Atoms and Molecules

Date: 11/28/2001 at 19:54:25
From: Jenny
Subject: Word problems

A certain molecule contains twice as many atoms of hydrogen as oxygen 
and one more atom of carbon than hydrogen. If there are 21 atoms 
altogether in the molecule, how many atoms of carbon are there?

Date: 11/29/2001 at 12:42:30
From: Doctor Ian
Subject: Re: Word problems

Hi Jenny,

Suppose there is 1 hydrogen atom. Then there must be half as many 
ogygen atoms... oops! So there must be at least two hydrogen atoms. 

Okay, suppose there are 2 hydrogen atoms. Then there must be half as 
many oxygen atoms: 1. And there must be 1 more carbon atoms than 
hydrogen atoms: 3. That gives us

  hydrogen   oxygen    carbon    total

    2         1          3        6

So that's not the molecule we're looking for. But we can find the one 
we're looking for just by varying the hydrogen count. Note that we 
have to have an even number of hydrogen atoms, or we can't get an 
integer for the number of oxygen atoms:

  hydrogen   oxygen    carbon    total

     2         1          3        6
     4         2          5       11
     6         3          7       16

With a little imagination, we could write an expression to compute the 
total number of atoms in terms of the number of hydrogen atoms:

  hydrogen   oxygen    carbon    total

     2         1          3        6
     4         2          5       11
     6         3          7       16

     h        h/2        h+1      h + h/2 + (h+1)     

Once we've done that, we can specify the total that we want, 

  h + h/2 + (h+1) = 21

and solve the equation to find h directly.  

(Then we have to add 1 to find the number of carbon atoms. Forgetting 
to do that would be an easy mistake to make, which is why the problem 
asks you for the number of _carbon_ atoms instead of the number of 
_hydrogen_ atoms. A well-constructed problem always gives you lots of 
different ways to make a mistake.)

For a problem like this, where there aren't a lot of atoms involved, 
the extra work to set up and solve the equation may or may not be 
worth the trouble, compared to a little judicious guessing. But for 
much larger numbers (for example, suppose the total number of atoms is 
251), using an equation is much, much quicker. Which is one of the 
things that makes equations so useful!

Does this help? 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/