Ch 2.2 Exponential Decay and Radioactivity

• Historians, geologists, archaeologists, palaeontologists and many others use dating procedures to establish theories within their disciplines.
• Unstable elements emit \( \alpha \)-particles, \( \beta \)-particles, or photons while decaying into isotopes of other elements.

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

• 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.

• 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.

• Start with word equation:

\[ \small{ \begin{Bmatrix} \mathrm{amount \, of } \\ \mathrm{radioactive \, material } \\ \mathrm{at \, time \,} t + \Delta t \end{Bmatrix} = \begin{Bmatrix} \mathrm{amount \, of } \\ \mathrm{radioactive \, material } \\ \mathrm{at \,time \,} t \end{Bmatrix} - \begin{Bmatrix} \mathrm{amount \, of } \\ \mathrm{radioactive \, material } \\ \mathrm{ decayed \, in \, time \,} \Delta t \end{Bmatrix} } \]

• Subract middle term to left side, and for small \( \Delta t \), we obtain

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

• Let \( N(t) \) = mass (grams) of radioactive nuclei at time \( t \).
• Rate of change of mass of decayed material is proportional to mass of nuclei.
• Proportionality constant \( k \) determined by experiment.

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

• Corresponding initial value problem (IVP):

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

• General IVP has the form:

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

• Solve IVP using separation of variables:

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

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

rk4plot <- function(f,x0,y0,h,n){
  x <- rep(0,n)
  y <- rep(0,n)
  x[1] <- x0
  y[1] <- y0
  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
  }
  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 general 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 \( \small{N(t)} \) = amount of \( \small{~^{14}} \) C per gram in the charcoal at time \( t \).
• Let \( t_0 = 0 \) denote the current time.
• Let \( \small{T} < 0 \) denote the time at which carbon was formed.
  
• 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

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

• Thus

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

-1/0.0001245*log(1.69/13.5)

[1] 16690.45

• The accuracy of the process depends on having prior knowledge of the exact ratio of \( ~^{14} \) C to the total C in the atmosphere, but this has changed over the years.
• There is a basic sinusoidal variation with an 8,000 year period.
• Volcanic eruptions and industrial smoke emit only \( ~^{12} \) C, from materials that are older than 100,000 years.
• Nuclear testing has resulted in a 100% increase in ratio in certain places, and are now factored into dating process.
• For recent past, a lead isotope with half-life of 22 years is used for dating, while Uranium-238, with half-life of billions of years, is used to date the Earth.