrochemists for ages and the still do not know how it works exactly. So if you are confused, welcome to the club.
You should rather ask how do high impedance voltmeters work? Such voltmeters work when there is no current flow and you are solely interested in potential difference between the electrodes.

Since there is no transfer of electrons, how does this apparatus
actually work?

This is perhaps a misconception. Potential difference only develops when electrons are transferred (only once) but the is no continuous flow of electrons (which constitutes current). If you are combing your dry hair in winter, a huge potential difference develops between your hair and the comb. You comb is still &quot;charged&quot; when you move it away from your hair. The electrons were transferred but only at one instant. There is no current flow between your comb and the hairs.