__Category__: Chemical Engineering Math, Differential Equations, Integral Calculus"Published in Newark, California, USA"

Radium decomposes at a rate proportional to the quantity of radium present. Suppose that it is found that in 25 years approximately 1.1% of a certain quantity of radium has decomposed. Determine approximately how long it will take for one-half the original amount of radium to decompose.

__Solution__:

The given word problem is about the decomposition of a substance in a certain period of time. This is exactly the opposite of exponential growth as we discussed in the previous topics. If the statement says "Radium decomposes at a rate proportional to the quantity of radium present.", then the working equation will be

or

The sign for constant of proportionality is negative since it is a decomposition process or an exponential decay. If it is an exponential growth, then the sign is positive.

Let

^{dx}/

_{dt}= be the rate of decomposition of a radium

x = be the amount of radium at time t

x

_{0}= be the initial amount of radium at time t = 0

Consider the above equation

Solve the above equation using the Separation of Variables, as follows

Integrate both sides of the equation, we have

Take the inverse natural logarithm on both sides of the equation

To solve for the value of C, we need the following condition: If x = x

_{0}at t = 0, then the above equation becomes

Substitute the value of C to the above equation, we have

Next, we need to solve for the value of k which is the constant of proportionality. If the next statement says "Suppose that it is found that in 25 years approximately 1.1% of a certain quantity of radium has decomposed.", then the following condition will be as follows

Let

x = (1 - 0.011)x

_{0}= 0.989x

_{0}

t = 25 years

Substitute the values of x and t to the above equation, we have

Take the natural logarithm on both sides of the equation

The final working equation will be

If x = 0.5x

_{0}, then radium will decompose in

Take natural logarithm on both sides of the equation