Question

$1g$ atom of $R{a}^{226}$ is placed in an evacuated tube of volume $5litre$. Assuming that each ${}_{88}R{a}^{226}$ nucleus is an $\alpha -emitter$ and all the contents are present in tube, calculate the total pressure of gases and partial pressure of $He$ collected in tube at ${27}^{\circ }C$ after the end of $800year$. $t_{1/2}$ of $Ra$ is $1600$ year. Neglect volume occupied by undecayed $Ra$.

Answer 1

Given :
Initial amount of radioactive species, $N_0=\left(1/226\right)$g- atom; decay period $t=1600$ year;

Volume, $V=5$: litre; Temperature, $T=300$ K; half life period, $t_{1/2}=1600$ year

Decay constant, $\lambda =\frac{0.693}{t_{1/2}}=0.693/1600$

The decay equation is as shown.

${}_{88}^{226}\text{Ra}{⟶}_{80}^{210}\text{X}+4{}_2^4\text{He}$

The rate law expression is $t=\left(\frac{2.303}{\lambda }\right)\log \left(\frac{N_o}{N}\right)$

$t=\frac{2.303\times 1600}{0.693}\log \frac{1}{226\times N}$

$\therefore N=2.21\times {10}^{-3}$ g -atom

Amount of Ra decayed$=\left(1/226\right)-2.21\times {10}^{-3}$
$=2.215\times {10}^{-3}$ g-atom

Amount of He formed $=4\times 2.215\times {10}^{-3}$ g-atom

The ideal gas equation is $PV=nRT$

The total pressure is $P=\frac{nRT}{V}=\frac{4\times 2.215\times {10}^{-3}\times 0.0821\times 300}{5}=4.36\times {10}^{-2}atm$

$\left[x\right]=\left[4.36\right]=4$
Ans - $4$