RADIOACTIVE DECAY AND HALF-LIFE

Radioactive decay is a process where unstable atomic nuclei transform into more stable nuclei by emitting radiation. This process occurs randomly, but on average, the rate of decay is proportional to the number of radioactive nuclei present. The rate of decay is often described by a first-order reaction.

The half-life of a radioactive substance is the time it takes for half of the radioactive nuclei to decay. It is a characteristic property of each radioactive isotope. For example, Carbon-14 (^14C) is a radioactive isotope that decays with a half-life of 5730 years.

FRACTION REMAINING AFTER A CERTAIN TIME

To determine the fraction remaining after a certain time, we can use the equation for a first-order reaction:

N(t) = N₀ * e^(-kt)

where:
- N(t) is the number of radioactive nuclei remaining at time t
- N₀ is the initial number of radioactive nuclei
- k is the rate constant for the reaction
- t is the time

In this case, we are given the half-life of ^14C as 5770 years. Since the half-life is the time it takes for half of the nuclei to decay, we can set up the following equation:

N(t) = (1/2) * N₀

Using the equation for a first-order reaction, we can substitute these values:

(1/2) * N₀ = N₀ * e^(-kt)

Simplifying the equation, we get:

(1/2) = e^(-kt)

Taking the natural logarithm of both sides, we have:

ln(1/2) = -kt

Solving for k, we get:

k = ln(1/2) / (-t₁/₂)

Substituting the known values, we can calculate the value of k:

k = ln(1/2) / (-5770)

CALCULATING THE FRACTION REMAINING AFTER 11540 YEARS

Now that we have the rate constant (k), we can calculate the fraction remaining after 11540 years:

N(t) = N₀ * e^(-kt)

N(11540) = N₀ * e^(-k * 11540)

Substituting the values of k and t, we can calculate N(11540):

N(11540) = N₀ * e^(-(ln(1/2) / 5770) * 11540)

Using a calculator, we find that N(11540) is approximately 0.249.

Therefore, the fraction remaining after 11540 years is approximately 0.249 (rounded to three decimal places).

CONCLUSION

In summary, radioactive decay is a first-order process, and the fraction remaining after a certain time can be calculated using the equation for a first-order reaction. For the given radioactive carbon wood sample with a half-life of 5770 years, the fraction remaining after 11540 years is approximately 0.249.