Should binding constants be unitless when deriving fractional occupancy equations from reactions?

Question

It is known that binding (aka association) constants are in fact unitless, as has been discussed here already. However, I'm not a chemist and am confused about when one should or should not use units when working with association constants. One source says:

$K_\text{eq}$ for a reaction with unequal numbers of reactants and products is always given with units, even in published papers.

But why is that? Why not always use unitless values? Is there something inherently wrong with never using units for binding constants?

Consider this example. Let $\ce{A}$ bind to $\ce{X}$ as an n-mer (e.g. $\ce{A}$ can be a transcriptional activator binding to gene promoter). This results in active state, $\ce{X_{A}}$ (e.g. a state leading to gene transcription):

$$\ce{X + nA ->[k_{\text{on}}][k_{\text{off}}] X_{A}}$$

Association constant: $K_\text{A} = \frac{k_\text{on}}{k_\text{off}} $

Assuming equilibrium and law of mass action:

$$K_\text{A} \cdot \text{X} \cdot \mathrm{A^n} = \mathrm{X_A}$$

Now, the fractional occupancy (active states to all states ratio) is:

$$y=\mathrm{\frac{X_A}{X+X_A}} = \frac{K_\text{A}\cdot \mathrm{A^n}}{1 + K_\text{A}\cdot \mathrm{A^n}}$$

As $y$ must be unitless (i.e. it can be interpreted as a probability), for this particular equation, $K_\text{A}$ must have units of $\mathrm{M^{-n}}$ (in general, $\text{concentration}^{-n}$) for the equation to work out, correct (assuming concentration of $\ce{A}$ has units $\mathrm{M}$)? So in this particular case, using unitless $K_\text{A}$ seems wrong, but is it really, or is there something I'm missing?

Now let's extend this equation so that it includes a half-saturation constant $h$, i.e. concentration of $\ce{A}$ required for $y=0.5$ (50% activation). If I'm doing this correctly, we get:

$$y=\frac{K_\text{A}\cdot \mathrm{A^n}}{K_\text{A}\cdot h^n + K_\text{A}\cdot \mathrm{A^n}} = \frac{\mathrm{A^n}}{h^n + \mathrm{A^n}}$$

Note that this is equivalent to the Hill equation for an activator. This generalized equation, unlike the previous one, works just fine regardless of whether or not $K_\text{A}$ is unitless.

Is my understanding of this correct, and does the choice of unitless vs non-unitless binding constant indeed depend on the formulation of a specific equation?

Answer 1

The equilibrium constant does not need to be unitless, because it is depending on its definition. See goldbook:

Equilibrium Constant Quantity characterizing the equilibrium of a chemical reaction and defined by an expression of the type $$K_x = \Pi_B x_B^{\nu_B},$$ where $\nu_B$ is the stoichiometric number of a reactant (negative) or product (positive) for the reaction and $x$ stands for a quantity which can be the equilibrium value either of pressure, fugacity, amount concentration, amount fraction, molality, relative activity or reciprocal absolute activity defining the pressure based, fugacity based, concentration based, amount fraction based, molality based, relative activity based or standard equilibrium constant (then denoted $K^\circ$ ), respectively.

The standard equilibrium constant is always unitless, as it is defined differently (goldbook)

Standard Equilibrium Constant $K$, $K^\circ$ (Synonym: thermodynamic equilibrium constant) Quantity defined by $$K^\circ = \exp\left\{-\frac{\Delta_rG^\circ}{\mathcal{R}T}\right\}$$ where $\Delta_rG^\circ$ is the standard reaction Gibbs energy, $\mathcal{R}$ the gas constant and $T$ the thermodynamic temperature. Some chemists prefer the name thermodynamic equilibrium constant and the symbol $K$.

In your example both would work if you straighten your definitions, involving concentrations \begin{aligned} K_A \cdot c(\ce{X}) \cdot c^n(\ce{A}) &= c(\ce{X_{A}}),\\ y=\frac{c(\ce{X_A})}{c(\ce{X})+c(\ce{X_{A}})} &= \frac{K_A\cdot c^n(\ce{A})}{1 + K_A\cdot c^n(\ce{A})}, \end{aligned}

or activities \begin{aligned} K^\circ_A \cdot a(\ce{X}) \cdot a^n(\ce{A}) &= a(\ce{X_{A}}),\\ y=\frac{a(\ce{X_A})}{a(\ce{X})+a(\ce{X_{A}})} &= \frac{K^\circ_A\cdot a^n(\ce{A})}{1 + K^\circ_A\cdot a^n(\ce{A})}. \end{aligned}

In both cases $y$ remains unitless.

In experimental chemistry it is much easier to observe concentrations instead of activities. Therefore to fist named constant is more often in use. A straightforward use of the thermodynamical equilibrium constant might sometimes prove a little tricky.

For reasonable dilutions one can simply substitute activity with a unitless concentration as given through $$a(\ce{Y})=\gamma\frac{c(\ce{Y})}{c^\circ},$$ with $\gamma\approx1$ for $c(\ce{Y})\to0$ and $c^\circ=1\:\mathrm{mol/L}$.