## Apparatus¶

About 200 small cubes, marked $$\odot$$ and $$\otimes$$ on two faces; two trays; 2 x $$500\text{ml}$$ beakers; periodic table of the elements; 2 sheets graph paper.

## Introduction¶

In this model of radioactive decay, the cubes represent atoms. They are either parent (not decayed) atoms or daughter (decayed) atoms. When a cube is thrown at random, if a marked face of the cube faces up, then the atom has decayed.

## Decay Model 1¶

(uses 1 tray and 1 beaker only)

$\begin{split}\text{Reaction: } & \qquad {^{12}_{5}B} \qquad \longrightarrow \qquad {^{A}_{Z}X + \beta^-} \\ &\widehat{\text{in the tray}} \qquad \qquad \quad \widehat{\text{discarded}}\end{split}$
1. Count the cubes. This is $$N_0$$, the number of parent atoms of $${^{12}_{5}B}$$ at time $$t=0$$.

2. Place all the cubes in a beaker and empty the beaker into a tray. Shake the tray until all the cubes lie flat (do not touch any cubes).

3. Each time you empty a beaker into a tray, 0.01s has elapsed. Record the time $$t=0.01$$s. Discard cubes showing $$\odot$$ or $$\otimes$$ (these are atoms of $${^{A}_{Z}X}$$ , the daughter atoms). Count and record $$N$$, the number of cubes left in the tray.

4. Place the cubes now in the tray into the beaker. Empty the beaker into the tray and shake as before. Record $$t=0.02$$s. Discard decayed atoms. Record the new number $$N$$ of cubes left in the tray.

5. Continue for $$t=0.03$$, $$0.04$$, $$0.05$$, ... $$2.5$$s, or until $$N=0$$.

## Analysis¶

1. Plot a graph of $$N$$ vs. $$t$$. From the graph find the half life $$T_{\frac{1}{2}}$$. The decay rate (lamda $$\lambda$$) is related to the half life as follows: $$\lambda = \ln{\frac{2}{T_{\frac{1}{2}}}}$$
2. Using the formula: $$\frac{dN}{dt} = -\lambda N$$, calculate the decay rate when $$t=0$$. Find the graph’s gradient at time $$t=0$$; is this the same (approximately) as the calculated value?
3. On the same sheet of graph paper, plot another curve showing the number of daughter atoms.
4. Find $$A$$, $$Z$$, and $$X$$. Is this atom stable?

## Decay Model 2¶

(uses 2 trays and 2 beakers)

In this experiment, tray #1 contains $${^{227}_{90}Th}$$ atoms and tray #2 contains $${^{A1}_{Z1}X}$$ daughter atoms. These daughter atoms decay again and are discarded.

$\begin{split}\text{Reactions: } \qquad {^{227}_{90}Th} \qquad & \longrightarrow \qquad {^{A1}_{Z1}X }+ \alpha \qquad \text{Half life } T_{\frac{1}{2}}\\ {^{A1}_{Z1}X } \qquad & \longrightarrow \qquad {^{A2}_{Z2}Y } + \alpha \qquad \text{Half life 11.7 days}\end{split}$
1. Place all the cubes into tray #1, count them, and record number $$N_0$$ at time $$t=0$$. Record for tray #2 that $$N_0 = 0$$ at $$t = 0$$.

2. Place tray#1 cubes into beaker #1, return to tray #1 and shake tray to settle the cubes. Move cubes showing $$\otimes$$ into tray #2. Record $$N$$ for tray #1 and tray #2 at this time $$t=5$$ days.

3. FIRST: Place cubes from tray #2 into beaker #2. Return to tray #2 and shake. Discard cubes showing $$\odot$$.

SECOND: Place cubes from tray #1 into beaker #1. Return to tray #1 and shake. Move cubes showing $$\otimes$$ to tray #2.

THEN: Count and record $$N$$ for trays #1 and #2 at $$t=10$$ days.

4. Continue repeating step 3, letting $$t = 15, 20, 25, ... \text{up to } 200$$ days. (Each time you perform step 3, $$t$$ advances by $$5$$ days).

## Analysis¶

1. On the same piece of graph paper plot $$N$$ vs. $$t$$ for trays #1 and #2 to obtain two curves.
2. Using the #1 curve, find $$T_{\frac{1}{2}}$$ for $${^{227}_{90}Th}$$. Calculate $$\lambda$$ and thus find $$N$$ at $$t = 40$$ days (use $$N = N_0 e^{- \lambda t}$$). Check that the value of $$N$$ at $$t = 40$$ days is about the same by using the graph, and note this value.
3. Explain carefully why the curve #2 has the shape that it does.
4. Use the reaction equations given above to determine $$A1$$, $$Z1$$, $$X$$ and also $$A2$$, $$Z2$$, and $$Y$$.

## Questions¶

1. $${^{A2}_{Z2}Y }$$ is unstable and decays. There follows a whole series of decays, ending with a stable atom, as follows:

$\begin{split}{^{A2}_{Z2}Y} & \longrightarrow \text{?}+ \alpha & \quad T_{\frac{1}{2}}&= 3.92 \text{s}\\ \text{?} & \longrightarrow \text{?}+ \alpha & \quad T_{\frac{1}{2}} &= 1.8 \times 10^{-3} \text{s}\\ \text{?} & \longrightarrow \text{?}+ \beta^{-} & \quad T_{\frac{1}{2}}&= 36.1 \text{min}\\ \text{?} & \longrightarrow \text{?}+ \alpha & \quad T_{\frac{1}{2}}&= 2.15 \text{min}\\ \text{?} & \longrightarrow \text{?}+ \beta^{-} & \quad T_{\frac{1}{2}}&= 4.8 \text{min}\end{split}$

Write down the above set of reactions, deducing each of the ?s, giving atomic mass, atomic number, and symbol in each case.

2. A sample of $$\ {^{227}_{90}Th},$$ when left for 30 days, is found to contain a lot of $${^{227}_{90}Th},$$ $${^{A1}_{Z1}X},$$ and the final stable isotope. There is very little of $${^{A2}_{Z2}Y}$$ and the four intermediate isotopes. Why?

3. Draw a decay chain to map the complete series of seven decays from $${^{227}_{90}Th}$$ to the stable isotope.