When energy is produced in nuclear reactions, we are reminded of the equation e=mc². However, since no energy can come from just nowhere, even in chemical reactions the energy released must come from some conversion of mass to energy. But the Wikipedia article on the Conservation of Mass mentions that "Einstein pointed out that the change in mass of systems for which the chemical amounts of energy were allowed in or out of systems, was predicted by his theory to be so small that it could not be measured with available instruments."

When hydrogen is burned it gives off 242 kiloJoules per mole of hydrogen (H₂ -- so it's Avogadro's number of molecules -- not atoms). How much does the molecular weight of water differ from half the total of the molecular weights of two molecules of hydrogen and one of oxygen?

The reaction is 2H₂ + O₂ → 2H₂O or H₂ + (1/2)O₂ → H₂O.

(In reply to solution by Justin)

Interesting problem. I don't really perceive it as a conversion of mass to energy in a sort of annihilation process, but rather a slowing down of the components and the consequential release of energy.

However:

starting here:      
http://hyperphysics.phy-astr.gsu.edu/hbase/molecule/boneng.html      
it seems that each oxygen molecule in the reaction produces 5.7 eV (other sites give a slightly smaller value)      
1eV = 1.602177×10^-19 Joules      

The critical step is that the eV is convertible directly to mass by:
([mass] [length]^2 [time]^(-2))/ c^2([length]^2/ [time]^2) =[mass]/c^2
     
c^2=8.9875517873681764 × 10^16m^2/s^2      
(1.602177×10^-19 Joules)/(8.9875517873681764 × 10^16m^2/s^2)      
=1.78266×10^-36 kg   

There are 6.022141*10^23 molecules in a mol.      
      
Now we have 1.78266×10^-36 kg*5.7*6.022141*10^23      
or 6.119 nanograms.   

Cross checking: (1.602177×10^-19 Joules*5.7*6.022141*10^23 gives 550kJ.  

The figure given in the puzzle was 242*2=484kJ, which is not far off (figures elsewhere go as high as 572kJ) 

Posted by broll on 2013-01-31 07:05:16