[isears]

I'm self-studying chem right now and was hoping someone could help me out with a specific electrochemistry problem:

Given a galvanic cell @ 25 C^o:
Pt(s)|H2(g, 1.0 bar)|H+(pH = ?)||Cl-(aq, 1.0M)|AgCl(s)|Ag(s), E = +0.30V

I need to find the PH on the left side of the cell.

(1) find half-reactions + combined rxn

AgCl(s) + e^-  Cl^-(aq) + Ag(s)  (E^o=+0.22V)
0.5H₂(g) H⁺(aq) + e^- (E^o=0V)

so

AgCl(s) + 0.5H₂(g) Cl^-(aq) + Ag(s) + H⁺(aq)

(2) Use Nernst Equation to find [H⁺]

ΔE = E^o - (RT/nF) * ln(Q)

(Expanding Q)

ΔE = E^o - (RT/nF) * ln([Cl^-][H⁺] / [H₂]^0.5)

Knowns:
ΔE = +0.30V (given)
E^o = E_R - E_L = +0.22V - 0V = +0.22V
n = 1
[Cl^-] = 1.0 M (given)
[H₂] = 1.0 bar  (given, but use P/RT = n/V to get in terms of M)  = 1.0 / (0.08314 * 298.15) M
T = 298.15 K
R, F = constants

(3) Solve Nernst Equation symbolically and take -log to solve for unknown pH

-log( e^{((ΔE - Eo) / -(RT/F))} * ([H₂]^0.5 / [Cl^-])) = pH

This results in a calculated pH of 2, but the solution guide I'm using tells me the pH is 1. Is my issue here a fundamental misunderstanding of how to use the Nernst equation or do you think I'm just making a simple computation mistake somewhere when I plug in the values?

Any comments much appreciated!

[Borek]

I can be missing something, but at first sight nothing sticks out as obviously wrong.

Full cell equation doesn't matter, and to be honest in most cases I think it is just an unnecessary complication. It is much easier to deal with each half cell separately.

I admit I just skimmed, but I am not convinced pH 1 (nor 2) is a correct answer. If the potential difference for the cell is 0.30V, the hydrogen electrode must be at -0.08 V. Nernst equation for the hydrogen electrode (after canceling everything that cancels out, and after simplifying everything that can be simplified) takes a very simple form of

[tex]E = 0.059 log[H^+] = -0.059 pH[/tex]

I don't see how you can get 1 or 2 for pH when plugging -0.08 into that formula.

[mjc123]

Your mistake is in trying to use the concentration of hydrogen in the Nernst equation. The standard state for gases is a pressure of 1 atm, so you should use the pressure in atm.

Really what should appear in the Nernst equation is not concentration or pressure, but C/C° or P/P°, where ° indicates the standard state. This is because the equation is all about deviations from standard conditions. We usually omit these in practice because the standard state for solutions is 1M and for gases 1 atm. However, the concentration of a gas in the standard state at 298K is ca. 0.04 M. So putting [H₂] = 0.04 in the equation will give the wrong answer, you need [H₂]/[H₂]° = 0.04/0.04 = 1. Because hydrogen is in the standard state, it doesn't affect E.

I agree with Borek that the book answer of pH 1 is wrong. I don't know how they got that.

[isears]

Thank you both for taking a look

Your mistake is in trying to use the concentration of hydrogen in the Nernst equation. The standard state for gases is a pressure of 1 atm, so you should use the pressure in atm.

This is my crucial misunderstanding. I don't think I ever really fully understood this concept when I was going over equilibrium. Everything tends to just be in Molarity...until it's not.

FWIW, when I use the activity of H (H / H^o) rather than the concentration using the full Nernst equation, I get a pH of 1 (rounded to one sig fig): https://www.wolframalpha.com/input/?i=-log_10(e%5E((0.3-0.22)+%2F+-0.025693)+*+(1%2F1))

That said, there's still obviously a discrepancy here between the (much simpler and more desirable) half-cell Nernst equation and the full equation. I haven't gotten to the bottom of that yet.

[mjc123]

This is my crucial misunderstanding. I don't think I ever really fully understood this concept when I was going over equilibrium. Everything tends to just be in Molarity...until it's not.

You can use concentration if you want, though it is simpler to use pressure in Nernst. The important thing is that everything is relative to its standard state. If the standard state does not have a value of 1 in the unit system you're using (e.g. the concentration of a gas in M), you must include it explicitly in the Nernst equation.