## Wednesday, March 13, 2013

### Simple Chemical Conversion

Category: Chemical Engineering Math, Differential Equations

"Published in Newark, California, USA"

For a substance C, the time rate of conversion is proportional to the square of the amount x of unconverted substance. Let k be the numerical value of the constant of proportionality and let the amount of unconverted substance be x0 at time t = 0. Determine x for all t ≥ 0.

Solution:

The given word problem is another good application of first order, first linear equation where a simple chemical conversion of a substance is involved. Let's analyze the given problem as follows

for                           C ────────────> Product

initial                       x0                                     0
at time t                 x0 - x                                  x

From the word problem, "the time rate of conversion is proportional to the square of the amount x of unconverted substance", the working equation will be

The above equation is negative because the value of x is decreasing as time is increasing. Using the method of separation of variables, we have

Integrate on both sides of the equation

To solve for the value of C, substitute x = x0 and t = 0 to the above equation, we have

Therefore, the amount of x at t ≥ 0 will be