The heaviest member of the alkaline earth metals is radium...

The heaviest member of the alkaline earth metals is radium (Ra), a naturally radioactive element discovered by Pierre and Marie Curie in $1898 .$ Radium was initially isolated from the uranium ore pitchblende, in which it is present as approximately $1.0 \mathrm{~g}$ per $7.0$ metric tons of pitchblende. How many atoms of radium can be isolated from $1.75 \times 10^{8} \mathrm{~g}$ pitchblende $(1$ metric ton $=$ $1000 \mathrm{~kg}) ?$ One of the early uses of radium was as an additive to paint so that watch dials coated with this paint would glow in the dark. The longest-lived isotope of radium has a half-life of $1.60 \times 10^{3}$ years. If an antique watch, manufactured in 1925, contains $15.0 \mathrm{mg}$ radium, how many atoms of radium will remain in $2025 ?$

Key Concepts

Unit Conversion

This concept involves converting between different units of mass, such as grams, kilograms, and metric tons. In problems like these, unit conversion is essential for ensuring that all quantities are expressed in compatible units before performing further calculations, such as scaling content per mass of material.

Mole Concept and Avogadro's Number

This concept is used to relate a given mass of an element to the number of atoms present. By using the atomic mass of an element to determine the number of moles and then using Avogadro's number (approximately 6.022 × 10^23 atoms/mol), one can calculate the total number of atoms in the sample.

Radioactive Decay and Half-Life

This concept describes the process by which unstable isotopes transform into more stable ones over time. The half-life is the time required for half of the atoms in a sample to decay. This property is crucial in calculating how many atoms remain after a certain period, using the decay formula based on the half-life.

Exponential Decay Calculation

This involves applying the exponential decay formula, N = N0(1/2)^(t/T1/2), where N0 is the initial number of atoms, t is the elapsed time, and T1/2 is the half-life of the radioactive element. This calculation is fundamental for determining the remaining quantity of a radioactive substance after a given time interval.