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While giving a test, I encountered a question which was as follows

Lead reduces $\ce{NO_3^-}$ into $\ce{NO}$ and $\ce{NO_2}$ depending on the concentration of $\ce{HNO_3}$ in solution. Assuming that $\ce{[Pb^{2+}]} = \pu{0.1 M}$ and $p_\ce{NO} = p_\ce{NO2} = \pu{0.001 bar}$ and use the following data:

\begin{align} E^\circ_\ce{Pb^2+/Pb} &= \pu{-0.13 V}\\ E^\circ_\ce{NO3^-/NO} &= \pu{0.96 V}\\ E^\circ_\ce{NO3^-/NO2} &= \pu{0.79 V}\\ \end{align} and at $\pu{298K}$ $\frac{(2.303)RT}{F} = \pu{0.06 V}$, find the concentration of $\ce{HNO3}$ at which thermodynamic tendency for reduction of $\ce{NO3^-}$ into $\ce{NO}$ and $\ce{NO2}$ by lead is the same.

In this question what is the meaning of thermodynamic tendency? Is it represented by $E_\mathrm{cell}$ or $\Delta G$? Quoting a reliable source would help.

The answer is $\pu{10^{0.625} M}$ which comes by equating the $E_\mathrm{cell}$ of the reactions. My intuition told me to equate $\Delta G$ but it seems I am wrong.

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    $\begingroup$ How is equating Ecell different from equating ΔG? $\endgroup$ Aug 7, 2021 at 9:56
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    $\begingroup$ The number of electron that are involved in the reaction are different hence equating $\ce{E_{cell}}$ and equating $\ce{\Delta{G}}$ are two different things. $\endgroup$
    – ecneics
    Aug 7, 2021 at 10:32
  • $\begingroup$ Guys have tendency to be attracted to ( in their eyes ) the most beautiful girl regardless of the success. Similarly, TD systems have tendency to reach the state with the minimal Gibbs energy, regardless of if they reach it ( e.g. due kinetic reasons ). Diamonds have TD tendency to become graphite, staying being diamonds for billions years. $\endgroup$
    – Poutnik
    Aug 7, 2021 at 15:25
  • $\begingroup$ I am unable to understand that what are you wanting to imply? $\endgroup$
    – ecneics
    Aug 7, 2021 at 19:52
  • $\begingroup$ The Hess law is applicable on $\Delta G_\mathrm{halfr}$ ( implicitly referring to SHE as the 2nd half reaction), or on its equivalent $-nF.E_\mathrm{redox}$. // As $\Delta U = q \cdot \Delta E$ (energy, charge, potential difference ) $\endgroup$
    – Poutnik
    Aug 12, 2021 at 9:53

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For solving this problem, the $\Delta G$ values cannot be compared, or equalized, because they do not correspond to the same equation. Both reactions start from the same products, $\ce{Pb + NO3^-}$. But the first reaction leads to $\ce{NO}$ and the other one to $\ce{NO2}$, and these two nitrogen oxides do not have the same $G^\circ_\mathrm{form}$ values. So the Gibbs energy values are no use for solving this problem. It is much better to compare potentials through the Nernst equation.

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  • $\begingroup$ I am not able to understand that if the $\Delta{G^o_form}$ values are different then why can't we compare the $\Delta{G}$ of both the reactions? Also then, under what circumstances the $\Delta{G}$ values of two reactions can be compared? $\endgroup$
    – ecneics
    Aug 7, 2021 at 13:04

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