Question

The molar mass of carbon is 12.0107 g/mol. This molar mass includes all isotopes of Carbon in the fractions that are naturally found in the atmosphere and living organisms. A 12.0 g sample of carbon from living matter decays at the rate of 184 decays/minute due to the radioactive 14C in it.

a) What fraction of these carbon atoms were 14C?

b) A second identical sample is put in a spaceship that is traveling with a speed of 0.8c relative to observers on earth.

i. What would the people on the spaceship determine the activity of this second sample to be?

ii. What would people on earth determine the activity of this second sample to be?

c) In analyzing an archaeological specimen containing 500 mg of carbon, you observe 174 decays in one hour. What is the age of this specimen, assuming that its activity per unit mass of carbon when it died was the average value of that in air?

Answer 1

a] A = 184 decays/minute = 184/60 decays/s = 3.0667 Bq

Half life of C-14 is: T_1/2 = 5730 years = 5730 x 365 x 24 x 3600 seconds

so, N = 8 x 10¹¹

these are the number of C-14 atoms in the sample.

Now, 12 grams of C has (12/12.0107) x 6.023 x 10²³ = 6.017 x 10²³ atoms

the fraction of C-14 atoms is: 1.329 x 10^-12.

b]

i] Activity remains the same for the spaceframe.

ii] Here, due to time dilation, the half-life will be:

so, the activity will be:

A = 1.84 Bq.