"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
