ajedrezlaproa6j

2022-01-02

Radium decomposes at a rate proportional to the amount present. In 100 years, 100 mg of radium decompose to 96mg. How many mg will be left after another 100 years What is the "half life" (the time required to decompose half the initial amount) of radium?

Neil Dismukes

We have $t=100yr$, the initial amount of radium $No=100mg$, the final amount of radium id $N=96mg$.
The decay equation for 100 years:
$N={N}_{0}{e}^{-\lambda t}$
$96=100{e}^{-100\lambda }$
${\left(96\right)}^{2}={\left(100{e}^{-100\lambda }\right)}^{2}$
${96}^{2}={10}^{4}{e}^{-200\lambda }$ (A)
Let $\lambda$ be the decay constant.
${N}^{\prime }={N}_{0}{e}^{-200\lambda }$
${N}^{\prime }=100{e}^{-200\lambda }$ (B)
$\frac{{96}^{2}}{{N}^{\prime }}=\frac{{10}^{4}{e}^{-200\lambda }}{10{e}^{-200\lambda }}$
${N}^{\prime }=92.16$
$N={N}_{0}{e}^{-\lambda t}$
$96=100{e}^{-100\lambda }$
$0.96={e}^{-100\lambda }$
$\mathrm{ln}\left(0.96\right)=\mathrm{ln}\left({e}^{-100\lambda }\right)$
$-100\lambda =\mathrm{ln}\left(0.96\right)$
$\lambda =0.0004$
The expression for half-life $\left({T}_{\frac{1}{2}}\right)$ is,
${T}_{\frac{1}{2}}=\frac{0.693}{\lambda }$
$=\frac{0.693}{0.0004}$
$=1732.5$ years

Tiefdruckot

, k = proportionality constant. =

$\mathrm{ln}\left(R\right)=kt+C$ (1), C is integral constant.
if $t=100$ years, amount of radium = 96mg.
$⇒\mathrm{ln}\left(100\right)=k\left(0\right)+C$
$⇒\mathrm{ln}\left(100\right)=C$
$⇒C=4.6051702$
$\mathrm{ln}\left(R\right)=kt+4.6051702$ (2)
$⇒\mathrm{ln}\left(96\right)=k\left(100\right)+4.6051702$
$⇒\frac{\mathrm{ln}\left(96\right)-4.6051702}{100}=k$
$⇒k=-0.000408219$
$\mathrm{ln}\left(R\right)=-0.000408219t+4.6051702$ (3)
a) Another 100 years
$⇒\mathrm{ln}\left(R\right)=-0.000408219\left(200\right)+4.6051702$
$⇒\mathrm{ln}\left(R\right)=4.523526196$
$⇒R+{e}^{4.523526196}$
$⇒R=92.16mg$
Amount of radium present after another 100 years is 92.16mg.
b): For half-life, $R=50mg$
$⇒\mathrm{ln}\left(50\right)=-0.000408219t+4.6051702$
$⇒t=\frac{\mathrm{ln}\left(50\right)-4.6051702}{-0.000408219}$
$⇒t=1697.97876$ years
$⇒t\approx 1698$ years

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