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

Answer: (A)

$T_{\frac 12}=\frac {ln\ 2}{λ}$

Answer 2

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 $(T_\frac 12)$.