a) Balance the following equation (10 marks) b) Balance the following equation (
ID: 883380 • Letter: A
Question
a) Balance the following equation (10 marks) b) Balance the following equation (10 marks) C,H2Cla + Ca(OH)2 C2HCla + CaCl2+ H2O In answering a) and b) set up the corresponding systems of linear equations, find the row reduced echelon forms of the corresponding augmented matrices, and in this way find so- lutions to the linear systems. Based on these solutions find the smallest positive integer coefficients that balance the systems. Double-check that you get the same solutions as the online balancer. You should use Mathematica or Wolfram Alpha for calculating the reduced row echelon forms of the augmented matrices and simply record the results.1 For the next two parts of this problem, we won't worry about what is specifically possible in real life. Instead, we'll start with the letters A, B, C, and D (standing for four diferent mystery atoms) and we'll consider all possible ways to make up unbalanced equations. For example (Basically, just go wild with combinations of these letters and subscripts on the left of the arrow and then do the same on the right -anything goes and qualifies as a fantasy unbalanced equation c) Is it possible to give a fantasy equation that cannot be balanced? Give an example of such an equation if it is possible. Or, give a convincing argument why such an equation cannot exist. (Hint: Read again very carefully what it means to balance an equation.) (5 marks) d) Is it possible to make up a fantasy equation that can be balanced in two fundamentally different ways (that is, where the solutions are not just multiples of a smallest solution)? Give an example, of such an equation if it is possible. Or, give a convincing argument why such an equation cannot exist. (5 marks)Explanation / Answer
First, we form the chemical composition matrix
NH3
O2
N2
H2O
N
1
0
2
0
H
3
0
0
2
0
0
2
0
1
Matrix
1
0
2
0
3
0
0
2
0
2
0
1
Second, transpose.
1
3
0
0
0
2
2
0
0
0
2
1
An identity matrix is a matrix that has all ones along its diagonal, and zeroes everywhere else. In this step, an appropriately sized identity matrix to the right hand side of transposed matrix. (An appropriate size means it is the same height.)
1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
1
1
3
0
1
0
0
0
0
0
2
0
1
0
0
2
0
0
0
0
1
0
0
2
1
0
0
0
1
Fourth, matrix into row echelon
1
0
0
0
0
0
1/2
0
0
1
0
0
1/3
0
-1/6
0
0
0
1
0
0
1/2
0
0
0
0
0
0
2/3
1/2
-1/3
-1
Fifth, examine the row echelon form's left side for rows of all zero. Here, there's exactly one— the last row:
This means that the right hand part contains the coefficients that balance r equation. Putting all the positive terms on the left and the negative terms on the right,:
2/3NH3 + 1/2O2 1/3N2 + H2O
When the augmented chemical composition matrix is put in row echelon form, there are no rows of zeros. Hence, the equation can't be balanced.
c.
FeS2+HNO3Fe2(SO4)3+NO+H2SO4
the augmented chemical composition matrix is put in row echelon form, there are no rows of zeros. Hence, the equation can't be balanced.
D. No, it is not possible to create such equation. balancing will be only done by smaller multiples solutions.
NH3
O2
N2
H2O
N
1
0
2
0
H
3
0
0
2
0
0
2
0
1
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