Ch 2.2 Exponential Decay and Radioactivity

• Radioactive decay provides setting to apply modeling steps.
  • Background, Problem Statement, Assumptions, Compartment Diagram, Word Equation, Variables/Parameters, Initial Value Problem, Analytical and Numerical Solutions, Discussion of Results
• Henri Becquerel's photographic plate (left), fogged by exposure to uranium salt (@1900, Wikipedia).
• Radio dating methods used to determine age of artifacts.

What did the finger say to the thumb?

I'm in glove with you.

• Historians, geologists, archaeologists, palaeontologists and many others use dating procedures to establish theories.
• Unstable elements emit \( \alpha \)-particles, \( \beta \)-particles, or photons while decaying into isotopes of other elements.
• Decay modes are sensitive to chemical and environmental effects that change electronic structure of atom. https://en.wikipedia.org/wiki/Radioactive_decay

• The disintegration of one nucleus is a random event, so for small numbers of nuclei we could use probability functions to model the amount of material as a function of time.
• For large numbers of nuclei we can assume that a proportion of nuclei will decay in any time interval.
• Thus we model process as continuous with fixed decay rate.

Develop a general IVP-based model for the amount of radioactive material as a function of time.

• Amount of element present is large enough so that we are justified in ignoring random fluctuations.
• Process is continuous in time.
• Rate of decay for an element is fixed.
• No increase in mass of the body of material.
• Rate of change of mass of decayed material proportional to mass of nuclei.

• We consider the process in terms of a compartment model.
• For radioactive decay, there is an output term over time, but no input term.

From the Balance Law, we know that

\[ \small{ \begin{Bmatrix} \mathrm{net \, rate \, of \, change} \\ \mathrm{of \, a \, substance} \end{Bmatrix} = \begin{Bmatrix} \mathrm{rate~in} \end{Bmatrix} - \begin{Bmatrix} \mathrm{rate~out} \end{Bmatrix} } \]

Word equation:

\[ \small{ \begin{Bmatrix} \mathrm{rate \, of \, change \, of} \\ \mathrm{mass \, of \,radioactive } \\ \mathrm{material \, at \, time \,} t \end{Bmatrix} = - \begin{Bmatrix} \mathrm{rate \, of \, change \, of} \\ \mathrm{mass \, of \,decayed } \\ \mathrm{material \, at \, time \,} t \end{Bmatrix} = - k \begin{Bmatrix} \mathrm{mass \, of \,radioactive } \\ \mathrm{material \, at \, time \,} t \end{Bmatrix} } \]

• Let \( N(t)= \) mass (grams) of radioactive nuclei at \( t \) years.
• Let \( t_0= \) specified time of interest.
• Let \( N(t_0)=n_0= \) mass of material at time \( t_0 \).
• Let \( k= \) decay rate constant, in units of 1/years.

Using the assumptions, word equation and specification of variables and parameters, our IVP is:

\[ \small{\frac{dN}{dt} = - kN, \,\,\, N(t_0) = n_0 } \]

Solve IVP using separation of variables:

\[ \small{ \begin{aligned} \frac{dN}{dt} &= - kN, \,\,\, N(t_0) = n_0 \\ \int \frac{dN}{N} &= - k \int dt \\ \ln N & = -kt + C \\ N(t) &= e^{-kt+C} = A e^{-kt}, \,\, A = e^C\\ N(t_0) &= Ae^{-kt_0} = n_0 \,\, \Rightarrow \,\, A = n_0 e^{kt_0}\\ N(t) & = \left(n_0 e^{kt_0}\right) e^{-kt} \\ N(t) &= n_0 e^{-k(t-t_0)} \end{aligned} } \]

Note that when \( \small{ t_0 = 0, \, N(t) = n_0 e^{-kt} } \).

rk4plot <- function(f,x0,y0,h,n){
#f is the slope formula from the differential equation
#x0 is the initial value for x
#y0 is the initial value for y
#h is the step size
#n is the number of steps to take (number of iterations)

#Initialize vectors
x <- rep(0,n+1)  #x is vector of n+1 zeros
y <- rep(0,n+1)  #y is vector of n+1 zeros
x[1] <- x0
y[1] <- y0

#RK4 Loop
for(i in 1:n) {
  a <- h*f(x[i], y[i])
  b <- h*f(x[i] + h/2, y[i] + a/2)
  c <- h*f(x[i] + h/2, y[i] + b/2)
  d <- h*f(x[i] + h, y[i] + c)
  y[i+1] <- y[i] + 1/6*(a + 2*b + 2*c + d)
  x[i+1] <- x[i] + h
}

#Plot numerical solution
return(plot(x,y,type="l", col="blue"))

• Half-life \( \tau \) is the time required for half of the radioactive nuclei to decay.
• The half-life \( \tau \) is more commonly known than the value of the rate constant \( k \) for radioactive elements.
• The half-life \( \tau \) can be used to determine \( k \).

• Recall analytical solution \( \small{N(t) = n_0 e^{-k(t-t_0)} } \)

• Let \( t = \tau \) denote the half-life. Then

\[ \small{ \begin{align*} N(t+\tau) & = 0.5N(t) \\ n_0 e^{-k(t+\tau-t_0)} &= 0.5 n_0 e^{-k(t-t_0)} \\ n_0e^{-kt}e^{-k\tau}e^{kt_0} &= 0.5 n_0 e^{-kt}e^{kt_0 } \\ e^{-k\tau} &= 0.5 \\ -k\tau & = \ln\left(2^{-1}\right) = -\ln(2) \\ k & = \frac{\ln(2)}{\tau} \end{align*}} \]

• Note that \( k \) and \( \tau \) are independent of \( n_0 \) and \( t_0 \).

• The Lascaux Cave paintings are believed to be prehistoric.
• Using Geiger counter, a decay rate of \( ^{14} \) C in charcoal fragments from cave measured 1.69 disintegrations per minute per gram of carbon.
• In comparison, for living tissue in 1950, the measurement was 13.5 disintegrations per minute per gram of carbon.
• How long ago was the radioactive carbon formed?

• Let \( N(t)= \) amount (grams) of \( ^{14} \) C per gram in charcoal at time \( t \) years.
• Let \( t_0=0 \) denote the current time.
• Let \( N(t_0)=n_0= \) mass of material at time \( t_0 \).
• Let \( T<0 \) denote the time at which charcoal was formed.
  
• Let \( N(T)= \) amount of \( ^{14} \) C in charcoal at time \( t=T \); i.e., when the charcoal was formed.
• Let \( k= \) decay rate constant, in units of 1/years.

Assume our previous IVP model applies here:

\[ \small{ \frac{dN}{dt} = - kN, \,\,\, N(0) = n_0, \,\,\, t > T } \]

The solution to the IVP for \( \small{^{14}} \) C is

\[ \small{ N(t) = n_0 e^{-kt}, \,\,\, k \cong 0.0001245 } \]

From \( \small{N(t) = n_0 e^{-kt}} \), we substitute \( \small{ t } \) = \( \small{T} \) and use algebra to solve for \( \small{T} \) as follows:

\[ \small{ \begin{aligned} N(T) & = n_0 e^{-kT} \\ e^{-kT} &= \frac{N(T)}{n_0} \\ -kT &= \ln\left(\frac{N(T)}{n_0} \right) \\ T &= -\frac{1}{k} \ln\left(\frac{N(T)}{n_0} \right) \end{aligned} } \]

From \( \small{N(t) = n_0 e^{-kt}} \), we have

\[ \small{N'(t) = -k n_0 e^{-kt} = -k N(t)} \]

From previous slide,

\[ \small{ T = -\frac{1}{k} \ln\left(\frac{N(T)}{n_0} \right) } \]

The values of \( \small{N(T)} \) and \( n_0 \) are not known. However,

\[ \small{ \frac{N'(T)}{N'(0)} = \frac{- k N(T)}{-k N(0)} = \frac{N(T)}{n_0} } \]

Recall that a decay rate of \( ^{14} \) C in charcoal fragments from cave measured 1.69, and for living tissue it was 13.5. Then

\[ \small{ \frac{13.5}{1.69} = \frac{N'(T)}{N'(0)} = \frac{N(T)}{n_0} } \]

Thus

\[ \small{ T = -\frac{1}{k} \ln\left(\frac{N(T)}{n_0} \right) = -\frac{1}{k} \ln\left(\frac{13.5}{1.69}\right) \cong -16,692 } \]

-1/0.0001245*log(13.5/1.69)

[1] -16690.45

• From previous slide, we have

\[ \small{ \frac{13.5}{1.69} = \frac{N(T)}{n_0} \,\, \Rightarrow n_0 \cong 0.125 N(T)} \]

1.69/13.5

[1] 0.1251852

• Thus amount \( n_0 \) today is 12.5% of original amount \( N(T) \).
• This is where most homework problems start, rather than providing Geiger counter measurements.
  
• We will also start this way for an upcoming class activity.

• From Wikipedia and History.com, Lascaux Cave Paintings date back to 17,000 years ago, which matches our result.
• The accuracy of the process depends on knowing the ratio of \( \,^{14} \) C to \( \,^{12} \) C in atmosphere; see textbook and weblink http://www.webexhibits.org/pigments/intro/dating.html
• This has changed over the years, following a basic sinusoidal variation with an 8,000 year period (see textook).

• Volcanic eruptions and industrial smoke emit only \( ^{12} \) C from materials that are older than 100,000 years.
• Nuclear testing increases ratio of \( \,^{14} \) C to \( \,^{12} \) C in atmosphere.
• These considerations are now factored into dating process.
  

• Radioactive decay provides setting to apply modeling steps.
  • Background, Problem Statement, Assumptions,
  • Compartment Diagram, Word Equation,
  • Variables/Parameters, Initial Value Problem,
  • Analytical and Numerical Solutions, Discussion of Results

• Radiocarbon dating used to determine age of artifacts.
• For recent past, Lead-210 (22 year half-life) is used, while Uranium-238 (billions of years half-life), is used to date Earth. https://en.wikipedia.org/wiki/Lead%E2%80%93lead_dating