Polonium-209, an alpha emitter, has a half-life of 102 years. How many alpha particles are emitted in 1.0 s from a 1.0 ngsample of 209Po?

Answer: number of alpha particles emitted = 620.80798 particles

Explanation:

Given from the question that the element Polonium with a molecular weight of 208.98243 ≈ 209 g/mol has a half life of 102 years.

i. e. Half life = 102 years = 102 * 365 days * 24 hrs * 60 mins *

60 sec = 3216672000 = 3.217 * 10⁹ sec

also given that Avogadro's number = 6.022 x 1023 atoms.

where half life t ½ = in2 / λ

where radioactive decay constant λ = in2 / t ½

λ = in2 / (3.217 * 10⁹) = 2.1546 * 10⁻¹⁰ s⁻¹

the mass of P₀ decayed = M₀ - M ... (1)

where M = mass of P₀ remaining, given as M = M₀ (e∧-λt)

and M₀ = mass of sample = 1 ng

time is 1 sec

from equation (1), we input the expression of M

∴ mass of P₀ decayed = M₀ - M = M₀ (1 - e∧-λt)

mass of P₀ decayed = 1.0 ng * (1-e∧ (-2.1546 * 10⁻¹⁰s⁻¹ * 1 sec))

mass of P₀ decayed = 1.0 ng * (1 - e∧ (-2.1546 * 10⁻¹⁰))

mass of P₀ decayed = 1.0 ng * (1 - 0.9999999998)

mass of P₀ decayed = 1.0 ng * 2.1546 * 10⁻¹⁰ = 2.1546 * 10⁻¹⁰ ng

= 2.1546 * 10⁻¹⁹g

from basic knowledge on solving for no of moles of a substance;

No of moles = mass of a substance / molar mass of the substance

∴ Moles of alpha particles = mass given / molar mass in grams

Moles of alpha particles = 2.1546 * 10⁻¹⁹g / 209 g/mol

Moles of alpha particles = 1.0309 * 10⁻²¹ moles

Which gives the no of alpha particle emitted = 1.0309 * 10⁻²¹ moles * 6.022 x 10²³ = 620.80798 particles