The table that follows shows some measurements of the decay rate of a sample of `.^(128)I`, a radio nuclide often used medically as a tracer to measure the rate at which iodine is absorbed by the thyroid gland.
`|{:("Time(min)",A (counts//s),"Time(min)",A(counts)//s,),(4,392.2,132,10.9,),(36,161.4,164,4.56,),(68,65.5,196,1.86,),(10,26.8,218,1.00,):}|`

The half life `t_(1//2)` for this radio nuclide.

Explanation

The disintegration constant `lamda` determines the exponenetial rate at which the decay rate R decreases with time t (as indicated by Eq. 40 -17, `R=R_(0)e^(-lamda)` ). Therefore we should be able to determin `lamda` by plotting the measurements of R against the measurement times t. However, obtaining `lamda` from a plot of R versus t is difficult because R decreases exponentially with l, according to Eq. 40-17. A neat solution is to transform eq. 40 -17 into a linear function of t, so that we can easily find `lamda`. To do so we take the natural logarithms of both sides of Eq. 40 -17.
Calculations: We obtain
In `~r="In"(R_(0)e^(-lamdat))="In" R_(0)+"In"(e^(-lamdat))`
`="In" R_(0)-lamdat` (40-20)
Because eq. 40 -20 9s of the form y=b+mx with b and m constants, itis a linear equation giving the quantity in R as a function of t. Thus, we plot In R (instead of R) versus t, we should get a straight line. Further, the slope of the line should be equal to `-lamda)`.
Figure 40 -11 shows a plot of In R versus time t for the given measurements. The slope of the straight line that fits through the plottd points is
slope `=(0-6.2)/(225"min"-0)=-0.276"min"^(-1)`
Thus `-lamda-0.0276"min"^(-1)`
or `lamda=0.0276"min"~~1.7h^(-1)`
The time for the decay rate R to decrease by 1/2 is related to the disintegration constant `lamda` via Eq. 40 -19 `(T_(1//2)=("In"2)//lamda)`. From the equation. We find
`T_(1//2)=("In"2)/(lamda)=("In"2)/(0.0276"min"^(-1))~~25` min.