Why is Mixing Units in the Nernst Equation Considered Accurate?

[Acut]

One thing that always puzzled me on the Nerst equation is that you may mix the units of concentration and pressure. This, however, seems to be rather arbitrary. How can using the units of mol/L and atm in the same equation result in accurate results? It seems to be rather arbitrary! Particularly, when you use PV = nRT and calculate what the "concentration" of gas molecules in a sample of gas at 1atm and 298K, one does not get 1 mol/L.

I could agree if people used something on the lines of k*(pressure in atmospheres) in Nernst equation, where k is a correction factor. But setting k = 1 (which is what is done in chemistry textbooks) seems to be arbitrary, requiring a tremendous coincidence between the units mol/L and atm!

Can anyone explain me why mixing units in the Nernst equation is fine?

 

[Borek]

This is not about the Nernst equation, this is about a reaction quotient in general.

You will find several approaches to the explanation, the one I stick to is that you don't have concentrations nor pressures in the reaction quotient, but unitless activities. We approximate activities by pressures and concentrations. Technically each concentration (or pressure) should be multiplied by the activity coefficient, and activity coefficient has units that cancel out units of the concentration (or pressure). For ideal solution activity coefficients equal 1, but for the real solutions they are usually smaller. At least for some cases their values can be calculated from the first principles (for example for diluted ionic solutions google Debye–Hückel theory), sometimes they have to be determined experimentally.

 

[Acut]

So we don't actually use pressures or concentrations, but we use unitless activities in the reaction quotient. And for (ideal) solutions, the activity coefficients are equal to one. Equally, each pressure should be multiplied by the activity coefficients. This is similar to what I suggested in the first post, and it makes perfectly sense that concentrations or pressures should be corrected. That's not what I'm struggling with.

In the chemistry books I've looked up, when authors need to include a gas in the reaction quotient, they simply write the gas' pressure expressed in atm - which would imply they are considering that the activity coefficient for the gas is also one. That's what bugging me. I don't really care about the dimensions, but the fact that my books will do the following

- expressing the concentration of a solution in mol/L, the activity coefficient can be considered equal to one (considering it an ideal solution, of course)

- expressing the pressures in atm, the activity coefficient (would it be the fugacity coefficient?) can also be considered equal to one.

It seems as if one is measuring an object with two rulers, one graduated in centimeters and the other in inches, plugging the results in the same equation and magically getting the right result. Moles per litter and atmospheres are wildly different - how come one can simply plug them in the same equation without "interconverting" them in some way, by using a convenient numerical factor of some sort? I'm not talking about dimensions, but their absolute numerical value, only.

 

[Borek]

how come one can simply plug them in the same equation without "interconverting" them in some way, by using a convenient numerical factor of some sort?

But they are interconverted. You can express the pressure any way you want, you will get different value of K - but as long as you are using this value and you express the pressure in the same units, you are OK. Problems will start when you use K value determined using one unit for calculations using other units.

It happens we assumed 1 atm to be the reference, but we could select any other unit for that. That would change all K values - but they will be still consistent. Just different.

 

[Acut]

But they are interconverted. You can express the pressure any way you want, you will get different value of K - but as long as you are using this value and you express the pressure in the same units, you are OK. Problems will start when you use K value determined using one unit for calculations using other units.

It happens we assumed 1 atm to be the reference, but we could select any other unit for that. That would change all K values - but they will be still consistent. Just different.

So K values will change based on the unit we choose to measure the pressure and as long we use the same units, we will keep consistency. But ΔG = -RT ln K. So, if the value of K changes based on the units we chose to express pressure, the values of ΔG will be different for the same reaction - even though we are not changing the unit we use to express ΔG, since all the information about the units we chose for calculating K is lost when we establish that K is adimensional. I don't remember my old chemistry book ever making a distinction whether K was calculated using concentrations in mol/L, pressures in atm or a combination of both - it just uses SI for R and T and expresses ΔG in Joules.

 

[Borek]

But ΔG = -RT ln K. So, if the value of K changes based on the units we chose to express pressure, the values of ΔG will be different for the same reaction

No. We scale the result using R value.

 

[Acut]

No. We scale the result using R value.

Then my old chemistry book is wrong. Which is good, because this part didn't make sense .

But get to a concrete example. Suppose we have a very simple reaction, whose ΔG can be expressed as:

ΔG = - RT ln P₀ (equation 1)

where P0 is its pressure expressed in atmospheres. Suppose now I want to measure the pressure in an arbitrary unit, mta, such that 1 mta = 2 atm, but we keep measuring the temperature in the same units. Then, the new pressure P in mtas will be P0/2. And to keep units consistent, we will need a new value for the universal gas constant R'. It follows (from the ideal gas law) that R' = R/2 if we use mtas (and keep the same unit for temperature in both cases).

Then, using this new set of units,

ΔG = - R'T ln P

But knowing R' = R/2 and P = P₀/2, we have:

ΔG = - \frac{R}{2}*T*ln\frac{P0}{2}

Which is not equal to equation 1. Where is my reasoning wrong?

 

[Borek]

Thermodynamics is not my thing, perhaps it is a good moment for someone else to help. However, note that ΔG means change. That means subtracting final from initial - and my feeling is that the factor \frac {RT \ln 2} 2 will cancel out.

Off topic rambling: I always wonder why people mix LaTeX with text in their equations.