From: quad <quadricode@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Sent: Sun, April 24, 2011 1:41:16 PM Subject: Re: [math-fun] Balancing chemical equations by hand On Sun, Apr 24, 2011 at 2:22 PM, Henry Baker <hbaker1@pipeline.com> wrote:
Marc:
It's been a while (nearly 50 years) since I've done this; could you give an example of what you mean?
He means, given a chemical "equation", determine the "coefficients" of each molecule. Example: Given KMnO4 + HCl ---> KCl + MnCl2 + H2O + Cl2 determine a, b, ..., f so that a*KMnO4 + b*HCl ---> c*KCl + d*MnCl2 + e*H2O + f*Cl2 "balances", i.e., there are equal numbers of each type of atom on each side. The balanced equation here is 2 KMnO4 + 16 HCl ---> 2 KCl + 2 MnCl2 + 8 H2O + 5 Cl2 because we have equal numbers of each atom. Look at O for example. On the left, we have 2*4 = 8 O, on the right, we have 8*1 O. Anyway, I think you get it. :) - Robert ________________________________ That's a nice example, and I'll demonstrate how I would solve it manually. Balancing the K and Mn, there are equal amounts of KMnO4, KCl and MnCl2. Next, balance the electrons. Mn+7 accepts 5e to become Mn+2, so for each Mn, 5 Cl- are oxidized to neutral Cl. So it's 2 KMnO4 and 5 Cl2. On the RHS there are 6 more Cl for the KCl and MnCl2, so we need 16 HCl. Finally, since the H and O do not participate in oxidation or reduction, they should automatically balance, and indeed they do with 8 H20. -- Gene
At 01:13 PM 4/24/2011, Marc LeBrun wrote:
Can anyone come up with a nice way to balance chemical equations manually?
All the web seems to offer is either vague "fiddle around until it works" or the nuclear option "translate into a simultaneous linear system and solve".
Is there anything in between? It need not be theoretically optimal, just easy to apply by hand to small solvable cases.
I'm imagining a well-defined procedure repeatedly "adjusting" coefficients until "done", then dividing out their common factor, perhaps akin to an n-D raster line drawing algorithm that somehow manages to hill climb onto a scaled solution.
It should be more clever than, say, mindlessly trying all the possible cases in some fixed order, yet stay grounded in the problem domain.
It might even be "morally equivalent" to Gaussian elimination but performed directly on the chemical equations. Longhand division is kind of like this. There's a little eyeballing and maybe even some backtracking estimating the digits, but it's a reasonably effective way to arrive at the answer by hand. Crunching determinants for simple chemistry is analogous to using Newton's method on everyday division problems.
Any ideas?