Question

Experiments show that the concentration of substance X in a first order chemical reaction at future time (in seconds) t i.e. [X] is described by the following differential equation:

d[X]/ dt= .001[X],
find:
(a) Find an expression for the concentration [X] after t seconds if the initial concentration is W by solving the differential equation.

Answer 1

dx/dt = .001x
dx/x = .001 dt
lnx = .001t+c
x(t) = ce^.001t
x(0) = W, so
x(t) = We^.001t

Answer 2

d[X]/dt = (0.001)*[X]
=> d[X]/[X] = (0.001)*dt

Integrating both sides,

∫(1/[X])*d[X] = (0.001)*∫dt
=> log[X] = 0.001*t + C

Now, the question says that when t = 0, X = W

=> log[W] = 0 + C = C

Putting this value of C in the original equation,

log[X] = 0.001*t + log[W]
=> log([X]) - log([W]) = 0.001*t
=> log([X]/[W]) = 0.001*t
=> [X]/[W] = e^(0.001*t)

=> [X] = [W]*e^(0.001*t)