Question

Biotransformation of an organic compound having concentration (x) can be modeled using an ordinary differential equation $\frac{dx}{dt}+k{x}^2=0$, where k is the re-action rate constant. If x = a at t = 0, the solution of the equation is

• A. x = $a{e}^{-kt}$
• B. $\frac{1}{x}=\frac{1}{a}+kt$
• C. x = a $(1-e^{-kt})$
• D. x = a + kt

Answer 1

The correct option is B $\frac{1}{x}=\frac{1}{a}+kt$

Differential equation for biotransformation of an organic compound is

$\frac{dx}{dt}+k{x}^2=0$ ...(i)

$\Rightarrow \frac{dx}{x^2}+Kdt=0$

Integrating, we get

$-\frac{1}{x}+Kt=C$ ...(ii)

Using x(0) = a $\Rightarrow C=-\frac{1}{a}$

Hence, using eq. (ii)

$-\frac{1}{x}+Kt=-\frac{1}{a}$

or $\frac{1}{x}=\frac{1}{a}+Kt$