pH meter mechanism - connecting potential differences

Question

This question was asked here: pH probe bulb - what is happening within the glass? and How does a pH glass electrode work?.

But, the mechanism remains unanswered. How does a potential difference between the inner and outer surfaces of the glass bulb result in a measurable difference between the two electrodes? These are two different electrical systems. I am unable to find in the literature how the the two systems are connected.

Given:

• There is an electrolytic solution connecting the electrodes through openings in the tubes housing each electrode.
• A very small electric current flows between the reference electrode to the pH electrode through a high value resistor for voltage measurement. The electrodes maintain ion balance through the chemical reactions occurring at each electrode and the movement of ions through the openings.

From a-cyclohexane-molecule answer in pH probe bulb - what is happening within the glass?:

A few side remarks.

• Equilibrium is reached when the favorable binding of protons to the glass surface is balanced by the unfavorable electrostatic repulsion and chemical potential gradient that result from diffusion into the hydrated gel layer. This provides an equation relating the potential difference to the pH of the solution and allows for pH measurement.
• Something has to be able to move through the inner layer of the glass membrane to conduct a current and hence allow for a measurement of the potential difference. It turns out that sodium ions can move through this inner layer, but only sluggishly---the resistance of the glass membrane is about 108Ω.

A hypothesis is:

• When H+ invades the porous glass, the side with the more H+ can repulse H+ out of the opposite side. An acidic solution being measured, ejects H+ out of the inner bulb surface and an alkaline solution allows more H+ to enter into the inner glass surface.
• The positive ion concentration near the electrode next to the inner glass surface changes.
• This change in positive ion concentration can not be compensated by the chemical reactions at each electrode that are maintaining a electrically neutral solution.
• The change in positive ion concentration near the pH electrode tip is causing a change in the pH meter potential.
• This change in the pH meter potential results in a measurable difference in voltage between the electrodes.

Answer 1

A glass electrode is a galvanic cell, where the membrane is a thin glass bulb, at the end of a long glass tube. The bulb is filled with some diluted HCl solution. And the bulb is made of a special glass where the protons $\ce{H+}$ are nearly free to move from one site to another one, like the electrons in a metal. I saw once that this glass contains Lanthanum $\ce{La}$. So if such a bulb (filled with a dilute HCl solution) is dipped in a more diluted acidic solution, the $\ce{H+}$ have the tendency to get out of the bulb through the membrane. But they cannot, because of the $\ce{Cl-}$ ions who attract them, and they cannot get out of the bulb.

There is a way to take care of the $\ce{Cl-}$ ions. In the center of the bulb, the end of a long silver wire (covered with insoluble $\ce{AgCl}$) is introduced to dip into the $\ce{HCl}$ solution. The unhappy $\ce{Cl-}$ ions do react with the silver wire, according to : $$\ce{Cl- + Ag -> AgCl + e-}$$ The emitted electron got through the silver wire to an external circuit containing a voltmeter and then go to another glass tube containing an electrode made of a silver wire covered by $\ce{AgCl}$ dipping in a $\ce{KCl}$ solution. The same electrode as before, but a different solution. Here, the external electrons react with $\ce{AgCl}$ producing the opposite reaction $$\ce{e- + AgCl -> Ag + Cl-}$$ Now this second glass tube has a small hole in its lower part, which is blocked by a porous ceramic. Ions can cross this barrier if there is an attraction from outside. And here there is an attraction, if this second electrode is dipped in the same weakly acidic outer solution as the first electrode (with the bulb). In this case, the $\ce{H+}$ ions having crossed the glass membrane are able to attract the $\ce{Cl-}$ ions coming out of the second electrode. This whole system of two electrodes is a galvanic cell, and the measured potential is defined by Nerns't law. The voltmeter gives a tension which is $$\pu{E = ~constant + 0.059 log[H^+] = ~ constant - 0.059 pH}$$ So the voltmeter gives directly the pH value of the outer solution.

Usually the second electrode is placed in a muff around the tube of the first electrode, taking care not to cover the thin bulb.