Using Matrix (Gaussian Elimination) Iron (III) sulphate (Fe_2(SO_4)_3) reacts wi

Question

Using Matrix (Gaussian Elimination)

Iron (III) sulphate (Fe_2(SO_4)_3) reacts with potassium thiocyanate (KSCN) to form potassium iron (III) thiocyanate (K_3Fe(SCN)_6) and potassium sulfate (K2_SO_4) in accordance with the following (unbalanced) chemical equation Fe_2(SO_4)_3 + KSCN right arrow K_3Fe(SCN)_6 + K_2SO_4. Using linear algebra, balance the above reaction. In other words, you have to find x, y, z, and w such that xFe_2(SO_4)3 + yKSCN right arrow zK_3Fe(SCN)_6 wK_2SO_4. Although you can balance the equation by inspection, you must write down a system of linear equations and find all possible solutions for x, y, z, and w. Of the infinitely many solutions, select the one that yields the smallest positive integer values for x, y, z, and w.

Explanation / Answer

The reaction is,

x Fe2(SO4)3 + y KSCN ----> z K3Fe(SCN)6 + w K2SO4

Balancing Fe in LHS and RHS,

2x = z ----------- (1)

Balancing O (oxygen atom),

12x = 4w

=> 3x = w ------------- (2)

Balancing CN,

=> y = 6z ------------- (3)

Balancing 'S',

=> 3x + y = 6z + w --------- (4)

Balancing 'K',

=> y = 3z + 2w

=> 6z = 3z + 2w

=> 3z = 2w ---------- (5),

From the above equations we get,

x = z/2 = w/3 = y/12

Substituting, y = 12

we get x = 1, z = 2 and w = 3 which are the smallest positive integers.

The balanced equation is,

Fe2(SO4)3 + 12 KSCN ----> 2 K3Fe(SCN)6 + 3 K2SO4

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