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):

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

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

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

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 of [M], [L], [n], [T], [I] etc. and see if it is all consistent.