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
