Decay measured in the laboratory

Problem

A detector measures the activity of a radioactive sample at regular intervals, obtaining the table below.

$t$ (hours)	$0$	$2$	$4$	$6$	$10$	$20$
$A$ (kBq)	$1200$	$850$	$600$	$425$	$213$	$37{,}5$

• (a) Verify that the data follow an exponential decay law.
• (b) Determine the half-life $T_{1/2}$ and identify the isotope between technetium-99m ($T_{1/2} \approx 6{,}0\ \text{h}$) and iodine-123 ($T_{1/2} \approx 13\ \text{h}$).

Solution

(a) Verifying the exponential trend. In an exponential decay, the ratio between activities at equal time intervals is constant. Let us calculate the ratio every $2$ hours:

$\frac{850}{1200} \approx 0{,}71;\qquad \frac{600}{850} \approx 0{,}71;\qquad \frac{425}{600} \approx 0{,}71$

The ratio is constant ($\approx 0{,}71$ every $2$ hours): this confirms the exponential law $A(t) = A_0\,e^{-\lambda t}$.

(b) Half-life. From a factor of $0{,}71$ every $2$ hours we obtain $T_{1/2}$ by imposing $A/A_0 = 0{,}71 = 2^{-2/T_{1/2}}$, i.e.:

$T_{1/2} = \frac{2\,\ln 2}{\ln(1/0{,}71)} = \frac{2\cdot 0{,}693}{0{,}342} \approx 4{,}0\ \text{h}$

Check against the last data point. At $t = 20\ \text{h}$ with $T_{1/2} = 4\ \text{h}$: $A = 1200\cdot 2^{-20/4} = 1200/32 = 37{,}5\ \text{kBq}$, in perfect agreement with the table.

Identification. Of the two options proposed, the measured value $\approx 4\ \text{h}$ is closer to technetium-99m ($6\ \text{h}$) than to iodine-123 ($13\ \text{h}$). The isotope is therefore technetium-99m.

$\ev{T_{1/2}\approx 4\ \text{h}\Rightarrow\text{technetium-99m}}$

Note on the data

The values in the table strictly give $T_{1/2}\approx 4\ \text{h}$, not $6\ \text{h}$ as stated in the textbook’s answer key. The numerical data are therefore slightly inconsistent with the nominal value for technetium-99m ($6\ \text{h}$): although technetium-99m remains the closer isotope of the two options, the half-life extracted from the data does not exactly match the tabulated value.

Collegamenti

Argomenti: Fisica nucleare Concetti: Decadimento radioattivo Competenze: Lettura dei grafici · Uso delle scale logaritmiche