Its cross ketols this time!

Chemistry Level 3

Your Situation

After having fun with the cross aldol problem you start experimenting with $n$ mixed ketones: such that the substituent on one ketone is not present on any of the $n-1$ ketones. Now you mix all these $n$ ketones in a large trough with $dil.$ $NaOH$ solution.

Hypothetical again !

$n=1000$

Now find the no. of $CROSS$ ketol condensation products that are formed.

DO NOT consider stereoisomers

all substituents have atleast one alpha-hydrogen.

The answer is 1998000.

1 solution

Vignesh S
Apr 12, 2016

Since they are mixed ketones we have $2n$ distinct carbanions,and $n$ substrates to attack. Total no. ketol condensation products are $\dbinom{2n}{1}$ $\dbinom{n}{1}$ .For the ketol(non-cross) There are $2$ carbanions for each ketone and there are $n$ such substrates, therefore the total non-cross ketols are $2n$ . By P.I.E. the no.of cross ketol products are $\dbinom{2n}{1}$ $\dbinom{n}{1}$ $-$ $2n$ .Putting $n=1000$ we get the above answer.
For better understanding check the solution of this problem

Nice mathematical question in chemistry

Arunava Das - 3 years, 4 months ago

Just saying, you're unnecessarily bringing in a very strong theorem from combinotorics, principle of inclusion and exclusion (P.I.E.)

A Former Brilliant Member - 5 years, 2 months ago

I just thought it would give better clarity , and anyways I have used the the same to exclude the non-cross ketols so I guess there isn't any mistake on.my part in mentioning P.I.E. I'm just using a much simpler understandable version of it

Vignesh S - 5 years, 2 months ago

Its cross ketols this time!

The answer is 1998000.

1 solution

0 pending reports