Question

The luminous dials of watches are usually made by mixing a zinc sulphide phosphor with an $\alpha -particle$ emitter. The mass of radium (mass number $226$, half-life $1620$ years) that is needed to produce an average of $10$ $\alpha -particles$ per second for this purpose is

• A. $2.77mg$
• B. $2.77g$
• C. $2.77\times {10}^{23}g$
• D. $2.77\times {10}^{-10}g$

Answer 1

The correct option is D $2.77\times {10}^{-10}g$

Rate of decay $=10$
$\frac{dN}{dt}=10$

First Order decay:
$\frac{dN}{dt}=\lambda N$
$\lambda N=10$
$\lambda =\frac{ln2}{t_{1/2}}$
$N=\frac{10}{\lambda }=\frac{10\times (1620\times (365\times 24\times 3600)}{ln2}=7.3\times {10}^{11}$

$Moles=N/N_A=1.223\times {10}^{-12}moles$
$Mass=N\times molarmass=\times 226=2.76\times {10}^{-10}gms$