Answer:
8 half-lives of polonium-210 occur in 1104 days.
0.1174 g of polonium-210 will remain in the sample after 1104 days.
Step-by-step explanation:
Initial mass of the polonium-210 = 30 g
Half life of the sample, = [tex]t_{\frac{1}{2}}=138 days[/tex]
Formula used :
[tex]N=N_o\times e^{-\lambda t}\\\\\lambda =\frac{0.693}{t_{\frac{1}{2}}}[/tex]
where,
[tex]N_o[/tex] = initial mass of isotope
N = mass of the parent isotope left after the time, (t)
[tex]t_{\frac{1}{2}}[/tex] = half life of the isotope
[tex]\lambda[/tex] = rate constant
[tex]\lambda =\frac{0.693}{138 days}=0.005021 day^{-1}[/tex]
time ,t = 1104 dyas
[tex]N=N_o\times e^{-(\lambda )\times t}[/tex]
Now put all the given values in this formula, we get
[tex]N=30g\times e^{-0.005021 day^{-1}\times 1104 days}[/tex]
[tex]N=0.1174 g[/tex]
Number of half-lives:
[tex]N=\frac{N_o}{2^n}[/tex]
n = Number of half lives elapsed
[tex]0.1174 g=\frac{30 g}{2^n}[/tex]
[tex]n = 7.99\approx 8[/tex]
8 half-lives of polonium-210 occur in 1104 days.
0.1174 g of polonium-210 will remain in the sample after 1104 days.
