$$C_2H_2 + O_2\rightarrow H_2O + CO_2$$

How do I solve this equation problem with linear algebra? How do I take vectors and matrix so what should I do at here? But the answer is: $$2C_2H_6 + 7O_2\rightarrow 6H_2O + 4CO_2$$

Would someone show me this?

closed as off-topic by Dietrich Burde, Davide Giraudo, amWhy, Aweygan, tilper May 8 '17 at 15:32

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  • No need for linear algebra, see here. – Dietrich Burde May 8 '17 at 13:57
  • @DietrichBurde: $ax=b$ is linear algebra, and for more complex reactions, the equations may not be that trivial. – Yves Daoust May 8 '17 at 14:32
up vote 3 down vote accepted

The solution is of the form

$$a\,C_2H_2 + b\ O_2\rightarrow c\ H_2O + d\ CO_2$$ leading to the matter conservation equations

$$\begin{cases}2a-d=0,\\2a-2c=0,\\2b-c-2d=0,\end{cases}$$

which is a system of three linear equations in four unknowns.

As the system is homogeneous, the unknowns are indeterminate to a constant factor, that you can choose to obtain integer values.

Here,

$$a=2,b=5,c=2,d=4.$$

  • 1
    I'd comment that the first equation is conservation of carbon, the second equation is conservation of hydrogen, and the third equation is conservation of oxygen. – Ian May 8 '17 at 14:30

This may help(for combustion reactions): $$C_xH_y + \big{(}x+\frac{y}{4} \big{)} O_2\rightarrow xCO_2+\frac{y}{2}H_2O$$

  • Do you suggest that they should learn this by heart instead of understanding how balancing a chemical reactiion works? – Did May 15 '17 at 6:32
  • No, but what I'm suggesting is that this equation is usable in cases like OP has mentioned @Did – The Dead Legend May 19 '17 at 6:25