Time to decay : 2040 years
Further explanation
The atomic nucleus can experience decay into 2 particles or more due to the instability of its atomic nucleus.
Usually radioactive elements have an unstable atomic nucleus.
The main particles are emitted by radioactive elements ,so that they generally decay, are alpha (α), beta (β) and gamma (γ) particles
General formulas used in decay:
[tex]\tt Nt=No.\dfrac{1}{2}^{\dfrac{T}{t\frac{1}{2} }[/tex]
T = duration of decay
t 1/2 = half-life
N₀ = the number of initial radioactive atoms
Nt = the number of radioactive atoms left after decaying during T time
moles zirconium :
[tex]\tt \dfrac{15.4}{91}=0.169[/tex]
moles niobium :
[tex]\tt \dfrac{2.2}{91}=0.0242[/tex]
the half-life of niobium-91 is 680 years
Reaction
⁹¹₄₁Nb ⇒⁹¹₄₀Zr + ₁⁰e
Amount of Nb from reaction : 0.169
Amount of initial Nb = 0.0242 + 0.169 = 0.1932
Time to decay (T) :
[tex]\tt 0.0242=0.1932.\dfrac{1}{2}^{\dfrac{T}{680}}\\\\0.1252=\dfrac{1}{2}^{\dfrac{T}{680}}\\\\\dfrac{1}{2}^{3}=\dfrac{1}{2}^{\dfrac{T}{680}}\\\\T=680\times 3=2040[/tex]
