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Degree: Chemical Engineering with Industrial experience (Other) - Manchester University
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Balancing equations is usually fairly simple. However some of them involve several steps.
You may have come across balanced equations in data booklets that look intimidating. The balanced equation for reduction of Mn7+ to Mn2+ is one such equation.
Initially one might write:
Mn7+ + 5e- --> Mn2+
Although technically balanced (since the ox state of Mn in MnO4- is +7), this equation does not represent the full reaction that takes place which involves H2O molecules and H+ ions.
The fully balanced equation is:
MnO4- + 8H+ + 5e- --> Mn2+ + 4H2O
To balance this, the following steps must be followed:
Step 1: Write only what's given.
MnO4- --> Mn2+
Step 2: Balance all atoms except for H and O.
MnO4- --> Mn2+ (Mn atoms are already balanced; one on each side).
Step 3: Balance Oxygen atome by adding H2O to the side where more oxygen atoms are needed. At this stage, we have 4 O atoms on the left hand side and need 4 on the RHS. One water molecule contains one O atom, so we need 4 water molecules.
Therefore, MnO4- --> Mn2+ + 4H2O
Notice that now we have 4 O atoms on each side but 8 H atoms on the RHS.
Step 4: Balance H atoms by adding the required number of H+ ions to the side that is short of H atoms. At this stage we have (4 x 2) 8 H atoms on the RHS and none on the LHS.
Therefore, MnO4- + 8H+ --> Mn2+ + 4H2O
Step 5: Balance the charges by adding an electron, e-. At this stage, the LHS has a (-1 +8) +7 charge. The RHS has a +2 charge. If we add 5e- to the LHS, the charge becomes 7 - 5 = +2
Therefore, MnO4- + 8H+ + 5e- --> Mn2+ + 4H2O
By the end of step 5, we have obtained the fully balanced equation.
You can balance any equation using these steps, however, there is a slight adjustment that has to be made to step 4 sometimes.
In acidic solutions, to balance H atoms you just add H+ to the side lacking H atoms but in a basic solution, there is a negligible amount of H+ present. Instead, OH- is abundant. In this case, you add H2O to the side lacking H atom(s) and a OH- to the opposite side. The net effect is that you end up adding 1 H atom to the side that lacks a H atom. If a side lacks 'n' number of H atoms, add 'n' number of H2O molecules to that side and 'n' number of OH- ions to the opposite side.see more
Before solving this question, lets look at a second order polynomial.
Let the polynomial x^2 +5*x +4 be f(x)
Using factorisation, we can write f(x) as (x+1)(x+4)
If you were to expand (x+1)(x+4) you'd get x^2 +5*x +4.
To solve f(x) = 0, we can write (x+1)(x+4) = 0 and then set each one of the factors to zero.
x+1 = 0 --> x = - 1 and x+4 = 0 --> x = - 4
Check that: f(-1) = 0 and that f(-4) = 0
We can conclude that,
- setting the factors of a polynomial f(x) to zero gives the roots of the equation f(x) = 0
- plugging the roots (values of x) back in the polynomial expression will lead to f(x) equalling zero.
Coming back to the question, we know that the factor of polynomial p(x) is (x+2).
Setting x+ 2 equal to zero gives x = - 2
This is a one of the three roots/solutions of the equation p(x) = 0 and plugging it into the polynomial expression should give zero,
i.e, p(-2) = 0
p(-2) = -2^3 - 2*(-2) + a = 0
-8 + 4 +a = 0
-4 + a = 0
a = 4
- setting factor of p(x) to zero gives root of the equation p(x)=0
-Plugging root into p(x) expression will satisfy p(x)=0