The velocity of a chemical reaction is given by (dx/dt) = k(a-x) , where x is the amount transfered in time t, k is a constant and a is the concentration at time t=0 when x=0.

Question

Solve the equation and determine x in terms of t.

Answer 1

Solution:
The equation can be solved using separation of variables.

Integrating both sides of the equation, we get:

∫(dx/dt)dt = ∫k(a-x)dt

On the left side, we have the integral of the velocity, which is the displacement x. On the right side, we have the integral of the rate of reaction, which is the amount of reaction.

Therefore,

x = kt(a-x) + C

where C is the constant of integration.

Rearranging the equation, we get:

x = (ka/k+1) + (C/k+1)e-kt

Therefore, x in terms of t is given by:

x = (ka/k+1) + (C/k+1)e-kt

Answer 2

Is -e^-kt = e^kt ?

Answer 3

the bot's logic is, as usual opaque or just wrong.
x = ce^(-kt) + a