Mathematics AI HL
Sample Internal Assessment
4/7
2,245 Words
English
Free

# Modelling the radioactive decay of the Chernobyl Incident

## Introduction -

The Chernobyl nuclear power plant accident took place on April 26th, 1986 with widespread impact affecting regions in Belarus, Ukraine and Russia. The explosion of the plant released an enormous amount of detrimental radioactive material into the atmosphere with harmful local and global contamination. The study of the radioactive decay process is essential in understanding how the radiation emitted from the accident behaves over time. This Internal Assessment aims to explore the concept of radioactive decay and model it using the data from the Chernobyl incident. For my Internal Assessment, I decided to investigate the time it will take for Chernobyl to be completely free of radioactivity in order to reduce health risks and environmental hazards, along with determining whether the area can be reused in the upcoming future. First, I will investigate the rate of radioactive decay, I will examine what the initial impact of the accident was and the area it covered, this will be denoted as R0. I have adapted to two different methodologies to investigate this decay rate, my first approach is to calculate the number of years it will take for Chernobyl to be free of radioactivity by calculating the linear or geometric regression of the Nucli through area coverage, this will indicate the reduction in the total area. My second approach consists of examining the amount of radioactive atoms in the two isotopes: cesium - 137 and strontium - 90.

## Method 1 -

To calculate the radioactive decay from the Chernobyl incident, we will use the following data -

• The Chernobyl incident released approximately 85 petabecquerels (PBq) of radioactive material into the atmosphere, of which about 50 PBq was cesium- 137 and about 5 PBq was strontium-90.
• The half-life of caesium-137 is approximately 30 years, and the half-life of strontium - 90 is approximately 28 years.
• We will assume that the rate of release of radioactive material from the Chernobyl incident was constant over time, and the rate of decay of the radioactive material is also constant over time.
• To calculate the amount of cesium-137 and strontium-90 present at any given time after the Chernobyl incident, we will use the formula for exponential decay -

N(t) = N0e −λt

where N(t) is the amount of radioactive material remaining at time t, N0 is the initial amount of radioactive material, λ is the decay constant, and t is time.

The decay constant is related to the half-life of the radioactive material by the formula -

$$\lambda=\frac{In(2)}{t_\frac{1}{2}}$$

For cesium - 137,

$$\lambda= \frac{In(2)}{30}$$

λ = 0.0231 per year

For strontim - 90

$$\lambda=\frac{In(2)}{28}$$

λ = 0.0247 per year

We can now calculate the amount of cesium - 137 and strontium - 90 present at any given time after the Chernobyl incident. We will assume that the initial amount of cesium - 137 and strontium - 90 released from the Chernobyl incident was 50 PBq and 5 PBq, respectively.

Time (years)
Cesium-137 (PBq)
Strontium-90 (PBq)
0
50.00
5.00
10
37.61
3.95
20
28.26
2.97
30
21.19
2.23
40
15.92
1.68
50
11.96
1.26
60
8.99
0.95
70
6.76
0.71
80
5.08
0.54
90
3.83
0.40
100
2.88
0.30
110
2.17
0.23
120
1.63
0.17
130
1.23
0.13
140
0.93
0.10
150
0.70
0.07
160
0.53
0.06
170
0.40
0.04
180
0.30
0.03
190
0.23
0.02
Figure 1 - Table On Shows The Amount Of Cesium - 137 And Strontium - 90 Present At Various Times After The Chernobyl Incident.
Figure 2 - Calculating The Function Of The Graph using G.D.C

f(x) in an exponential decay is represented by f(x) = aebx

The function of the Cesium - 137 regression line is -

f(x) = 49.5e − 0.028x

The function of the Strontium-90 regression line is -

f(x) = 5.27e − 0.029x

Through the use of the GDC, the value of r2 was derived, which is the representation of the variation in the dependent variable on the independent variable. The laws of exponential decay state that f(x) = aebx where f(0) = a. However, this does not hold true for the function and data presented above since 50 ≠ 49.5, hence.

The percentage error is

$$|\frac{vA-vE}{vE}|.100%$$%

$$|\frac{50-49.5}{49.5}|.100$$%

= 1.01%

The causes for the inaccuracy could vary from factors such as the values of decay was an approximate round off to three significant figures which hinder with the accuracy of results.

Analysis of the graph -

• The figure is a scatter diagram of the amount of cesium-137 and strontium 90 present over the years.
• There is an exponential decay in the presence of the two isotopes.
• The horizontal asymptote is given as y = 0 as N approaches and never reaches 0. There is no vertical asymptote as x reaches 0.
• An increase in x (independent variable) results in a decrease in the value of y, indicating decay.

## Method 2

Top IB Math Tutor: 45/45 IBDP, 7/7 Further Math, 7 Yrs Exp, Medicine Student

• ### Geometric progression

Geometric progression is a sequence of numbers in which each term is obtained by multiplying the previous term by a fixed number called the common ratio. It can be used to model the decay of a radioactive substance by assuming that the decay rate is proportional to the amount of the substance remaining.

Let us consider a radioactive substance with an initial amount N0 that decays at a rate proportional to the amount remaining. Then the amount remaining after t time units can be modeled by a geometric progression with a common ratio r, where

r = e-λt

Here, λ is the decay constant, which is a measure of the rate of decay. The decay constant is related to the half-life $$T_{\frac{1}{2}}$$ by the equation -

$$\lambda=\frac{In(2)}{T_{\frac{1}{2}}}$$

We can use the formula for a geometric progression to calculate the decay rate at any time t. The formula for the nth term of a geometric progression with first term a and common ratio r is given by -

un = arn-1

In our case, the first term a is N0 and the common ratio r is e −λt. Therefore, the amount remaining after t time units can be written as -

N(t) = N0e-λt

The rate of decay, or the rate at which the amount of substance is decreasing, is given by the derivative of N(t) with respect to t -

$$\frac{dN}{dt}=-\lambda N_0e^{-\lambda t}$$

This formula gives us the instantaneous decay rate at any time t. We can use it to calculate the decay rate at a specific time by substituting the value of t into the formula.

### Calculus

Calculus can be used to analyse the rate of change of the amount of radioactive material over time. We can use differential equations to model the radioactive decay process. The rate of change of the amount of radioactive material is proportional to the amount of radioactive material present. We can write this as -

$$\frac{dN}{dt}=-\lambda N$$

where N is the amount of radioactive material present at time t, λ is the decay constant, and $$\frac{dN} {dt}$$ is the rate of change of the amount of radioactive material with respect to time.

We can solve this differential equation using the separation of variables -

$$\frac{dN}{dt}= -\lambda dt$$

Integrating both sides, we get -

ln|N| = −λt + C

where C is the constant of integration. To determine the value of C, we use the initial condition that N = N0 at t = 0 -

ln|N0| = C

Substituting this into the previous equation, we get -

ln|N| = −λt + ln|N0|

Solving for N, we get -

N(t) = N0 × e −λt

This is the same formula we used earlier to calculate the amount of radioactive material present at any given time after the Chernobyl incident. To analyse the rate of decay using calculus, we can take the derivative of the formula for N with respect to time -

$$\frac{dN}{dt}=-\lambda ×N_0× e^{-\lambda t}$$

This is the rate of change of the amount of radioactive material with respect to time. At t = 0, this is equal to -

$$\frac{dN}{dt}=-\lambda N_0$$

This is the initial rate of decay, which is proportional to the initial amount of radioactive material present. This proves that the radioactive decay can be measured by both exponential decay and calculus.

### Calculating Half-life -

We can also use calculus to determine the half-life of the radioactive material. The half-life is the time it takes for the amount of radioactive material to decrease to half its initial value. We can find this by setting N = N0/2 in the formula for N and solving for t -

$$\frac{N_0}{2}×e^{-\lambda t_\frac{1}2}$$

Dividing both sides by N0 and taking the natural logarithm, we get -

$$In(\frac{1}2)=\lambda t\frac{1}{2}$$

Solving for $$t_\frac{1}2$$, we get -

$$t_\frac{1}2=\frac{In(2)}\lambda$$

This is the same formula we used earlier to calculate the decay constant from the half- life. Using calculus, we can also analyse the behaviour of the rate of decay over time. The rate of decay is proportional to the amount of radioactive material present, so it decreases over time as the amount of radioactive material decreases. This means that the rate of decay is not constant over time but decreases exponentially.

We can calculate the rate of change of the rate of decay by taking the derivative of the formula for $$\frac{dN}{dt}$$ with respect to time -

$$\frac{d^2N}{dt^2}=-\lambda^{2N_0}× e^{-\lambda t}$$

This is the rate of change of the rate of decay with respect to time. At t = 0, this is equal to -

$$\frac{d^2N}{dt^2}=-\lambda ^{2N_0}$$

This is the initial rate of change of the rate of decay, which is proportional to the square of the decay constant and the initial amount of radioactive material present.

In this process, we have shown how calculus can be used to model and analyse the radioactive decay process. We have used differential equations to model the rate of change of the amount of radioactive material over time, and we have shown how this formula can be used to calculate the initial rate of decay, the half-life, and the behaviour of the rate of decay over time.

Top IB Math Tutor: 45/45 IBDP, 7/7 Further Math, 7 Yrs Exp, Medicine Student

• ## Connection with the Chernobyl incident -

Exponential decay is a process in which the rate of decay of a quantity is proportional to the amount of that quantity. The half-life of a substance is the amount of time it takes for half of the substance to decay.

To calculate the exponential decay based on the half-life of a substance, we use the following formula -

$$N(t)=N_0× \frac{1^{\frac{t}{t_{\frac{1}{2}}}}}{2}$$

where N(t) is the amount of substance remaining at time t, N0is the initial amount of substance, t is the elapsed time, and $$t_\frac{1}{2}$$ is the half-life of the substance.

In this formula, $$\frac{1^{\frac{t}{t_\frac{1}{2}}}}{2}$$ is the exponential decay factor, which describes how much the amount of substance decreases over time. For example, if $$\frac{t}{t_\frac{1}{2}}=1$$, then the amount of substance will have decreased to half its initial value, because $$\frac{1^1}{2}=\frac{1}2$$.

To use this formula to calculate the amount of substance remaining at a specific time t, we simply substitute the value of t into the formula and evaluate it.

Verification -

To verify the accuracy of the formula, we can use the information from the table to calculate the half - life of the two isotopes.

$$N(50)=50×\frac{1^{25}}{2}$$

= 0.000001

This proves that the formula works as the time at which the concentration of Cesium - 137 is 50 is 0.

This can also be verified for the Strontium - 90 as well using the formula N(t) = N0 * (1/2)^(t/t_half).

$$N(5)=5× \frac{1^{2.5}}{2}$$

= 0.8

In this case the time is an approximate round of 1, hence indicating a flaw in the equation.

It is difficult to estimate how long it will take for the areas around the Chernobyl accident to be cleared of radioactive material. Radioisotopes released during a disaster have different half-lives and decay rates.

Some of the released radioactive isotopes, such as iodine - 131 and cesium - 137, have relatively short half-lives, ranging from days to decades. These isotopes decay quickly and pose no significant long - term risk.

However, other isotopes such as plutonium - 239 and strontium - 90 have much longer half-lives, ranging from decades to thousands of years. These isotopes persist in the environment for much longer and continue to pose risks to human health and the environment.

According to the United Nations Scientific Committee on the Effects of Ionizing Radiation (UNSCEAR), the exclusion zone around the Chernobyl site will remain contaminated with radioactive material for decades. However, the decay rate of radioactive material gradually decreases the level of contamination over time.

The Ukrainian government has implemented plans to gradually reduce the exclusion zone and develop it for other uses. However, the speed of this process will depend on many factors, including the rate at which radioactive material decays, the effectiveness of decontamination efforts, and people's willingness to return to the area. In short, when will the areas around the Chernobyl accident be free of radioactive material?

## Bibliography -

https://world-nuclear.org/information-library/safety-and-security/safety-of-plants/chernobyl-accident.aspx