Question

Polonium-209, an α emitter, has a half-life of 102 years. How many α particles are emitted in 1.8 s from a 1.4 ng sample of 209Po?

Answer 1

Sol:-

Given Half life = 102 years = 102 x 31556926 = 3.219 x 10⁹ s

Decay constant = k = ln 2 / half life

Decay constant = k = ln 2 / 3.219 x 10⁹ = 2.1534 x 10^-10 s^-1

Mass of Po remains = M = M₀ e^-kt

Mass of Po decayed = M₀ - M = M₀ (1- e^-kt )

Mass of Po decayed = 1.4ng x (1 - e (-2.1534 x 10-10 s-1 x 1.8 s) )

Mass of Po decayed = 1.4 ng ( 1 - e ^{(-3.876 x 10-10}))

Mass of Po decayed = 1.4 ng ( 1- 0.99999999961240)

Mass of Po decayed = 1.4 ng ( 3.876 x 10^-10)

Mass of Po decayed = 5.4264 x 10^-10 ng = 5.4264 x 10^-19 ng

So

Moles of alpha particles = given mass / gram molar mass

Moles of alpha particles = 5.4264 x 10^-19 g / 209 g/mol = 2.5964 x 10^-21 mol

and Number of alpha particles = 2.5964 x 10^-21 mol x 6.022 x 10²³ = 15.63 x 10² = 1563 particles