# Units for dissociation constant and relationship to Gibbs Free Energy

How does one "convert" between an equilibrium constant calculated from the Gibbs free energy for a dissociation reaction and experimentally determined dissociation constants?

$$\Delta G^\circ=-RT\log(K_{eq})$$

I think I understand the derivation of this equation and why the equilibrium constant should be unitless if it's defined in terms of activities.
But I have a binding reaction that has an estimated free energy change of $-32~\mathrm{kcal/mol}$, which yields a dissociation constant on the order of $10^{-24}$ at $298~\mathrm{K}$.
Is it correct to just multiply the equilibrium constant by the standard concentration (1 molar) to get a more typical dissociation constant with units? A yoctomolar dissociation constant seems kind of strange to me, but I'm not sure where I've made a mistake.

• Is it $\ce{AB<=>A + B}$, with $\Delta G^\circ_{\mathrm{diss}}(\ce{AB})=-32~\mathrm{kcal/mol}$? Multiplying with the standard concentration just adds a unit to the same value. Also your definition of $K$ is slightly incorrect: goldbook.iupac.org/S05915.html – Martin - マーチン Jul 3 '14 at 3:04

## 1 Answer

free energy change of -32 kcal/mol, which yields something on the order of 10^-24 at 298K.

Seems the main mistake is a sign error. Also "log" must mean "natural logarthim" the way the equation in the question is written to be correct.

If the free energy change is negative, the equillbrium constant is $>1$

Should be $10^{\frac{32\frac{kcal}{mol}}{2.303\times0.00199\frac{kcal}{Kmol}\times{298K}}} = 10^{23}$

• Sorry, that was my mistake, I was referring to the free energy change of the binding reaction as being negative, and so the dissociation constant should in fact be < 1. In any case, my concern is with the order of magnitude of the constant, which doesn't seem biologically or even physically reasonable. If my calculation looks correct otherwise, does that mean that my estimated free energy change is incorrect? Thanks for taking the time to answer my question! – Jazzy Jul 2 '14 at 20:24
• It is certainly physically possible. For say dinitrogen to dissociate into two nitrogen atoms, the equilibrium constant would be much smaller than 10^-24. Without knowing more about your system, I don't see any way to rule out it being correct. – DavePhD Jul 3 '14 at 11:34