# Equation for the equilibrium constant for the hydration of carbonyls using UV spectroscopy

I was looking through the literature when I found a paper by Greenzaid et al.  which states that the equilibrium constant $$K_\mathrm{hyd}$$ is:

$$K_\mathrm{hyd} = [\text{hydrate}]/[\text{carbonyl}] = (\varepsilon_0^\mathrm{w} - \varepsilon^\mathrm{w})/\varepsilon^\mathrm{w}, \label{eqn:1}\tag{1}$$

where $$\varepsilon^\mathrm{w}$$ is the molar absorption coefficient measured under conditions of hydration, and $$\varepsilon_0^\mathrm{w}$$ is the molar absorption coefficient of the carbonyl compound in water in the absence of hydration.

(In water solvent, $$\varepsilon_0^\mathrm{w}$$ usually taken as the molar extinction coefficient in cyclohexane.)

How did the researchers get to the above equation in terms of the molar absorption coefficients?

As $$[\text{hydrate}]$$ is equal to $$[\text{carbonyl}]_0 - [\text{carbonyl}],$$ the equation \eqref{eqn:1} turns to:

$$K_\mathrm{hyd} = \frac{[\text{carbonyl}]_0 - [\text{carbonyl}]}{[\text{carbonyl}]}$$

However, the Beer–Lambert law states that

$$A = c\varepsilon l,$$

where $$A$$ is the optical density, $$c$$ is the concentration, $$\varepsilon$$ is the extinction coefficient and $$l$$ is the path length.

Denoting the corresponding parameters for carbonyl without hydration as $$A_0$$ and $$\varepsilon_0$$, and the parameters for carbonyl with hydration as $$A$$ and $$\varepsilon$$, the following can be written:

\begin{align} K_\mathrm{hyd} &= \frac{[\text{carbonyl}]_0 - [\text{carbonyl}]}{[\text{carbonyl}]} \\ &= \frac{\frac{A_0}{\varepsilon_0l} - \frac{A}{\varepsilon l}}{\frac{A}{\varepsilon l}} \\ &= \frac{\frac{A_0}{\varepsilon_0} - \frac{A}{\varepsilon}}{\frac{A}{\varepsilon}} \\ \therefore &= \frac{\varepsilon}{A}\left(\frac{A_0}{\varepsilon_0} - \frac{A}{\varepsilon}\right) \\ &= \frac{\varepsilon}{A}\frac{A_0\varepsilon - A\varepsilon_0}{\varepsilon\varepsilon_0} \\ &= \frac{A_0\varepsilon - A\varepsilon_0}{A\varepsilon_0} \\ \end{align}

But we require

$$K_\mathrm{hyd} = \frac{\varepsilon_0 - \varepsilon}{\varepsilon}$$

Shouldn't this mean that $$K_\mathrm{hyd}$$ is actually proportional to a fraction containing the $$1/\varepsilon$$ values?

### References

1. Greenzaid, P.; Rappoport, Z.; Samuel, D. Limitations of Ultra-Violet Spectroscopy for the Study of the Reversible Hydration of Carbonyl Compounds. Trans. Faraday Soc. 1967, 63 (0), 2131–2139. DOI: 10.1039/TF9676302131.
• The assumption discussed by the paper (which they argue should be discarded because it is not a good one) is that the absorption of carbonyl species is independent of solvent. That is the same as saying the concentration of carbonyl is uniquely determined by absorbance. The other assumption is that the only thing that affects absorbance of a compound is the extent to which its hydrated: hydrated carbonyls are assumed not to absorb at all, and apparent per-molecule absorbance of carbonyl compounds in water is thus lowered by an amount proportional to the equilibrium constant. – Curt F. Apr 18 '19 at 18:17
• Under those assumptions, there is no real difference between $A$ and $\epsilon$, they are both direct measures of the concentration of carbonyl in a solution. They key is that the units of $\epsilon$ are L/mol/cm, where mol is mole of total carbonyl containing compound. – Curt F. Apr 18 '19 at 18:19

The law is not as you state although it often gets reported as this. The '$$A$$' you quote is the optical density. The Beer-Lambert law is $$I_{tr}=I_0e^{-\epsilon_\lambda [C]\ell}$$ where $$I_{tr}$$ is the intensity of transmitted light for a molecule at concentration $$[C]$$ at wavelength $$\lambda$$ and cell path length $$\ell$$, and $$\epsilon_\lambda$$ is the extinction coefficient at wavelength $$\lambda$$.
From the definition you can see where the expression you ask about comes from even with their different notation; it looks as if $$w=-{\epsilon_\lambda [C]\ell}$$ where $${\epsilon_\lambda [C]\ell}$$ is the optical density.
• The equation you quote has $\epsilon^{+w}$ and from Beers law it should be $e^{-\epsilon [C]L}$ so $-w$, it may be my misunderstanding from what you wrote I has assume that it was a typo and $\epsilon^{w}$ should actually be $e^w$. – porphyrin Nov 19 '18 at 12:08