Exponential Law of Growth and Decay Module 11 PDF

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Differential Equations: Exponential Law of Growth and Decay...

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Cagayan State University–Carig Campus COLLEGE OF ENGIN ENGINEERING EERING DEPARTMENT OF CIVIL ENGINEERING

LECTURE MODULE 11

DIFFERENTIAL EQUATIONS: CHAPTER 3 TOPIC 11 EXPONENTIAL LAW OF GROWTH AND DECAY

In Chapter 2, we have studied all the different solutions in solving a differential equation in the first order and first degree, including linear differential equations and Bernoulli’s equations. These solutions are unique in their own forms but provide same answers to a particular equation as long as applicable. Having mastered these solutions, we now proceed to the physical applications of first-order first-degree differential equations. The entire Chapter 3 will focus on the understanding of differential equations to different physical applications by applying the solutions studied on the previous chapter.

At the end of this lesson, the student will be able to: 1. understand the concept of exponential law of growth and decay; 2. analyze problems related to growth and decay by modeling differential equations.

1.1 Radioactive Decay and Bacterial Growth Radioactive decays and bacterial growths are the usual problems involved in the study of exponential law of growth and decay. It is very useful in the field of science like Chemistry, Nuclear Physics and Nuclear Medicine in analyzing half-life and propagation. The exponential law of growth and decay states that the rate of change of the number of bacteria or radioactive present at any time 𝑡 is directly proportional to the number of bacteria present at any time 𝑡. That is, 𝑑𝑃 ∝𝑃 𝑑𝑡 𝑑𝑃 = 𝑘𝑃 𝑑𝑡 𝑑𝑃 = ∫ 𝑘 𝑑𝑡 ∫ 𝑃 ln|𝑃| = 𝑘𝑡 + 𝐶 𝑃 = 𝑒 𝑘𝑡+𝐶 𝑃 = 𝑒 𝑘𝑡 𝑒 𝐶 𝑃 = 𝐶𝑒 𝑘𝑡

The number of bacteria or radioactive present at any time 𝑡 is directly proportional to the number of bacteria present at any time 𝑡 where 𝑷 is the population or number present at time 𝒕. Introduce constant of proportionality 𝒌.

Solve the equation through separation of variables.

𝑒 𝐶 is a pure constant equal to 𝐶.

Assume that the initial number or population is 𝑃0 . That is at initial time 𝑡 = 0, 𝑃 = 𝑃0 . Let us substitute the condition from our solution. 𝑃 = 𝐶𝑒 𝑘𝑡 𝑃0 = 𝐶𝑒 𝑘(0) 𝑃0 = 𝐶(1) 𝐶 = 𝑃0 𝑷 = 𝑷𝟎 𝒆𝒌𝒕

where 𝑃 is the population or amount present at any time 𝑡.

The solution above represents the formula for exponential growth and decay. While some may choose to obtain the equation and find for the solution, it would save us time to use derived formulas.

In addition, some books are using +𝑘 for growth problems (increasing 𝑃) and −𝑘 for decay problems (decreasing 𝑃). The lecturer opted to present general solution for either case to avoid error and miscalculation. Let us try some examples! ENGR

ARISTON C

TALOSIG

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Cagayan State University–Carig Campus COLLEGE OF ENGIN ENGINEERING EERING DEPARTMENT OF CIVIL ENGINEERING

DIFFERENTIAL EQUATIONS: CHAPTER 3

LECTURE MODULE 11

Example 1: Radioactive Decay Radium decomposes at a rate proportional to the amount present at any time. If of 100 mg set aside now, there will be left 96 mg 100 years hence, find a. how much will be left after 75 years. b. how much will be left after 2.58 centuries. c. the half-life of radium. Analysis and Solution: From our problem, the initial amount of radium is 100 mg which means that 𝑃0 = 100 mg. It is evident that the material decomposes or decays since its amount becomes 96 mg after 100 years. That is, at 𝑡 = 100, 𝑃0 = 96 mg.

Apply initial condition to define 𝑒 𝑘 . Solve for 𝑒 𝑘 so that the function becomes a pure function of time 𝑡.

Substitute 𝑒 𝑘 to the working equation.

a. How much will be left after 75 years? (𝑡 = 75 years)

b. How much will be left after 2.58 centuries? Our units must be consistent in solving any type of problem. Since we have used years as the unit of our time 𝑡 in deriving the solution of this problem, we must use this unit throughout. Let us convert centuries to years by noting that 1 century = 100 years. Therefore, 𝑡 = 2.58 centuries = 258 years.

c. Find the half-life of radium. Half-life is a property of radioactive nuclides which measures their degree of radioactivity. Half-life is defined as the time it takes for a radioactive material to decay half of its original value. For example, Carbon-14 (C-14) has a half life of 5700 years. This means that if initially there are 100 grams of C-14, then its amount after 5700 years will be

half of its initial amount or 50 grams. ENGR ARISTON C TALOSIG

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