initial avtivity of a living organism is 15 decays / min

when dead its activity reduced to 9 decays/min

given half life of C14 = 5730 years

then how many years ago it was alive

Ans 4224 years

Please give detailed solution if possible with concepts i am not able to solve such qns

To determine how many years ago the organism was alive based on its radioactive decay, we can use the concept of half-life and the relationship between the initial and final activity of the carbon-14 (C-14) isotope. Let's break this down step by step.

Understanding Radioactive Decay

Carbon-14 is a radioactive isotope that is formed in the atmosphere and taken up by living organisms. When an organism dies, it stops absorbing C-14, and the existing C-14 in its body begins to decay at a known rate, characterized by its half-life. The half-life of C-14 is approximately 5730 years, meaning that after this period, half of the original amount of C-14 will have decayed.

Initial and Final Activity

In your scenario, the initial activity of the living organism is 15 decays per minute, and when it is dead, the activity reduces to 9 decays per minute. This reduction in activity indicates that some of the C-14 has decayed since the organism's death.

Calculating the Fraction of C-14 Remaining

To find out how much C-14 remains, we can express the final activity as a fraction of the initial activity:

• Initial activity (A₀) = 15 decays/min
• Final activity (A_t) = 9 decays/min

The fraction of C-14 remaining is:

Fraction remaining = A_t / A₀ = 9 / 15 = 0.6

Using the Half-Life Formula

The relationship between the remaining quantity of a radioactive substance and time can be expressed using the formula:

N(t) = N₀ * (1/2)^(t/T_1/2)

Where:

• N(t) = remaining quantity after time t
• N₀ = initial quantity
• T_1/2 = half-life of the substance

In our case, we can rearrange this formula to find the time (t) since the organism died:

0.6 = (1/2)^(t/5730)

Solving for Time

To solve for t, we can take the logarithm of both sides:

log(0.6) = (t/5730) * log(1/2)

Now, we can isolate t:

t = 5730 * (log(0.6) / log(0.5))

Calculating the Values

Using a calculator, we find:

• log(0.6) ≈ -0.2218
• log(0.5) ≈ -0.3010

Now substituting these values into the equation:

t = 5730 * (-0.2218 / -0.3010) ≈ 5730 * 0.736

t ≈ 4224 years

Final Thoughts

This calculation shows that the organism died approximately 4224 years ago. By understanding the principles of radioactive decay and the half-life of C-14, we can effectively determine the age of ancient organic materials. This method is widely used in archaeology and geology to date artifacts and fossils, providing valuable insights into our past.