Hi! I am taking an online CLEP chemistry course to prepare for the CLEP exam. The exam covers material you would expect from a first-year chemistry course in college.
This link has more info on the exam from the course website.I just reached this chapter which introduces the idea of the reaction quotient (which I'll label as [itex]Q_c[/itex] for the purpose of this post). I understand how to solve basic problems with reaction quotients, but I am seriously confused by my book's definition. I'll try to explain how the book defines [itex]Q_c[/itex] in clear terms, and then I'll explain what confuses me.
To define [itex]Q_c[/itex], my book gives an example of a reversible reaction:
[tex]m\text{A} + n\text{B} \rightleftarrows x\text{C} + y\text{D}.[/tex]
Each chemical in the formula ([itex]\text{A}[/itex], [itex]\text{B}[/itex], [itex]\text{C}[/itex], and [itex]\text{D}[/itex]) has a certain molar concentration. The book's convention is to write the molar concentration of a chemical in square brackets. For example, the molar concentration of [itex]\text{A}[/itex] would be [itex][\text{A}][/itex].
Now, the book defines the reaction quotient [itex]Q_c[/itex] as a dimensionless quantity. This quantity is calculated by taking the product-to-reactant ratio of the molar concentrations raised to their coefficients' powers:
[tex]Q_c = \frac{ [\text{C}]^x [\text{D}]^y }{ [\text{A}]^m [\text{B}]^n }.[/tex]
The book puts serious emphasis on the fact that [itex]Q_c[/itex] is a dimensionless quantity.
Here's what confuses me. If my understanding is correct, [itex][A][/itex], [itex][B ][/itex], [itex][C][/itex], and [itex][D][/itex] are not dimensionless quantities! They are measured in units of amount per volume (like [itex]\frac{\text{mol}}{\text{L}}[/itex], for example). So, the dimension of [itex]Q_c[/itex] could be something like [itex]\frac{\text{mol}^2}{\text{L}^2}[/itex] or [itex]\frac{\text{mol}^3}{\text{L}^3}[/itex] or even [itex]\frac{\text{L}}{\text{mol}}[/itex]. For example, given the equation
[tex]2\text{NO} + \text{Cl}_2 \rightleftarrows 2\text{NOCl},[/tex]
let's assume we have the molar concentrations shown below:
[tex][\text{NO}] = 0.0500 \frac{\text{mol}}{\text{L}},[/tex]
[tex][\text{Cl}_2] = 0.0155 \frac{\text{mol}}{\text{L}},[/tex]
[tex][\text{NOCl}] = 0.500 \frac{\text{mol}}{\text{L}}.[/tex]
Now, going purely by my book's definition, the reaction quotient for this scenario is
[tex]Q_c = \frac{ \left(0.500 \frac{\text{mol}}{\text{L}}\right)^2 }{ \left(\left(0.0500 \frac{\text{mol}}{\text{L}}\right)^2 \left(0.0155 \frac{\text{mol}}{\text{L}}\right)\right) },[/tex]
which I calculated as [itex]6450 \frac{\text{L}}{\text{mol}}[/itex]. Compare with my book's answer of
[tex]Q_c \approx 6450.[/tex]
Why is it acceptable to toss out units in this way? What if I were given the exact same molar concentrations, but they were written in this form instead?
[tex][\text{NO}] = 50 \frac{\text{mol}}{\text{m}^3},[/tex]
[tex][\text{Cl}_2] = 15.5 \frac{\text{mol}}{\text{m}^3},[/tex]
[tex][\text{NOCl}] = 500 \frac{\text{mol}}{\text{m}^3}.[/tex]
That would make the reaction quotient [itex]6.45 \frac{\text{m}^3}{\text{mol}}[/itex], which is the same as what I got before. Clearly, it wouldn't be okay to simply chop off the units and write
[tex]Q_c \approx 6.45[/tex]
because that's a totally different constant!
So, what's going on here? What makes [itex]\frac{\text{mol}}{\text{L}}[/itex] such a special unit, and why do I only get right answers on the homework problem when I use measurements in [itex]\frac{\text{mol}}{\text{L}}[/itex]?