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In a sample of wood, the reading of a co...

In a sample of wood, the reading of a counter is 32 dpm and in a fresh sample of tree it is 122dpm . Due to error counter gives the reading 2 dpm in absence of `.^(14)C` . Half life of `.^(14)C` is 5770 years .
The approximate age (in years) of wood sample is :

A

7997.2

B

57570

C

11540

D

15140

Text Solution

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The correct Answer is:
To find the approximate age of the wood sample, we can use the principles of radioactive decay, specifically the decay of Carbon-14. Here’s a step-by-step solution: ### Step 1: Understand the given data - Counter reading for the wood sample: 32 dpm (decays per minute) - Counter reading for a fresh sample of tree: 122 dpm - Counter error in absence of C-14: 2 dpm - Half-life of C-14: 5770 years ### Step 2: Adjust the readings for the error Since the counter gives an error of 2 dpm, we need to adjust the readings: - Adjusted reading for the fresh sample: 122 dpm - 2 dpm = 120 dpm - Adjusted reading for the wood sample: 32 dpm - 2 dpm = 30 dpm ### Step 3: Determine the decay constant (k) The decay constant (k) can be calculated using the half-life formula: \[ k = \frac{0.693}{t_{1/2}} \] Substituting the half-life of C-14: \[ k = \frac{0.693}{5770} \approx 0.0001209 \text{ years}^{-1} \] ### Step 4: Use the radioactive decay formula The formula for radioactive decay is: \[ T = \frac{1}{k} \ln \left( \frac{R_0}{R_t} \right) \] Where: - \( R_0 \) = initial activity (adjusted fresh sample reading) = 120 dpm - \( R_t \) = current activity (adjusted wood sample reading) = 30 dpm ### Step 5: Substitute the values into the formula Substituting the values into the decay formula: \[ T = \frac{1}{0.0001209} \ln \left( \frac{120}{30} \right) \] ### Step 6: Calculate the natural logarithm Calculate the natural logarithm: \[ \ln \left( \frac{120}{30} \right) = \ln(4) \approx 1.386 \] ### Step 7: Calculate the age (T) Now substitute the value of the natural logarithm back into the equation: \[ T = \frac{1}{0.0001209} \times 1.386 \approx 11457 \text{ years} \] ### Conclusion The approximate age of the wood sample is about **11457 years**. ---
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