The half-life for radioactive decay (a first-order process) is 24,000 years. How many years does it take for one mole of this radioactive material to decay so that just one atom remains?

it will take 1.9 × 10⁶ yrs for one mole of the radioactive material to decay so that just one atom remains

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