Question

The relation between  \(λ\) and \(T_\frac 12\)is: 
\((T_\frac 12 = half\  life, \ λ→decay\  constant)\)

• A. \(T_{\frac 12}=\frac {ln\ 2}{λ}\)
• B. \(T_{\frac12} ln\ 2=λ\)
• C. \(T_{\frac12}=\frac 1λ\)
• D. \((λ+T_{\frac 12})=\frac {ln\ 2}{2}\)

Answer 1

The Correct Option is A

The decay of atoms over time can be described using the formula \(N_t = N_0 . e^{(-λt)}\), 
where
\(N_t\) represents the number of atoms at time \(t\) 
\(N_0\) is the initial number of atoms, 
\(λ\) is the decay constant.
For a specific case where \(t\) equals \(T_\frac12\):
\(2N_o = N_0 . e^{(-λ .T_\frac12)}\)
Simplifying this expression, we find:
\(2 = e^{(-λ . T_\frac  12)}\)
Taking the natural logarithm (ln) of both sides, we get:
\(ln\ (2) = -λ . T_\frac 12\)
Now, solving for \(T_\frac12\)
\(T_\frac 12 = \frac {ln\ (2)}{λ}\)
This equation establishes the relationship between the decay constant \((λ)\) and the half-life \((T_\frac 12)\).