Standard hydrogen electrode

Nernst equation for SHE

The hydrogen electrode is based on the redox half cell corresponding to the reduction of two hydrated protons, 2 H⁺(aq), into one gaseous hydrogen molecule, H₂(g).

General equation for a reduction reaction:

${\displaystyle \underset{\text{ox}}{\text{(oxidant)}}+z{\text{e}}^{-}\stackrel{-\rightharpoonup }{\leftharpoondown -}\underset{\text{red}}{\text{(reductant)}}}$

The reaction quotient (Q_r) of the half-reaction is the ratio between the chemical activities (a) of the reduced form (the reductant, a_red) and the oxidized form (the oxidant, a_ox).

${\displaystyle Q_r=\frac{a_{\text{red}}}{a_{\text{ox}}}}$

Considering the 2 H⁺ / H₂ redox couple:

${\displaystyle 2{\text{H}}_{(\text{aq})}^{+}+2{\text{e}}^{-}\stackrel{-\rightharpoonup }{\leftharpoondown -}{\text{H}}_{2(\text{g})}^{}}$

at chemical equilibrium, the ratio Q_r of the reaction products by the reagents is equal to the equilibrium constant K of the half-reaction:

${\displaystyle K=\frac{a_{\text{red}}}{a_{\text{ox}}}=\frac{a_{{\mathrm{H}}_2}}{a_{{\mathrm{H}}^{+}}^2}=\frac{p_{{\mathrm{H}}_2}/p^0}{a_{{\mathrm{H}}^{+}}^2}=\frac{x_{{\mathrm{H}}_2}p/p^0}{a_{{\mathrm{H}}^{+}}^2}}$

where

• ${\displaystyle a_{\text{red}}}$ and ${\displaystyle a_{\text{ox}}}$ correspond to the chemical activities of the reduced and oxidized species involved in the redox reaction
• ${\displaystyle a_{{\mathrm{H}}^{+}}}$ represents the activity of H⁺
  .
• ${\displaystyle a_{{\mathrm{H}}_2}}$ denotes the chemical activity of gaseous hydrogen (H
  ₂), which is approximated here by its fugacity ${\displaystyle p_{{\mathrm{H}}_2}.}$
• ${\displaystyle p_{{\mathrm{H}}_2}}$ denotes the partial pressure of gaseous hydrogen, expressed without unit; ${\displaystyle p_{{\mathrm{H}}_2}=x_{{\mathrm{H}}_2}\cdot p,}$ where
  • ${\displaystyle x_{{\mathrm{H}}_2}}$ is the H
    ₂ mole fraction
  • ${\displaystyle p}$ is the total gas pressure in the system
• p⁰ is the standard pressure (1 bar = 10⁵ pascal) introduced here simply to overcome the pressure unit and to obtain an equilibrium constant without unit.

More details on managing gas fugacity to get rid of the pressure unit in thermodynamic calculations can be found at thermodynamic activity#Gases. The followed approach is the same as for chemical activity and molar concentration of solutes in solution. In the SHE, pure hydrogen gas (${\displaystyle x_{{\mathrm{H}}_2}=1}$) at the standard pressure ${\displaystyle p}$ of 1 bar is engaged in the system. Meanwhile the general SHE equation can also be applied to other thermodynamic systems with different mole fraction or total pressure of hydrogen.

This redox reaction occurs at a platinized platinum electrode. The electrode is immersed in the acidic solution and pure hydrogen gas is bubbled over its surface. The concentration of both the reduced and oxidised forms of hydrogen are maintained at unity. That implies that the pressure of hydrogen gas is 1 bar (100 kPa) and the activity coefficient of hydrogen ions in the solution is unity. The activity of hydrogen ions is their effective concentration, which is equal to the formal concentration times the activity coefficient. These unit-less activity coefficients are close to 1.00 for very dilute water solutions, but usually lower for more concentrated solutions.

As the general form of the Nernst equation at equilibrium is the following:

$${\displaystyle E_{\text{cell}}=E_{\text{cell}}^{\ominus }-\frac{RT}{zF}\ln K}$$

and as ${\displaystyle E_{\text{cell}}^{\ominus }=0}$ by definition in the case of the SHE,

The Nernst equation for the SHE becomes:

${\displaystyle E=0-\frac{RT}{2F}\ln \frac{p_{{\mathrm{H}}_2}/p^0}{a_{{\mathrm{H}}^{+}}^2}}$

${\displaystyle E=-2.303\frac{RT}{F}\left(\mathrm{p}\mathrm{H}+\frac{1}{2}\log \frac{p_{{\mathrm{H}}_2}}{p^0}\right)}$

Simply neglecting the pressure unit present in ${\displaystyle p_{{\mathrm{H}}_2}}$, this last equation can often be directly written as:

${\displaystyle E=-2.303\frac{RT}{F}\left(\mathrm{p}\mathrm{H}+\frac{1}{2}\log p_{{\mathrm{H}}_2}\right)}$

And by solving the numerical values for the term

${\displaystyle -2.303\frac{RT}{F}=-2.303\left(\frac{8.314\times 298.15}{96,485}\right)=-0.0591\mathrm{v}\mathrm{o}\mathrm{l}\mathrm{t}\mathrm{s},}$

the practical formula commonly used in the calculations of this Nernst equation is:

${\displaystyle E=-0.0591\left(\mathrm{p}\mathrm{H}+\frac{1}{2}\log p_{{\mathrm{H}}_2}\right)}$ (unit: volt)

As under standard conditions ${\displaystyle p_{{\mathrm{H}}_2}=1\text{ bar,}}$ ${\displaystyle \log p_{{\mathrm{H}}_2}=\log 1=0,}$ the equation simplifies to:

${\displaystyle E=-0.0591\mathrm{p}\mathrm{H}}$ (unit: volt)

This last equation describes the straight line with a negative slope of −0.0591 volt/ pH unit delimiting the lower stability region of water in a Pourbaix diagram where gaseous hydrogen is evolving because of water decomposition.

where:

• ⁠${\displaystyle a_{{\mathrm{H}}^{+}}}$⁠ is the activity of the hydrogen ions (H⁺) in aqueous solution, ${\displaystyle a_{{\mathrm{H}}^{+}}={\gamma }_{{\mathrm{H}}^{+}}\frac{C_{{\mathrm{H}}^{+}}}{C^0}}$
  with:
  • ⁠${\displaystyle {\gamma }_{{\mathrm{H}}^{+}}}$⁠ is the activity coefficient of hydrogen ions (H⁺) in aqueous solution
  • ⁠${\displaystyle C_{{\mathrm{H}}^{+}}}$⁠ is the molar concentration of hydrogen ions (H⁺) in aqueous solution
  • C⁰ is the standard concentration (1 M) used to overcome concentration unit
• ⁠${\displaystyle p_{{\mathrm{H}}_2}}$⁠ is the partial pressure of the hydrogen gas, in bar (1 bar = 10⁵ Pa)
• R is the universal gas constant: 8.3144 J⋅K⁻¹⋅mol⁻¹ (rounded here to 4 decimal)
• T is the absolute temperature, in kelvin (at 25 °C: 298.15 K)
• F is the Faraday constant (the charge per mole of electrons), equal to 96,485.3 coulomb·mol⁻¹
• p⁰ is the standard pressure: 1 bar = 10⁵ Pa

Note: as the system is at chemical equilibrium, hydrogen gas, H_2(g), is also in equilibrium with dissolved hydrogen, H_2(aq), and the Nernst equation implicitly takes into account the Henry's law for gas dissolution. Therefore, there is no need to independently consider the gas dissolution process in the system, as it is already de facto included.