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In modern times the gradually increasing population has increased the demand for resources globally. One of these resources includes energy (i.e. electricity), in recent history, the primary source of energy was the use of unsustainable sources such as fossil fuels and nuclear power plants. Nuclear power plants use the heat produced during nuclear fission to heat water and produce steam which is then spun using large turbines to generate electricity* (Nuclear power plants - U.S. Energy Information Administration (EIA). (2022).).* However, as a by-product of the nuclear fission process radioactive substances are released into the surrounding environment. Under normal conditions radioactive substances decay naturally however during decay they release alpha and beta particles as well as gamma rays

In the case of Chernobyl, Ukraine in April 1986 the nuclear power plant had a flawed reactor which resulted in a steam explosion and fires, this released at minimum 5% of the radioactive reactor core (radioactive substances) into the surrounding environment * (Chernobyl | Chernobyl Accident | Chernobyl Disaster - World Nuclear Association . (2022).).* Whilst studying biology and chemistry I learned more about the effect of radioactive particles on the environment as well as humans exposed to an increased abundance of such a substance in an uncontrolled environment. One of the manifestations of such exposure in humans includes the development of cancer (mutated malignant cells). Because this disease is incurable when it's developed I am interested in researching about preventative measures that can be taken to stop the development of the disease. I choose to focus my research using math ai which can guide my calculation of decay rates of substances found in Chernobyl and model this data to predict to a degree of accuracy when it will be safe to live in Chernobyl, guided by the research question -

The half-life function is used to determine the time required for a substance to reduce to half of its initial value * (Half-Life Formula - What is Half-Life Formula? Examples. (2022).)* In radioactive modeling, it is used along with a function to ascertain how quickly a radioactive isotope undergoes radioactive decay and the length of time to which a radioactive isotope stays stable. It is generally displayed in the form of an exponential graph as the function follows the exponential decrease in the percentage of radioactive substances remaining against the number of half-lives (age of sample).

Radioactive decay occurs when an unstable radioactive substance spontaneously releases radiation in the form of energy (wavelengths) in order to achieve a more stable state ** (Radioactive Decay | US EPA. (2015).) **The energy emitted from radioactive substances during decay occurs in the form of a beta particles, gamma rays and alpha particles. In particular, alpha particles are extremely harmful to humans and surrounding living organisms as the materials can be inhaled, swallowed, or absorbed into the body

In the case of Chernobyl for which there are various radioactive isotopes initially released, I will calculate the half-life and model the functions of the 3 radioactive isotopes most concentrated in the affected area as well as the most harmful to the surrounding environment and to living organisms, these isotopes are Strontium-90, caesium-137, Krypton-85 (intext). Using the half-life formula to create data points of 5-year intervals between the release of radioactive substances into surrounding environments.

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

Using this formula where N(t) is the quantity remaining, N0 is the initial quantity of the substance, t is the time elapsed and\( t\frac{1}{2}\) is the half-life of the substance. I will vary the year I am calculating the radioactive substance remaining using 5-year intervals between the years of the desired data point. I will also use estimations made at the time of the initial quantity of the radioactive substance released in Chernobyl in 1986, along with pre-existing data on the average half-life calculated for the radioactive isotopes I am investigating.

Using the publication made by the international atomic energy agency (IAEA) produced in 2006 on the environmental consequences of the Chernobyl accident as well as the remediation made between 1986 and 2006. The publication also presented the initial quantity data following the Chernobyl accident that measured the initial substance quantities of the radioactive isotopes leaked into the environment following the explosion.* ( et al., 2006)*

Dr. Adam Nazha

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

Radioactive isotope

Quantity released (PBq)

Quantity released (Bq)

Physical half-life

Strontium- 90

115

1.15×10^{7}

28 years

Caesium- 137

85

8.5×10^{6}

30 years

Krypton-85

33

3.3×10^{6}

10.76 years

The use of this data which was conducted and recorded by an independent intergovernmental organization can be used to provide the initial quantities of these radioactive substances measured following the Chernobyl accident which can be used in the radioactive decay formula in order to estimate the quantity of radioactive substances at various times.

Below are samples of my calculations and use of the half-life formula to calculate the quantities remaining of the radioactive substances indicated. These sample calculations display the

methodology used to calculate the quantity of radioactive substances 5 years after it was released in the environment. I used the same methodology to calculate the quantity released for the indicated years below.

- Strontium -90

\(N(t) = 115(\frac{1}{2})^{\frac{5}{28}}\)

N(t) = 101. 61 PBq ~ 2 decimal places

- Caesium -137

\(N(t) = 85(\frac{1}{2})^{\frac{5}{30}}\)

N(t) = 75. 73 PBq ~ 2 decimal places

- Krypton - 85

\(N(t) = 33(\frac{1}{2})^{\frac{5}{10.76}}\)

N(t) = 23. 93 PBq ~ 2 decimal places

Using these data points I can create a scatter plot and then further determine if the function is linear or exponential and create a function accordingly

When creating this scatter plot diagram I was able to see the general negative trend that all radioactive isotopes over a period of time display. When using an analytical graphing tool I was able to manipulate the trendline to to follow a linear function or exponential function. When doing this I was able to derive that the exponential function was better suited to these data points as the trendline cut across all the data points and displayed the general correlation, while using a linear trendline some data points were not bisected by the trendline. Therefore to ensure that the equation derived is more accurate I choose to use an exponential trendline and derive exponential functions using the analytical graphing tool as the exponential trendline bisects the most data points and is therefore more accurate. The exponential function derived for Strontium-90 is** y = 115 ^{e-0.025x}** , the exponential function derived for Caesium- 137 is

Dr. Adam Nazha

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

Using the above equations generated from graphing I can graph the exponential functions to assimilate the point of intersection where the Y value (quantity of radioactive substance) becomes zero.

Firstly, modeling strontium-90 (displayed by the red curve), caesium-137 (displayed by the orange curve) and krypton-85 (displayed by the purple curve) in the graph I was unable to derive a y- intercept as each exponential function produced an asymptote line where the curve and the line approach zero but never intersect. Using the asymptote line I was able to estimate that the time at which the substances will no longer be found in the surrounding environment is; 202.42 years for strontium-90, 201.21 years for caesium-137 and 70.45 years for krypton-85.

However, the asymptote line has a larger margin of error, making this estimation less accurate. I began researching other functions that can model the properties of exponential functions, that may be used to derive a y- intercept producing more reliable data. When studying numbers and algebra as well as functions I learnt that logarithmic functions can be used to solve exponential equations as they are able to explore the properties of an exponential function and when graphed produce X and Y intercepts* (Logarithmic functions, 2023)*. Because of this I decided to model the logarithmic function of the data I collected above and derive the logarithmic equation using the same methodology used to attain the exponential equations for the three radioactive substances, which is by using a graphing tool and modeling the raw data calculated and creating a function that conforms to this data, using this graphing calculator I was able to derive a logarithmic equation. When graphed the equation would have an intersection point at the X- axis which is more reliable than the asymptote line, validating the approximated number of years it would take for the radioactive substances to be less than or equal to 0.

Dr. Adam Nazha

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

The point of intersection with the X axis is 202.75, this displays that in 202.75 years the abundance of strontium-90 in the surrounding environment will be less than or equal to 0. Correspondingly I was able to get the other number of years that it will take the other radioactive substances to extinct from the environment and I recorded these in the table below.

Subtsance

Equation

Number of years (x intercept)

caesium-137

y = -20.55ln(x) +112.64

240.15

krypton-85

y = -10.64ln(x) +41.25

48.25

Synthesizing the findings above, the more volatile radioactive elements have a longer half-life despite being small in the quantity of initial emission as compared to other elements measured such as iodine-131 which was estimated to be 1370 PBq released with a half-life of 8.04 days, therefore in long term environmental and human effects, it was not a high contributor. While in the cases of strontium-90 and caesium-137 the radioactive half-life is very large for a radioactive isotope ranging between 28 and 30 years. This results in the time at which the last radioactive substances stop emitting energy in the surrounding environment being within the range of 202.75 and 240.15 respectively. Further despite krypton-85 having a radioactive half-life of 10.76 years and the amount of time for the radioactive isotope to stop emitting energy being 48.29 years, when krypton-85 decays it forms the naturally occurring isotope rubidium-85 which has the radioactive half-life of 48.8 x 109 years which emits beta particles with a relatively lower PBqSynthesizing the findings above, the more volatile radioactive elements have a longer half-life despite being small in the quantity of initial emission as compared to other elements measured such as iodine-131 which was estimated to be 1370 PBq released with a half-life of 8.04 days, therefore in long term environmental and human effects, it was not a high contributor. While in the cases of strontium-90 and caesium-137 the radioactive half-life is very large for a radioactive isotope ranging between 28 and 30 years. This results in the time at which the last radioactive substances stop emitting energy in the surrounding environment being within the range of 202.75 and 240.15 respectively. Further despite krypton-85 having a radioactive half-life of 10.76 years and the amount of time for the radioactive isotope to stop emitting energy being 48.29 years, when krypton-85 decays it forms the naturally occurring isotope rubidium-85 which has the radioactive half-life of 48.8 x 109 years which emits beta particles with a relatively lower PBqSynthesizing the findings above, the more volatile radioactive elements have a longer half-life despite being small in the quantity of initial emission as compared to other elements measured such as iodine-131 which was estimated to be 1370 PBq released with a half-life of 8.04 days, therefore in long term environmental and human effects, it was not a high contributor. While in the cases of strontium-90 and caesium-137 the radioactive half-life is very large for a radioactive isotope ranging between 28 and 30 years. This results in the time at which the last radioactive substances stop emitting energy in the surrounding environment being within the range of 202.75 and 240.15 respectively. Further despite krypton-85 having a radioactive half-life of 10.76 years and the amount of time for the radioactive isotope to stop emitting energy being 48.29 years, when krypton-85 decays it forms the naturally occurring isotope rubidium-85 which has the radioactive half-life of 48.8 x 109 years which emits beta particles with a relatively lower PBq (becquerel)* (Kendall, C. (n.d.). Isotope tracers -- resources (2022).) *Therefore within my lifetime, it will not be entirely safe for humans and other living organisms to inhabit Chernobyl because these radioactive isotopes continue to emit alpha and beta particles in the form of wavelength lengths that permeate the genetic makeup of living organisms to cause mutations.

To date, the effect of radioactive substance exposure on the general population of Chernobyl includes the immediate effects in 1986 on workers and people within the immediate area of exposure suffered from radiation poisoning (acute radioactive syndrome and radioactive sickness)* (Radiation sickness - Symptoms and causes. (2022). )*. The long-term effects on health which are still being felt today include thyroid cancer in reportedly over 20,000 cases as of 2015 in both adults and juvenescence. 5000 of these cases were a result of environmental contamination in water sources as well as trace contamination in animals that have consumed grass and other plant organisms. There has also been an increase in birth complications such as stillbirths and detrimental pregnancy outcomes which has negative implications on children's health

To conclude, using the half-life radioactive isotope formula to create both an exponential function as well as a logarithmic function I have been able to approximate at which time themost dangerous radioactive isotopes released in abundance to the surrounding environments would stop emitting energy in the form of alpha and beta particles. Despite the exponential functions which are commonly used to model the pattern of radioactive decay being unable to assimilate A y-intercept which has a lower margin of error than an asymptote line, I was able to develop my methodology and I widened my scope of research to discern which other type of function would be able to explore the properties of a exponential function, which lead me to derive and model logarithmic functions for each radioactive isotope explored and derive A y-intercept value which more accurately provides a time in years where there will be zero radioactive isotopes in the surrounding environment. Therefore, it can be assimilated that the effect on human and living organisms as well as environmental impacts would not fully dissipate within my lifetime and would take over 200 years (from the time of release in 1986) to fully dissipate enough the repercussions of being exposed to radioactive particles to be discontinued. When handling the adverse effects of radioactive exposure this area of knowledge needs to be fully explored with continuous radioactive testing. Through modeling I have been able to predict with a degree of accuracy the time at which it will be fully safe for Chernobyl to be inhabited. Further, using this method mathematicians can model the decay of radioactive isotopes released due to nuclear accidents or bombs in the case of Hiroshima to discern when the affected environment will no longer be contaminated by these radioactive isotopes making it safe for inhabitants.

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Dr. Adam Nazha

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

Dr. Adam Nazha

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