Respuesta :
Answer:
The age of living tree is 11104 years.
Explanation:
Given that,
Mass of pure carbon = 100 g
Activity of this carbon is = 6.5 decays per second = 6.5 x60 decays/min =390 decays/m
We need to calculate the decay rate
[tex]R=\dfrac{-dN}{dt}=\lambda N=\dfrac{0.693}{t_{\frac{1}{2}}}N[/tex]....(I)
Where, N = number of radio active atoms
[tex]t_{\frac{1}{2}}[/tex]=half life
We need to calculate the number of radio active atoms
For [tex]N_{12_{c}}[/tex]
[tex]N_{12_{c}}=\dfrac{N_{A}}{M}[/tex]
Where, [tex]N_{A}[/tex] =Avogadro number
[tex]N_{12_{c}}=\dfrac{6.02\times10^{23}}{12}[/tex]
[tex]N_{12_{c}}=5.02\times10^{22}\ nuclie/g[/tex]
For [tex]N_{c_{14}}[/tex]
[tex]N_{c_{14}}=1.30\times10^{-12}N_{12_{c}}[/tex]
[tex]N_{c_{14}}=1.30\times10^{-12}\times5.02\times10^{22}[/tex]
[tex]N_{c_{14}}=6.526\times10^{10}\ nuclei/g[/tex]
Put the value in the equation (I)
[tex]R=\dfrac{0.693\times6.526\times10^{10}\times60}{5700\times3.16\times10^{7}}[/tex]
[tex]R=15.0650\ decay/min g[/tex]
100 g carbon will decay with rate
[tex]R=100\times15.0650=1507\ decay/min[/tex]
We need to calculate the total half lives
[tex](\dfrac{1}{2})^{n}=\dfrac{390}{1507}[/tex]
[tex]2^n=\dfrac{1507}{390}[/tex]
[tex]2^n=3.86[/tex]
[tex]n ln 2=ln 3.86[/tex]
[tex]n=\dfrac{ln 3.86}{ln 2}[/tex]
[tex]n =1.948[/tex]
We need to calculate the age of living tree
Using formula of age
[tex]t=n\times t_{\frac{1}{2}}[/tex]
[tex]t=1.948\times5700[/tex]
[tex]t=11103.6 =11104\ years[/tex]
Hence, The age of living tree is 11104 years.
