A sample has a activity of 0.0065 Bq per gram of carbon. (a) Find the age of the sample, assuming that the activity per gram of carbon in a living organism has been constant at a value of 0.23 Bq. (b) Evidence suggests that the value of 0.23 Bq might have been as much as 49% larger. Repeat part (a), taking into account this 49% increase.

To calculate the age of the sample, we need to use the concept of radioactive decay and the equation for exponential decay. In this case, we'll be using the equation:

N(t) = N0 * e^(-λt),

where N(t) is the activity of the sample at time t, N0 is the initial activity, e is the base of the natural logarithm (approximately 2.718), λ is the decay constant, and t is the time passed.

(a) Assuming that the activity per gram of carbon in a living organism has been constant at a value of 0.23 Bq, we have:

N0 = 0.23 Bq (initial activity),
N(t) = 0.0065 Bq (activity of the sample),
λ = decay constant (to be determined),
t = age of the sample (what we want to find).

To find the decay constant λ, we can use the ratio of activities:

N(t) / N0 = e^(-λt).

Substituting the given values:

0.0065 Bq / 0.23 Bq = e^(-λt).

Dividing both sides by 0.23 Bq:

0.0283 = e^(-λt).

Now, we need to take the natural logarithm of both sides:

ln(0.0283) = -λt.

Rearranging the equation to solve for t:

t = -ln(0.0283) / λ.

Now, we need to find the value of λ. The decay constant for carbon-14 (C-14) is approximately 0.693 / half-life, where the half-life of C-14 is 5730 years.

λ = 0.693 / (5730 years) ≈ 1.209 x 10^(-4) years^(-1).

Substituting this value back into the equation for t:

t = -ln(0.0283) / (1.209 x 10^(-4) years^(-1)).

Using a calculator, we find that the age of the sample is approximately 50,560 years.

(b) Taking into account the 49% increase in the value of 0.23 Bq, we can calculate a new value for N0:

New N0 = 0.23 Bq + (0.23 Bq * 49%) = 0.23 Bq + (0.23 Bq * 0.49) ≈ 0.23 Bq + 0.113 Bq ≈ 0.343 Bq.

Using the same process as in part (a), but substituting the new value of N0, we can find the revised age of the sample.

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