Unit derivation concerning Nernst equation

We know that Nernst equation is given by:


$E=E^{\circ}+\frac{RT}{nF} \ln[\ce{M^{n+}}]$.


Now, A question asks me to derive the unit of $\frac{RT}{nF}$.


Which I think can easily be derived this way:


$\frac{RT}{nF}=\frac{E-E^{\circ}}{\ln[\ce{M^{n+}}]}$,


Now, the unit of $E-E^{\circ}$ is volts (since it is emf), and $\ln[\ce{M^{n+}}]$ is just a constant numerical value.


Hence, I think the unit of $\frac{RT}{nF}$ is Volts.


This was my approach. Can anyone tell me if it is a correct approach?




M. Farooq have given good description of how to analyse units in a equation. Therefore, your assumption of the unit of $\dfrac{RT}{nF}$ should be Volts is correct. However, your question is not to assume but to derive. What's that means is unit conversion. Let's see the units of all four variables:


* $R$ is a constant: $\pu{8.314 J mol-1 K-1}$
* $F$ is also a constant: $\approx \pu{96485 C mol-1}$
* $T$ is a variable: $\pu{K}$
* $n$ is number of total electrons in the redox reaction involved, and hence a variable but unit less.


Now lets see the unit conversion:


$$\frac{RT}{nF} = \frac{\pu{J mol-1 K-1}\times \pu{K}}{\pu{C mol-1}} = \frac{\pu{J}}{\pu{C}} $$


By definition (Ohm's law):


$$ V = \frac{\text{potential energy}}{\text{Charge}} = \frac{\pu{J}}{\pu{C}} $$


$$\therefore \:\; \frac{RT}{nF} = \frac{\pu{J}}{\pu{C}}= \pu{V} $$


 3

You should read the article **"Can One Take the Logarithm or the Sine of a Dimensioned Quantity or a Unit? Dimensional Analysis Involving Transcendental Functions"** [Here](https://pubs.acs.org/doi/10.1021/ed1000476)


The concepts mentioned there will help for life even if you don't pursue chemistry. In short, natural log factor ($\ln$) should be dimensionless. Now your equation is


$E=E^{\circ}+\frac{RT}{nF} \ln[\ce{M^{n+}}]$.


Now do a unit analysis:
The left hand side has units of volts $\pu{V}$.


$[\pu{V}] = [\pu{V}] + \frac{RT}{nF}$ times (dimensionless natural log)


So you cannot add apples to oranges. It implies that $\frac{RT}{nF}$ must have units of Volts.


Now do a proper dimensional analysis of the equation in terms of [SI base units](https://en.wikipedia.org/wiki/SI_base_unit) of [M], [L], [n], [T], [I] etc. and see if it is all consistent.


 2