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 g per 7.0 metric tons of pitchblende. How many atoms of radium can be isolated from \(1.75 \times 10^{8} \mathrm{g}\) pitch- blende \((1 \text { metric ton }=1000 \mathrm{kg})\) ? 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\(?\)

Step by step solution

01

Convert the mass of pitchblende to radium mass

First, we need to find the mass of radium present in 1.75 x 10^8 g of pitchblende using the proportion given: 1.0 g radium / 7.0 metric tons of pitchblende Be aware that 1 metric ton = 1000 kg, and we will first convert the mass of pitchblende to metric tons. mass_pitchblende_metric_tons = (1.75 x 10^8 g) * (1 kg / 1000 g) * (1 metric ton / 1000 kg) mass_pitchblende_metric_tons = 175 metric tons Now we can find the mass of radium: mass_radium = (mass_pitchblende_metric_tons) * (1.0 g radium / 7.0 metric tons of pitchblende) mass_radium = 175 * (1.0 / 7.0) mass_radium ≈ 25 g

02

Convert the mass of radium to the number of atoms

To convert the mass of radium to the number of atoms, we will use the atomic mass of radium (226 g/mol) and Avogadro's number (6.022 x 10^23 atoms/mol): Number of atoms = (mass_radium) * (1 mol / 226 g) * (6.022 x 10^23 atoms / 1 mol) Number of atoms ≈ (25 g) * (1/226) * (6.022 x 10^23) Number of atoms ≈ 6.66 x 10^21 atoms So, approximately 6.66 x 10^21 atoms of radium can be isolated from 1.75 x 10^8 g of pitchblende.

03

Find the initial number of radium atoms in the antique watch

The antique watch contains 15.0 mg of radium. We need to convert this mass to the number of atoms using the atomic mass of radium and Avogadro's number: Initial number of atoms = (15.0 mg) * (1 g / 1000 mg) * (1 mol / 226 g) * (6.022 x 10^23 atoms / 1 mol) Initial number of atoms ≈ (15.0 x 10^(-3) g) * (1/226) * (6.022 x 10^23) Initial number of atoms ≈ 3.99 x 10^20 atoms

04

Calculate the remaining number of radium atoms after 100 years

We will use the radioactive decay equation to find the remaining number of radium atoms after 100 years: Remaining number of atoms = Initial number of atoms * (0.5)^(time elapsed / half-life) The time elapsed is 2025 - 1925 = 100 years Half-life of radium = 1.60 x 10^3 years Remaining number of atoms = 3.99 x 10^20 * (0.5)^(100 / 1600) Remaining number of atoms ≈ 3.64 x 10^20 atoms Therefore, approximately 3.64 x 10^20 atoms of radium will remain in the antique watch in 2025.

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