I was looking through the literature when I found a paper by Greenzaid et al. [1] 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
- 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.
