Simple Chemical Conversion

From the results of chemical experimentation of substance converted into another substance, it was found that the rate of change of unconverted substance is proportional to the amount of unconverted substance.
 

If x is the amount of unconverted substance, then

$\dfrac{dx}{dt} = -kx$

with a condition that x = xo when t = 0.
 

$\dfrac{dx}{dt} = -kx$

$\dfrac{dx}{x} = -k \, dt$

$\ln x = -kt + \ln C$

$\ln x = \ln e^{-kt} + \ln C$

$\ln x = \ln Ce^{-kt}$

$x = Ce^{-kt}$
 

From the initial condition of x = xo when t = 0
$C = x_o$
 

Thus,

$x = x_oe^{-kt}$

 

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