Question

Calculate the Uranium X1 decay constant from its halflife. How long would it take to decompose 30 percent of the original sample? koi onn hai i am bored​

Answer 1

$\huge \underline \mathfrak \blue {Given:}$

$\sf• \: Uranium \: sample.$

$\sf• \: Percentage \: decomposition \: = 30 \% \:$

$\huge \underline \mathfrak \pink {To \: Find:}$

$\implies\begin{gathered} \sf{The \: decay \: constant \: and \: time \: to \: decompose \: 30 \% \: of \: original \: sample.} \end{gathered}$

$\huge \underline \mathfrak \orange {Calculation:}$

⇝ We know that the half life of Uranium, T1/2 = 4.5 billion years = 4.5 × 10⁹ years

⇝ As radioactive decay is first-order reaction, we have:

k = 0.693/T1/2

⇒ k = 0.693/4.5 × 10⁹

⇒ k = 0.154 × 10⁻⁹ yr⁻¹

⇝ After 30% of decomposition, sample left = 70% of original sample = 0.7[A]₀

⇝ Time taken = (2.303/k) log [A]₀/[A]

⇒ T = (2.303/0.154 × 10⁻⁹) log [A]₀/0.7[A]₀

⇒ T = (14.955 × 10⁹) log 1/0.7

⇒ T = (14.955 × 10⁹) (log 10 - log 7)

⇒ T = (14.955 × 10⁹) (1 - 0.845)

⇒ T = 14.955 × 10⁹ × 0.155

⇒ T = 2.318 × 10⁹ years = 2.318 billion years

『 So, the decay constant for Uranium is k = 0.154 × 10⁻⁹ yr⁻¹ and the it would take 2.318 billion years to decompose 30% of the original sample.』

Answer 2

Answer:

we have decay equation:

$N = N_{0}e^{-\lambda t}\\or\\\frac{N}{N_{0}} = e^{-\lambda t}$

let half-life time is $T_{h}$

therefore,

$0.5 = e^{-\lambda T_{h}}\\\\-\lambda T_{h} = ln(0.5)\\\\\lambda T_{h} = -ln(0.5) \ ...[1]$

let t be the time for 30% decomposition

therfore,

$0.3 = e^{-\lambda t}}\\\\-\lambda t} = ln(0.3)\\\\\lambda t = -ln(0.3) \ ...[2]$

eqn 2 divided by eqn 1

$\frac{\lambda t}{\lambda T_{h}} = \frac{-ln(0.3)}{-ln(0.5)}\\\\t = \frac{ln(0.3)}{ln(0.5)}\ T_{h}$

Final Relation:

$t = \frac{ln(0.3)}{ln(0.5)}\ T_{h}$