Understanding the Calculation of Hydrogen Electrode Potential (SHE)

The calculation of the hydrogen electrode potential (SHE) is fundamental in electrochemistry. It defines the zero point for measuring electrode potentials.

This article explores detailed formulas, common values, and real-world applications of SHE calculations. It serves as a comprehensive technical guide.

¡Hola! ¿En qué cálculo, conversión o pregunta puedo ayudarte?

Pensando ...

Calculate the SHE potential at 25°C with 1 atm H₂ and pH 0.
Determine the effect of temperature on SHE potential at 50°C.
Compute the electrode potential for a hydrogen electrode at pH 7.
Analyze the impact of hydrogen partial pressure on SHE potential.

Comprehensive Table of Common Values for Hydrogen Electrode Potential (SHE)

Parameter	Symbol	Typical Value	Units	Notes
Standard Hydrogen Electrode Potential	E°_SHE	0.000	V (by definition)	Reference potential at standard conditions
Temperature	T	298.15 (25°C)	K	Standard temperature for SHE
Hydrogen Ion Activity (approximate concentration)	a_H+	1	mol/L	At pH 0, standard state
Hydrogen Gas Partial Pressure	P_H2	1	atm	Standard pressure for SHE
Gas Constant	R	8.314	J·mol^-1·K^-1	Universal gas constant
Faraday Constant	F	96485	C·mol^-1	Charge per mole of electrons
Number of Electrons Transferred	n	2	unitless	For hydrogen electrode reaction
Standard Electrode Potential of SHE	E°	0.000	V	By definition, zero volts
pH	pH	0 – 14	unitless	Hydrogen ion concentration scale

Fundamental Formulas for Calculating Hydrogen Electrode Potential (SHE)

The hydrogen electrode potential is calculated based on the Nernst equation, which relates the electrode potential to temperature, pressure, and ion activity.

General Nernst Equation for SHE:

E = E° – (RT)/(nF) ln Q

Where:

E = Electrode potential (V)
E° = Standard electrode potential (V), for SHE it is 0 V by definition
R = Universal gas constant = 8.314 J·mol^-1·K^-1
T = Temperature in Kelvin (K)
n = Number of electrons transferred in the half-reaction (2 for hydrogen)
F = Faraday constant = 96485 C·mol^-1
Q = Reaction quotient, dimensionless

For the hydrogen electrode half-reaction:

2H+ + 2e– ⇌ H2(g)

The reaction quotient Q is expressed as:

Q = (aH+)2 / PH2

Where:

a_H+ = Activity of hydrogen ions (approximate to concentration in dilute solutions)
P_H2 = Partial pressure of hydrogen gas (atm)

Substituting Q into the Nernst equation:

E = 0 – (RT / 2F) ln [ (aH+)2 / PH2 ] = – (RT / F) ln aH+ + (RT / 2F) ln PH2

Since pH = -log₁₀(a_H+), converting natural logarithm to base 10 logarithm:

E = – (2.303 RT / F) pH + (RT / 2F) ln PH2

At standard conditions (T = 298.15 K, P_H2 = 1 atm), the equation simplifies to:

E = -0.05916 × pH (V)

This shows the linear dependence of electrode potential on pH at standard pressure and temperature.

Detailed Explanation of Variables and Their Typical Values

R (Gas Constant): 8.314 J·mol^-1·K^-1, a universal constant used in thermodynamic calculations.
T (Temperature): Usually 298.15 K (25°C) for standard conditions, but can vary in practical applications.
n (Number of Electrons): 2 for the hydrogen electrode reaction, representing the two electrons involved in the reduction of protons to hydrogen gas.
F (Faraday Constant): 96485 C·mol^-1, representing the charge of one mole of electrons.
a_H+ (Activity of H⁺): Approximated by concentration in dilute solutions; at pH 0, a_H+ = 1.
P_H2 (Partial Pressure of Hydrogen): Standard pressure is 1 atm; deviations affect the electrode potential logarithmically.
pH: The negative logarithm of hydrogen ion activity, ranging typically from 0 to 14 in aqueous solutions.

Real-World Applications and Case Studies of SHE Potential Calculation

Case Study 1: Determining Electrode Potential at pH 7 and 1 atm Hydrogen Pressure

In biological and environmental electrochemistry, the hydrogen electrode potential at neutral pH is critical. Calculate the electrode potential of the hydrogen electrode at 25°C, pH 7, and 1 atm hydrogen pressure.

Given:

T = 298.15 K
pH = 7
P_H2 = 1 atm
E° = 0 V (by definition)

Calculation:

E = -0.05916 × pH = -0.05916 × 7 = -0.4141 V

Interpretation: The electrode potential is -0.4141 V relative to the standard hydrogen electrode at pH 0. This negative shift reflects the lower proton concentration at neutral pH.

Case Study 2: Effect of Hydrogen Partial Pressure on Electrode Potential at pH 0

Calculate the electrode potential of the hydrogen electrode at 25°C, pH 0, but with hydrogen gas partial pressure reduced to 0.5 atm.

Given:

T = 298.15 K
pH = 0
P_H2 = 0.5 atm
E° = 0 V

Calculation:

Using the Nernst equation:

E = – (RT / F) ln aH+ + (RT / 2F) ln PH2

At pH 0, a_H+ = 1, so ln a_H+ = 0.

Calculate (RT / 2F) ln P_H2:

(RT / 2F) = (8.314 × 298.15) / (2 × 96485) ≈ 0.01285 V

ln(0.5) = -0.6931

Therefore:

E = 0 + 0.01285 × (-0.6931) = -0.0089 V

Interpretation: The electrode potential decreases by approximately 8.9 mV due to the reduced hydrogen partial pressure, illustrating the sensitivity of SHE to gas pressure.

Additional Considerations in SHE Potential Calculations

While the standard hydrogen electrode is defined at 1 atm hydrogen pressure and pH 0, practical systems often deviate from these conditions. Accurate calculation of electrode potential requires consideration of:

Temperature Variations: The Nernst equation explicitly includes temperature, affecting the RT/F term. Elevated temperatures increase the magnitude of potential shifts.
Non-ideal Behavior: Activities rather than concentrations should be used for precise calculations, especially in concentrated solutions.
Pressure Effects: Partial pressure of hydrogen gas directly influences the electrode potential logarithmically.
Electrode Surface Conditions: Real electrodes may have overpotentials or kinetic limitations affecting measured potentials.

Summary of Key Equations for SHE Potential

Equation	Description
E = E° – (RT / nF) ln Q	General Nernst equation for electrode potential
Q = (a_H+)² / P_H2	Reaction quotient for hydrogen electrode half-reaction
E = – (RT / F) ln a_H+ + (RT / 2F) ln P_H2	Expanded Nernst equation for SHE
E = – (2.303 RT / F) pH + (RT / 2F) ln P_H2	Equation converting natural log to base 10 logarithm
E = -0.05916 × pH (at 25°C, 1 atm)	Simplified linear relation of potential with pH at standard conditions