Answer:
37.50g of ⁹¹Nb will remain after 2040 years.
Explanation:
The rate of decay of a radioactive isotope obeys the following formula:
[tex]Ln\frac{N}{N_0} = -Kt[/tex] (1)
Where N is moles of atoms in time t, N₀ is initial moles of atoms, K is decay constant and t is time.
300.0g of niobium are:
300.0g × (1mol / 91g) = 3.297 moles of ⁹¹Nb
It is possible to obtain decay constant from half-life, thus:
[tex]t_{1/2} = \frac{ln2}{K}[/tex]
680 years = ln 2 / K
K = 1.019x10⁻³ years⁻¹
Replacing these values in (1):
[tex]Ln\frac{N}{3.297 moles ^{91}Nb} = -1.019x10^{-3}years^{-1}*2040years[/tex]
N / 3.297 moles of ⁹¹Nb = -0.125
N = 0.4121 moles of ⁹¹Nb after 2040 years. In mass:
0.4121 moles of ⁹¹Nb × (91g / mol) = 37.50g of ⁹¹Nb will remain after 2040 years.
