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. [[1](https://doi.org/10.1039/TF9676302131)] 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](https://doi.org/10.1039/TF9676302131).




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.


 1