The decay constant of a radioactive substance is λ. The half life and mean life of substance are respectively given by

CONCEPT:

• The half-life of a radioactive element (T1/2): The time interval in which the mass of a radioactive substance or the number of atoms reduced to half of its initial value.
• The expression for the half-life is

\({T_{\frac{1}{2}}} =\frac{ln2}{\lambda}= \frac{{0.693}}{\lambda }\)
Where λ = is the decay rate constant

• Mean life: The average lifetime of all the nuclei of a particular unstable atomic species is called mean life.

\(τ=\frac{1}{λ}\)

where τ is mean life, and λ is radioactive decay constant

EXPLANATION:

• The correct relation between the half-life (T) of a radionuclide with its mean life (τ) is expressed as 

⇒ T_1/2 = τ ln2

\(\Rightarrow T_{1/2}=\frac{log_e2}{\lambda}\)               \([\because\, \, τ=\frac{1}{λ}]\)

• Therefore option 2 is correct.