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Question: The half-life period of \({C^{14}}\) is 5370 years. In a sample of a dead tree, the proportion \({C^...

The half-life period of C14{C^{14}} is 5370 years. In a sample of a dead tree, the proportion C14{C^{14}} is found to be 60% in comparison to a living tree. Calculate the age of the sample.

Explanation

Solution

The half-life period of a substance is defined as the time in which the concentration of reactant reduces to half of the initial concentration or we can say that half the life period is that time in which half of the substance has reacted.

Complete step by step answer:
As we know C14{C^{14}} isotope is radioactive and its radioactive disintegration follows first-order kinetics. Here given that the half-life period C14{C^{14}} is 5370 years. As we know the half-life period of the first order can be calculated as:
t1/2=0.693k{t_{1/2}} = \dfrac{{0.693}}{k}
Where decay or disintegration constant is represented by k. Now put the value of the half-life period of C14{C^{14}} in the above formula we get:
5370=0.693k5370 = \dfrac{{0.693}}{k}
⇒k=0.6935370\Rightarrow k = \dfrac{{0.693}}{{5370}}
⇒k=1.290×10−4\Rightarrow k = 1.290 \times {10^{ - 4}}
Suppose initial concentration .i.e Ro=100{R_o} = 100, and the final concentration .i.e. R=60R = 60. For first-order reaction time taken is calculated as:
t=2.303klog⁡[Ro][R]t = \dfrac{{2.303}}{k}\log \dfrac{{[{R_o}]}}{{[R]}}
On putting the value we get:
t=2.3031.290×10−4log⁡[100][60]t = \dfrac{{2.303}}{{1.290 \times {{10}^{ - 4}}}}\log \dfrac{{[100]}}{{[60]}}
As we know log⁡AB=log⁡A−log⁡B\log \dfrac{A}{B} = \log A - \log B. Hence the above equation becomes:
t=2.3031.290×10−4[log⁡10−log⁡6]t = \dfrac{{2.303}}{{1.290 \times {{10}^{ - 4}}}}[\log 10 - \log 6]
Here log⁡10=1\log 10 = 1 and log⁡6=0.778\log 6 = 0.778
⇒t=2.3031.290×10−4[1−0.778]\Rightarrow t = \dfrac{{2.303}}{{1.290 \times {{10}^{ - 4}}}}[1 - 0.778]
⇒t=2.3031.290×10−4×0.222\Rightarrow t = \dfrac{{2.303}}{{1.290 \times {{10}^{ - 4}}}} \times 0.222
⇒t=0.396×104\Rightarrow t = 0.396 \times {10^4}years

Additional information:
There are many elements like uranium, thorium, etc which spontaneously emit α,β  and  γ\alpha, \beta \;and\; \gamma particles; this process is known as natural radioactivity, and Radioactive disintegration is defined as the emission of particles by any radioactive element or any radioactive isotopes.

Note: As we know the whole of the radioactive isotopes never disintegrate and it means the total life period of a radioactive isotope is infinite. The average life of a radioactive isotope is also known as natural life and is inversely proportional to the disintegration constant. As smaller the value of disintegration constant i.e. slower the disintegration, the greater is the average or natural life.