Question

Solve using matlab and show code:

A reversible chemical reaction 2A + B C can be characterized by the equilibrium relationship K = c_c/c_a^2 c_b where the nomenclature ci represents the concentration of constituent i. Suppose that we define a variable x as representing the number of moles of C that are produced. Conservation of mass can be used to reformulate the equilibrium relationship as K = (c_c, 0 + x)/(c_a, 0 - 2x)^2 (c_b, 0 - x) where the subscript 0 designates the initial concentration of each constituent. If K = 0.016, ca, 0 = 42, cb, 0 = 28, and cc, 0 = 4, determine the value of x. Obtain the solution graphically. On the basis of (a), solve for the root with initial guesses of xl = 0 and xu = 20 to epsilon s = 0.5%. Choose either bisection or false position to obtain your solution. Justify your choice.

Answer 1

a)

Substitute all values

K = (Cc+x) / (Ca-2x)^2(Cb-x)

0.016 = (4+x)/ (42-2x)^2(28-x)

Provide an equation

0.016 (42^2 - 2*42x + 4x^2)*(28-x) = 4+x

((0.016)*(1764) - 84*0.016 x + 0.064x^2)(28-x) = 4+x

28*0.016*1764 - 84*0.016*28x +0.064*28x^2 - 0.016*1764x +  84*0.016x^2 - 0.064x^3 = 4+x

790.272 - 37.63x +1.792x^2 - 28.224x + 1.344x^2 - 0.064x^3 - 4 -x = 0

-0.064x^3 +(1.792+1.344)x^2 + x(-37.63-28.224) + (790.272-4) = 0

-0.064x^3 +3.136x^2 - 65.854x +786.272 = 0

multiply all by -

0.064x^3 - 3.136x^2 +65.854x - 786.272 = 0

solve graphically for "0"

x = 28.58 (approx)

2-666.596 3 -615.206 4 -568.936 5-527.402 6 -490.22 7 -457.006 8 -427.376 9 -400.946 10377.332 11 -356.15 12 -337.016 13 -319.546 14-303.356 15-288.062 16 -273.28 17-258.626 18-243.716 19 -228.166 20-211.592 21-193.61 22 -173.836 23 151.886 24 -127.376 25-99.922 69.14 27-34.646 27.5 -15.887 3.944 28.001 3.984768 200 2б

b)

Applying bisection rule...

you will find that 27.5 is the actual vlaue of the method of iteration