9. Chemical analysis shows that a bone fragment contains 10 g of C. Its activity is 50
counts/min. How old is the bone fragment?
N=mNAM=10⋅6.02⋅102312=5⋅1023,N=\frac{mN_A}M=\frac{10\cdot 6.02\cdot 10^{23}}{12}=5\cdot 10^{23},N=MmNA=1210⋅6.02⋅1023=5⋅1023,
N0=Nη=5⋅1023⋅1.3⋅10−12=6.5⋅1011,N_0=N\eta =5\cdot 10^{23} \cdot1.3\cdot10^{-12} = 6.5\cdot10^{11},N0=Nη=5⋅1023⋅1.3⋅10−12=6.5⋅1011,
A0=N0λ=6.5⋅1011⋅0.695730⋅365⋅24⋅3600=2.53 s−1,A_0=N_0\lambda = \frac{6.5\cdot10^{11 }\cdot 0.69 }{5730 \cdot 365 \cdot 24 \cdot 3600 }= 2.53~ s^{-1},A0=N0λ=5730⋅365⋅24⋅36006.5⋅1011⋅0.69=2.53 s−1,
A=A0e−λt, ⟹ A=A_0e^{-\lambda t},\impliesA=A0e−λt,⟹
λt=lnA0A=ln2.53⋅6050=1.11,\lambda t=\ln \frac{A_0}A=\ln \frac{2.53\cdot 60}{50}=1.11,λt=lnAA0=ln502.53⋅60=1.11,
t=1.11λ=1.11⋅53700.69=9180 years.t=\frac {1.11}{\lambda}=\frac{1.11\cdot 5370}{0.69}=9180~\text{years}.t=λ1.11=0.691.11⋅5370=9180 years.
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