[math-fun] Balancing chemical equations by hand
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?
Marc: It's been a while (nearly 50 years) since I've done this; could you give an example of what you mean? 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?
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
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?
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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?
In my high school chem class, we were taught the "Oxidation State" method. Going with quad's example ... KMnO4 + HCl ---> KCl + MnCl2 + H2O + Cl2 We use our chemistry knowledge to figure out that only Mn and Cl are changing oxidation state here. (Oxidation state = signed valence, from a math viewpoint. Usually, H = K = +1, O = -2, Cl = -1. Stable compounds have a total of 0, such as HCl, KCl, and H2O.) What's happening in this reaction is that Mn has an oxidation state of +7 in KMnO4 (potassium permanganate, an oxidizing agent), and it drops to +2 in MnCl2. The Mn is reduced, and some of the chloride (state -1) is oxidized to 0 (plain Chlorine, as Cl2). In theory, these represent electrons being shuffled around between atoms, filling completed shells. Fie. The deltas are Mn: +7 -> +2 and Cl: -1 -> 0. So we expect 1 "molecule" of KMnO4 to balance 5 atoms of Cl (as 2.5 * Cl2). Our first cut has 1 KMnO4 + ? HCl ---> ? KCl + ? MnCl2 + ? H2O + 2.5 Cl2 We can double this to clear the fraction; this is a matter of taste. 2 KMnO4 + ? HCl ---> ? KCl + ? MnCl2 + ? H2O + 5 Cl2 Now we fill in the obvious other coefficients, balancing K, Mn, and O from LHS to RHS: 2 KMnO4 + ? HCl ---> 2 KCl + 2 MnCl2 + 8 H2O + 5 Cl2 The HCl coefficient is the only thing left, and it must be 16 to make the H from H2O work. Then we check the Cl balance, and we're done. I've admired this method for its "power": it lets you do stuff in your head that would otherwise be hard. It's subject to criticism on both chemical and mathematical grounds: Reactions don't go to completion, how do you know what products to expect, why doesn't the Cl2 react with the water, ... . For mathematicians, a lot of reactions don't have a unique solution, and how do we know to look at Mn and Cl as the change agents? Why doesn't KMnO4 in water make peroxide (H2O2) or oxygen (O2)? There's so much in chemistry that seems vague & ambiguous. Rich ---------- Quoting quad <quadricode@gmail.com>:
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
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?
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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?
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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".
As an undergraduate, I discovered how to balance chemical reactions using determinants. Teacher response: This is MY course and we will do it MY way! Actually I had added conservation of charge to conservation of mass because redox equations were otherwise ambiguous. I wonder of thete is really any alternatice to conservation equations and linear algebra, even if most textbook equations can be solved by inspection. Of course the algebra part is rely mentioned in a chemistry course. -hvm
participants (6)
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Eugene Salamin -
Henry Baker -
Marc LeBrun -
mcintosh＠unam.mx -
quad -
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