Answer: it will take 1.9 × 10⁶ yrs for one mole of the radioactive material to decay so that just one atom remains
Step-by-step explanation:
Given that;
Half life t_1/2 = 24,000 years
initial amount of radio active element = 1 mole
now one mole of a substance has Avagadro number of atoms which is initially 6.023 × 10²³ which are present and finally only 1 atom is left
using the formula
t_1/2 = 0.693/k
In[ N₀/N_t] = kt
N₀ is the initial amount, N_t is the amount left after t time, t is the time, k is the rate constant.
now we substitute
t_1/2 = 0.693/k
k = 0.693/ t_1/2
k = 0.693 / 24,000 years
k = 0.000028875 yr⁻¹
so
In[ N₀/N_t] = kt
t = 1/0.000028875 yr⁻¹ In [ 6.023 × 10²³ / 1 ]
= 34632.0346 In[6.023 × 10²³]
= 34632.0346 × 54.7550
= 1896277.0545 yrs ≈ 1.9 × 10⁶ yrs
Therefore it will take 1.9 × 10⁶ yrs for one mole of the radioactive material to decay so that just one atom remains