$$t_{1/2} = \frac{\ln 2}{\lambda}$$
where:
- \(t_{1/2}\) is the half-life
- \(\lambda\) is the decay constant
The decay constant is a measure of how quickly the atoms in a radioactive sample decay. It can be calculated using the following formula:
$$\lambda = \frac{-\ln\frac{N_t}{N_0}}{t}$$
where:
- \(N_0\) is the initial number of atoms
- \(N_t\) is the number of atoms at time \(t\)
In this case, we are given that the initial number of atoms is \(3102\) and the present number of atoms is \(1020\). We can use these values to calculate the decay constant:
$$\lambda= -\frac{\ln(1020/3102)}{t}= \frac{\ln(0.33)}{t}= -\frac{1.1}{t}$$
We can then use the decay constant to calculate the half-life:
$$t_{1/2} =\frac{\ln2}{\lambda}= \frac{\ln2}{-\frac{1.1}{t}}= \frac{\ln 2}{t\times \frac{1.1}{t}}= 0.621t$$
Therefore the half life is 0.621 times the time elapsed
