Question

Linear substitution application

Answer 1

I need help balancing a chemical equation (now before you send me off to the chemistry section), the way to solve it involves math:

I have to balance this equation:
Cu + HNO3 = Cu(NO3)2 + NO + H2O

I find that the answer is
3Cu + 8HNO3 = 3Cu(NO3)2 + 2NO + 4H2O

But I have no idea how to get there

I found a wesite talking about using a linear substittuion, basically rewriting the equation like this

a*Cu + b*HNO3 = c*Cu(NO3)2 + d*NO + e*H2O

Since the same number of atoms exist on both sides, it becomes:

Cu: a=c
H: b=2e
N: b=2c+d
O: 3b=6c+d+e

Answer 2

@nincompoop @phi @TheSmartOne Any help please?

Answer 3

what is the question?

Answer 4

are you short an equation? b/c there are 5 unknowns

Answer 5

How to solve for each unknown

Answer 6

@peachpi I was thinking the same thing.

Answer 7

Here is where I found the technique http://www.chemicalaid.com/articles.php/view/1/how-to-balance-chemical-equations

Answer 8

@TheЅmartOne

Answer 9

need the matrix method to solve it???

Answer 10

I don't have 5 equations though

Answer 11

the example cleverly had enough eqautions to solve the problem.

Answer 12

I arrange it as \(x_1(Cu) +x_2(HNO_3)\rightarrow x_3(Cu(NO_3)_2)+x_4(NO)+x_5(H_2O)\)
Left hand side Righ hand side
Cu \(x_1\) = \(x_3\)
Hydrogen \(x_2\) = \(2x_5\)
Nitron \(2x_2\) = \(2x_3+x_4\)
Oxygen \(3x_2\) = \(6x_3+x_4+x_5\)

Answer 13

Cu: a=c
H: b=2e
N: b=2c+d
O: 3b=6c+d+e

Answer 14

a*Cu + b*HNO3 = c*Cu(NO3)2 + d*NO + e*H2O

Answer 15

Where do we go from there

Answer 16

It gives you:
\(x_1-x_3=0\\x_2-2x_5=0\\2x_2-2x_3-x_4=0\\3x_2-6x_3-x_4-x_5=0\)
hence the matrix is
\[\left[\begin{matrix}1&0&-1&0&0&0\\0&1&0&0&-2&0\\0&2&-2&-1&0&0\\0&3&-6&-1&-1&0\end{matrix}\right]\]

Answer 17

When I do rref, I get -.25 -2 -.25 and -3.5

Answer 18

Take row reduce echelon form, you have
\(x_1=(1/4)x_5\\x_2=2x_5\\x_3=(1/4)x_5\\x_4=(7/2)x_5\)
hence if \(x_5= 4\)
then
\(x_1= 1\\x_2=8\\x_3=1\\s_4=14\)

Answer 19

Where did you ger that that x_1=1/4x_5

Answer 20

Now, replace all
\(1(Cu) +8(HNO_3)\rightarrow 1(Cu(NO_3)_2)+14(NO)+4(H_2O)\)

Answer 21

That's not correct. It's 3Cu + 8HNO3 = 3Cu(NO3)2 + 2NO + 4H2O

Answer 22

from the matrix, take row reduce echelon form, you have the first row
1 0 0 0 -1/4 0
that is x1 -1/4 x5 =0

Answer 23

oh ok

Answer 24

I may make mistake at somewhere, but the technique is that. Do it carefully and you can get the answer

Answer 25

Thank you very much

Answer 26

ok

Answer 27

It' in the matrix, you put for Nitrogen, 2x instead of just x

Answer 28

Yes!!!!!!!!!!! it is. My bad. I am sorry

Answer 29

then the rref is \(x_1=(3/4)x_5\\x_2= 2x_5\\x_3=(3/4)x_5\\x_4=(1/2)x_5\)
if \(x_5=4\) then
\(x_1=3\\x_2=8\\x_3=3\\x_4=2\)
and the equation is the correct one as you had.

Answer 30

Thanks for the question!! I have an opportunity to recall my knowledge. :)