Degree of Unsaturation or Index of Hydrogen Deficiency [closed]








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I know the formula for the DOU, but don't really understand why the formula works. Could someone elaborate that to me?




The formula for calculating the Degree of Unsaturation (DU) is awkward. The derivation will come later. There is a better way to address this issue by what I shall call the Atom Replacement Method (ARM). The gist of the method is to replace all heteroatoms with C and/or H. This revised formula is compared with the most saturated hydrocarbon bearing the same number of carbon atoms. As an example, consider the molecular formula of the marine dye [Tyrian purple](https://en.wikipedia.org/wiki/Tyrian_purple): $\ce{C16H8Br2N2O2}$.  



![](https://i.stack.imgur.com/SN8Dn.jpg)  



Replace halogens with the same number of hydrogens and ignore divalent atoms: $\ce{C16H8Br2N2O2}$ --> $\ce{C16H10N2}$  

Replace each nitrogen with one carbon and one hydrogen: $\ce{C16H10N2}$ --> $\ce{C18H12}$ (**1**)  

The most saturated $\ce{C18}$ hydrocarbon is $\ce{C18H38}$ (**2**)  

Subtracting **1** from **2** gives 26.  

DU=26/2=13. There are 4 rings and 9 double bonds in Tyrian purple.


The following derivation of the formula is located [here](http://ursula.chem.yale.edu/%7Echem220/chem220js/STUDYAIDS/unsaturation/unsat.html). Here is a synopsis.  



Let c=#C, h=#H, n=#N and x=#Br, where "x" is the number of halogens.  
  

Ignoring divalent atoms and replacing xX with xH in CcHhNnXx  

gives CcHh+xNn  

Replacing nN with nC and nH yields Cc+nHh+x+n (**1**)  

The most saturated hydrocarbon is Cc+nH2(c+n)+2 (**2**)  

Subtracting **1** from **2** and simplifying gives 2c+2n+2-h-n-x  

Further simplification affords 2c+n+2-h-x  

Division by 2 gives  

DU = c+1+(n-h-x)/2  

Q.E.D.  
  

For a related discussions on ChemSE, see [this link](https://chemistry.stackexchange.com/questions/136377/importance-of-halogens-and-nitrogen-in-degree-of-unsaturation/136384#136384) and [this one](https://chemistry.stackexchange.com/questions/16046/index-of-hydrogen-deficiency-for-para-toluenesulfonylchloride?rq=1).


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