An archaeological sample (remains of an animal) containing 14C isotope of Carbon is found to give 10 beta decays per minute per gram of Carbon. It is known that the natural abundance of 14C in organic matter that is in equilibrium with the atmosphere today will give 15 beta decays per minute per gram of Carbon. The half life of 14C is known to be 5730 years. The estimated age of the sample (in years) isa)3010b)3350c)3500d)3800Correct answer is option 'B'. Can you explain this answer?

Question

Answer

Given information:
- The archaeological sample contains 14C isotope of Carbon.
- The sample gives 10 beta decays per minute per gram of Carbon.
- Natural abundance of 14C in organic matter in equilibrium with the atmosphere today gives 15 beta decays per minute per gram of Carbon.
- The half-life of 14C is 5730 years.

To find:
The estimated age of the sample.

Solution:

Understanding Beta Decay:
- Beta decay is a type of radioactive decay where a nucleus emits a beta particle (an electron or a positron) and transforms into a different element.
- In the case of 14C, it undergoes beta decay to transform into 14N.

Relationship between Beta Decay and Half-life:
- The rate of beta decay is proportional to the number of radioactive atoms present in a sample.
- The decay rate can be expressed as the number of decays per minute per gram of Carbon.
- As the number of radioactive atoms decreases over time due to decay, the decay rate also decreases.
- The half-life of a radioactive isotope is the time it takes for half of the radioactive atoms to decay.
- After one half-life, the decay rate is reduced to half of its initial value.

Using the given information:
- The sample gives 10 beta decays per minute per gram of Carbon.
- The natural abundance of 14C today gives 15 beta decays per minute per gram of Carbon.
- This means the sample has a lower decay rate compared to the natural abundance, indicating that some time has passed since the sample was in equilibrium with the atmosphere.

Calculating the age of the sample:
- Let's assume the original amount of 14C in the sample was x grams.
- After one half-life, the remaining amount of 14C will be x/2 grams.
- The decay rate after one half-life will be 15 beta decays per minute per gram of Carbon.
- Therefore, the decay rate of the sample after one half-life is 10/(x/2) = 20/x beta decays per minute per gram of Carbon.
- From this, we can calculate the value of x.

Using the formula for radioactive decay:
- The decay rate R at time t is given by R = Ro * (1/2)^(t/T), where Ro is the initial decay rate and T is the half-life.
- Plugging in the values, we have 20/x = 15 * (1/2)^(5730/T).
- Simplifying, we get x = 2 * 15 * (x/20) * (1/2)^(5730/T).
- Canceling out the x terms, we have 1 = (1/2)^(5730/T).
- Taking the logarithm of both sides, we get log(1) = log((1/2)^(5730/T)).
- Simplifying, we have 0 = (5730/T) * log(1/2).
- Rearranging the equation, we get T = 5730 / log(2).

Calculating the estimated age:
- The half-life T is given by T = 5730 / log(2).
- Substituting the value of T,

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