For a radioactive sample containing ^ 128 I , let R_0 be the...

According to the law of radioactive decay, the count rate $R$ at time $t$ is given by:

$R = R_0 e^{-\lambda t}$

where $R_0$ is the initial count rate and $\lambda$ is the decay constant.

Rearranging the equation, we get:

$\dfrac{R_0}{R} = e^{\lambda t}$

Taking the natural logarithm on both sides:

$\ln\left(\dfrac{R_0}{R}\right) = \lambda t$

Comparing this with the equation of a straight line $y = mx + c$, we have $y = \ln(R_0/R)$, $x = t$, $m = \lambda$, and $c = 0$.

Thus, the plot of $\ln(R_0/R)$ against $t$ is a straight line passing through the origin.