42.5.2 Nuclear Decay and Half-Lives

Rutherford discovered experimentally that for all types of radiation the number of radioactive atoms in a sample decreases exponentially with time. The reason is that radioactive decay—in which an unstable nucleus emits a form of radiation—is a random process. That is, we can predict only the probability that a nucleus will decay, not the exact moment. We encountered exactly this situation when we investigated the lifetimes of excited states of atoms in Section 41.7.

As we did with atoms, let $r$ be the decay rate, the probability of decay within the next second by the emission of an alpha or beta particle or a gamma-ray photon. Then the probability of decay within a small time interval $δ t$ is

Prob(decay in time interval δ t) = r δ t

(42.7)

This equation was also the starting point in our analysis of the spontaneous emission of photons by atoms. The mathematical analysis is exactly the same as in « SECTION 41.7, to which you should refer, leading to the exponential-decay equation

N = N_{0} e^{- t / τ}

(42.8)

where $τ = 1 / r$ is the lifetime of the nucleus.

FIGURE 42.16 shows the decrease of $N$ with time. The number of radioactive nuclei decreases from $N_{0}$ at $t = 0$ to $e^{- 1} N_{0} = 0.368 N_{0}$ at time $t = τ$ . In practical terms, the number decreases by roughly two-thirds during one lifetime.

The figure illustrates a plot of radioactive decay.

FIGURE 42.16 Full Alternative Text

NOTE An important aspect of exponential decay is that you can choose any instant you wish to be $t = 0$ . The number of radioactive nuclei present at that instant is $N_{0}$ . If at one instant you have 10,000 radioactive nuclei whose lifetime is $τ = 10 min,$ you’ll have roughly 3680 nuclei 10 min later. The fact that you may have had more than 10,000 nuclei earlier isn’t relevant.

In practice, it’s much easier to measure the time at which half of a sample has decayed than the time at which 36.8% has decayed. Let’s define the half-life $t_{1 / 2}$ as the time interval in which half of a sample of radioactive atoms decays. The half-life is shown in Figure 42.16.

The half-life is easily related to the lifetime $τ$ because we know, by definition, that $N = \frac{1}{2} N_{0}$ at $t = t_{1 / 2}$ . Thus, according to Equation 42.8,

\frac{N_{0}}{2} = N_{0} e^{- t_{1 / 2} / τ}

(42.9)

The $N_{0}$ cancels, and we can then take the natural logarithm of both sides to find

ln (\frac{1}{2}) = - ln 2 = - \frac{t_{1 / 2}}{τ}

(42.10)

With one final rearrangement we have

t_{1 / 2} = τ ln 2 = 0.693 τ

(42.11)

Equation 42.8 can be written in terms of the half-life as

N = N_{0} {(\frac{1}{2})}^{t / t_{1 / 2}}

(42.12)

Thus $N = N_{0} / 2$ at $t = t_{1 / 2}, N = N_{0} / 4$ at $t = 2 t_{1 / 2}, N = N_{0} / 8$ at $t = 3 t_{1 / 2}$ , and so on. No matter how many nuclei there are, the number decays by half during the next half-life.

NOTE Half the nuclei decay during one half-life, but don’t fall into the trap of thinking that all will have decayed after two half-lives.

FIGURE 42.17 shows the half-life graphically. This figure also conveys two other important ideas:

Nuclei don’t vanish when they decay. The decayed nuclei have merely become some other kind of nuclei, called the daughter nuclei.
The decay process is random. We can predict that half the nuclei will decay in one half-life, but we can’t predict which ones.

The figure illustrates the half-life graphically.

FIGURE 42.17 Full Alternative Text

Each radioactive isotope, such as $^{14} C$ , has its own half-life. That half-life doesn’t change with time as a sample decays. If you’ve flipped a coin 10 times and, against all odds, seen 10 heads, you may feel that a tail is overdue. Nonetheless, the probability that the next flip will be a head is still 50%. After 10 half-lives have gone by, ${(1 / 2)}^{10} = 1 / 1024$ of a radioactive sample is still there. There was nothing special or distinctive about these nuclei, and, despite their longevity, each remaining nucleus has exactly a 50% chance of decay during the next half-life.

EXAMPLE 42.2 The decay of iodine

The iodine isotope $^{131} I$ , which has an eight-day half-life, is used in nuclear medicine. A sample of $^{131} I$ containing $2.00 \times 10^{12}$ atoms is created in a nuclear reactor.

How many $^{131} I$ atoms remain 36 hours later when the sample is delivered to a hospital?
The sample is constantly getting weaker, but it remains usable as long as the number of $^{131} I$ atoms exceeds $5.0 \times 10^{11}$ . What is the maximum delay before the sample is no longer usable?

MODEL The number of $^{131} I$ atoms decays exponentially.

SOLVE

The half-life is $t_{1 / 2} = 8 days = 192 h$ . After 36 h have elapsed,

$N = (2.00 \times 10^{12}) {(\frac{1}{2})}^{(36 h) / (192 h)} = 1.76 \times 10^{12} nuclei$
The time after creation at which $5.0 \times 10^{11}^{131} I$ atoms remain is given by

$5.0 \times 10^{11} = 0.50 \times 10^{12} = (2.0 \times 10^{12}) {(\frac{1}{2})}^{t / 8 days}$

To solve for $t$ , we first write this as

\frac{0.50}{2.00} = 0.25 = {(\frac{1}{2})}^{t / 8 days}

Now we take the logarithm of both sides. Either natural logarithms or base-10 logarithms can be used, but we’ll use natural logarithms:

ln (0.25) = - 1.39 = \frac{t}{t_{1 / 2}} ln (0.5) = - 0.693 \frac{t}{t_{1 / 2}}

Solving for $t$ gives

t = 2.00 t_{1 / 2} = 16 days

REVIEW The weakest usable sample is one-quarter of the initial sample. You saw in Figure 42.17 that a radioactive sample decays to one-quarter of its initial number in 2 half-lives.

42.5.2 Nuclear Decay and Half-Lives

FIGURE 42.16 The number of radioactive atoms decreases exponentially with time.

FIGURE 42.17 Half the nuclei decay during each half-life.

EXAMPLE 42.2 The decay of iodine