Carbon`-14` used to determine the age of organic material. The procedure is absed on the formation of `C^(14)` by neutron capture iin the upper atmosphere.
`._(7)N^(14)+._(0)n^(1) rarr ._(6)C^(14)+._(1)H^(1)`
`C^(14)` is absorbed by living organisms during photosynthesis. The `C^(14)` content is constant in living organism. Once the plant or animal dies, the uptake of carbon dioxide by it ceases and the level of `C^(14)` in the dead being falls due to the decay, which `C^(14)` undergoes.
`._(6)C^(14)rarr ._(7)N^(14)+beta^(c-)`
The half`-` life period of `C^(14)` is 5770 year. The decay constant `(lambda)` can be calculated by using the following formuls `:`
`lambda=(0.693)/(t_(1//2))`
The comparison of the `beta^(c-)` activity of the dead matter with that of the carbon still in circulation enables measurement of the period of the isolation of the material from the living cycle. The method, however, ceases to be accurate over periods longer than 30000 years. The proportion of `C^(14)` to `C^(12)` in living matter is `1:10^(12)`.
A nuclear explosion has taken place leading to an increase in the concentration of `C^(14)` in nearby areas. `C^(14)` concentration is `C_(1)` in nearby areas and `C_(2)` in areas far away. If the age of the fossil is determined to be `T_(1)` and `T_(2)` at the places , respectively, then

Options

A. The age of the fossil will increase at the place where explosion has taken place and `T_(1)-T_(2)=(1)/(lambda)ln ``(C_(1))/(C_(2))`
B. The age of the fossil will decrease at the place where explosion has taken place and `T_(1)-T_(2)=(1)/(lambda)ln``(C_(1))/(C_(2))`
C. The age of fossile will be determined to be same.
D. `(T_(1))/(T_(2))=(C_(1))/(C_(2))`

Explanation

To solve the problem, we need to establish the relationship between the ages of fossils (T1 and T2) in relation to the concentrations of Carbon-14 (C1 and C2) in nearby and faraway areas, respectively.

### Step-by-Step Solution:

1. **Understanding the Decay Process**:
- Carbon-14 decays over time, and the decay can be described using the decay constant (λ). The relationship between the concentration of Carbon-14 and time can be expressed using the formula:
\[ N(t) = N_0 e^{-\lambda t} \]
- Where \(N(t)\) is the remaining quantity of Carbon-14 at time \(t\), \(N_0\) is the initial quantity, and \(λ\) is the decay constant.

2. **Using the Half-Life to Find the Decay Constant**:
- The half-life of Carbon-14 is given as 5770 years. The decay constant can be calculated using the formula:
\[ \lambda = \frac{0.693}{t_{1/2}} = \frac{0.693}{5770 \text{ years}} \approx 1.20 \times 10^{-4} \text{ year}^{-1} \]

3. **Establishing the Relationship Between Concentration and Age**:
- For a fossil in an area with concentration \(C_1\) (nearby the explosion), the age \(T_1\) can be expressed as:
\[ T_1 = \frac{1}{\lambda} \ln\left(\frac{C_0}{C_1}\right) \]
- For a fossil in an area with concentration \(C_2\) (far from the explosion), the age \(T_2\) can be expressed as:
\[ T_2 = \frac{1}{\lambda} \ln\left(\frac{C_0}{C_2}\right) \]

4. **Finding the Relation Between Ages and Concentrations**:
- To find the relationship between \(T_1\) and \(T_2\), we can take the ratio:
\[ \frac{T_1}{T_2} = \frac{\ln\left(\frac{C_0}{C_1}\right)}{\ln\left(\frac{C_0}{C_2}\right)} \]
- This implies that the ages \(T_1\) and \(T_2\) are related to the concentrations \(C_1\) and \(C_2\).

5. **Conclusion**:
- From the above analysis, we can conclude that:
\[ T_1 \propto \ln(C_0/C_1) \quad \text{and} \quad T_2 \propto \ln(C_0/C_2) \]
- Therefore, the relationship can be summarized as:
\[ \frac{T_1}{T_2} = \frac{\ln(C_0/C_1)}{\ln(C_0/C_2)} \]
- This indicates that the ages of the fossils are directly proportional to the logarithm of the ratio of their concentrations.