Well, I have a nice way to figure out a nice (heuristic) way. In the spirit of experimental math, get some common examples of equations, write them as a system, and proceed to note how you would solve each system intuitively. Then associate that method of solving with that kind of equation. For example, consider the single displacement reaction AgNO3 + Zn ---> Ag + Zn(NO3)2 Let X = NO3 just because it appears on both sides. AgX + Zn ---> Ag + ZnX2 I would start at the most complex molecule on the LHS, and the first atom of that molecule. Ag is balanced. Move to the next 'atom', X. X is not balanced, so balance it on the LHS with a 2. 2AgX + Zn --> Ag + ZnX2 Oh my we have to backtrack, look at the Ag, we have to balance on the right, 2AgX + Zn --> 2Ag + ZnX2 Now the first molecule is balanced, and we see the second is too. Looking back at the original eqn and adding coefs a*AgX + b*Zn ---> c*Ag + d*ZnX2 we have the system Ag... a = c X... a = 2d Zn... b = d Of course we have an infinite number of solutions. Pick '1' for the most complex molecule (let's say AgX, since with our heuristic way, we started with that one). So a=1. a=1 **given 1 = c 1/2 = d b = 1/2 so our eqn is 1*AgX + (1/2)*Zn ---> 1*Ag + (1/2)*ZnX2 Now multiply the sucker by lcm'(a,b,c,d) = 2 where lcm' = lcm of denominators 2AgX + Zn ---> 2Ag + ZnX2 The whole deal before about backtracking was to avoid having to have coefficients of 1/2. Personally, even when working by hand, I think writing the ridiculously simple equations out allows me to answer almost instantly what the correct balance is. Sorry for the ramble. I feel as if I havent answered anything. -Robert On Sun, Apr 24, 2011 at 2:13 PM, Marc LeBrun <mlb@well.com> 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?
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