Chemistry:Growth and decay problem

This article discusses an important problem from application of view. The document does not go into much depth but it has enough information to make you familiar with these problem.The result of this problem is used in many fields like nuclear chemistry, biology,etc. in one or another form. In our daily life, often we come across problems related with growth and decay like decay of organic matter, growth of plants,etc.But some growth and decay phenomenon can be described in terms of differential equations like decay of radioactive element like Uranium, growth of certain bacteria,etc. How this can be done, is described here.

Introduction

Growth and decay problem" is divided into two categories. Growth problem and decay problem. The data and requirement can be different but the general form is as follows: "The mass of a substance is M at a certain time and it changes in proportion to its mass at that moment. What will be its mass after t time?" At first consideration, it appears that if the substance is decaying then its mass will be definitely zero after certain time, but this will not happen. Now we are going to prove it using calculus and to also derive the general solution for this problem using differential equations.

Solution using differential equations

Let m be the mass of the substance at any moment. We know that the rate of change of mass of substance at any moment is proportional to mass of substance at that moment. Thus, we can write

dm/dt⋉m
dm/dt = km
where k is the constant of proportionality

Thus, we can write

∫ dm/m=∫kdt

Thus lnm = kt + lnc where lnc is the constant of proportionality

m = ce^(kt) . . . (1)

Now mass of substance initially(t=0) was M.Substituting this information in equation (1) and solving,we get c=M.Thus

m = Me^(kt).

Now by observing the solution, we can say that value of m is never going to be zero because the exponential function e^kt is never zero for any finite value of k or t.Thus the substance is never going to vanish. This is because in decaying the mass of substance decreases with time so does the rate of decay so even after a very long time, small amount of mass exists.

Applications

At first sight this problem may seem very trivial but its result is used in many fields like nuclear chemistry, medical field,etc. In nuclear chemistry many radioactive elements follow the above-mentioned differential equation. Hence, to know how much amount of substance will be left after particular amount of time, this result becomes very helpful. Also ¹⁴C is present in every living and non-living things and it is radioactive.Hence these result can be applied on it and by measuring its amount and rate of decay we can actually predict the age of fossils and the cosmic bodies that strike the earth. In biology, growth of some bacteria depend on this equation so their number can be calculated by this equation.This information can be helpful in development of drugs in medical field.

Moreover, in Chemistry some reaction follows this equation so the amount of product and reactant can be calculated by just measuring the rate of reaction and information about its feasibility and equilibrium can be obtained.

Conclusion

So, at the end we can conclude that even if a substance is decaying, it does not decay completely and even after a long time, some mass of it remains. Because of this, we are able to predict the ages of fossils and astronomical bodies coming to earth.

References

Differential Equations With Historical Notes and Applications by George F. Simmons