Question

If the cell potential is proportional to work and the standard reduction potential for the hydrogen ion is zero, does this mean that the reduction of the hydrogen ion requires no work?

Short answer

No, the reduction of the hydrogen ion does not necessarily require no work, despite its standard reduction potential being zero. The work depends on the cell potential of the coupled half-cell in the electrochemical cell. The cell potential difference drives the redox reaction, and using the relationship $W=-nF(E_{other})$, if the cell potential is non-zero, the work required for the reduction of the hydrogen ion will not be zero.

Step-by-step solution

1. Understand Cell Potential and Standard Reduction Potential

Cell potential (also called the electromotive force or emf) is a measure of the energy per unit charge in an electrochemical cell. It is this energy that drives the redox reaction to take place, causing electrons to move from one species to another, effectively resulting in a electric current. Standard reduction potential is defined for a given half-cell reaction and is a measure of the tendency of a chemical species to be reduced by accepting electrons. A positive standard reduction potential indicates that the species has a stronger tendency to be reduced, while a negative standard reduction potential indicates a weaker tendency. The standard reduction potential for the hydrogen ion is defined to be zero.

2. Relate Cell Potential to Work

The cell potential is proportional to work - that is, the greater the cell potential, the greater the amount of work that can be derived from the chemical energy in an electrochemical cell. Mathematically, the relationship between cell potential (E) and work (W) is: $W=-nFE$ where n is the number of moles of electrons transferred in the redox reaction, F is the Faraday constant (96,485 C/mol), and E is the cell potential.

3. Examine Work Required for the Reduction of Hydrogen Ion

Given that the standard reduction potential for the hydrogen ion is zero, we need to determine if the reduction of the hydrogen ion requires no work. To do this, we can analyze an example of an hydrogen ion half-cell coupled with an arbitrary half-cell with a non-zero standard reduction potential (let it be $E_{other}$). The cell potential of the whole cell would be the difference between the standard reduction potentials of the two half-cells, as follows: $E_{cell}=E_{other}-E_{H^{+}}$ Since the standard reduction potential of the hydrogen ion ($E_{H^{+}}$) is zero, the cell potential becomes: $E_{cell}=E_{other}$ Now, recall the relationship between work and cell potential: $W=-nFE$

4. Determine if Work is Required

Using the relationship between work and cell potential and substituting the cell potential calculated in step 3, we have: $W=-nF(E_{other})$ In this expression, the work required depends on the cell potential of the "other" half-cell, $E_{other}$. If the cell potential is non-zero, the work required for the reduction of the hydrogen ion will not be zero. Thus, the statement "the reduction of the hydrogen ion requires no work" is not necessarily true just because the standard reduction potential of the hydrogen ion is zero. The work depends on the cell potential of the coupled half-cell in the electrochemical cell.