Chapter 18, Problem 36QAP

Interpretation Introduction

Interpretation:

Whether the painting is made by Michelangelo, or made when the Michelangelo was alive should be determined.

Concept introduction:

Activity is defined as the product of rate constant and number of atoms. It is also defined as the rate of decay or expresses number of atoms decays in unit time.

The mathematical expression is given by:

$\text{A = kN}$

Where, A = activity, k = first order rate constant and N = number of radioactive nuclei.

Activity can be expressed in following units:

1 Bq = 1 atom/s

1 Ci = 3.700 × 10¹⁰ atoms/s

The relation between half-life and first order rate constant is given by:

$\text{k=}\frac{\text{0}\text{.693}}{{\text{t}}_{\text{1/2}}}$

Where, k = first order rate constant and ${\text{t}}_{\text{1/2}}$ = half-life

The prediction of the age of organic material is based on the rate of decay of carbon-14. Carbon-14 is an isotope of carbon which is released in the atmosphere in the form of interaction of neutrons from cosmic radiation with ordinary nitrogen atoms.

The intake of radioactive carbon stops when plant or animal dies. The reaction is given as:

${}_6^{14}\text{C}{\to }_7^{14}{\text{N+}}_{-1}^0\text{e}$

The half-life of radioactive decay of carbon-14 is 5730 years. The activity of sample is directly proportional to the amount of C-14. The mathematical expression is given by:

$\ln \frac{A_o}{A}=kt$

Where, $A_o$ = original activity

$A$ = measured activity

t= age of the sample