Question

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 ?

Short answer

For part 1, we find that \(1.75 \times 10^8 \mathrm{~g}\) of pitchblende can contain approximately \(1.25 \times 10^{23}\) atoms of radium. In 2025, the antique watch containing \(15.0 \mathrm{mg}\) of radium will have approximately \(3.93 \times 10^{18}\) remaining radium atoms.

Step-by-step solution

Calculate the amount of radium in grams

Find the amount of radium (in grams) that can be isolated from 1.75 x 10^8 g pitchblende, knowing that 1.0 g of radium can be obtained from 7.0 metric tons of pitchblende. First, determine the amount of pitchblende in grams and divide it by the total amount of pitchblende (7 metric tons) that contains 1 g of radium: 𝑃𝑖𝑡𝑐ℎ𝑏𝑙𝑒𝑛𝑑𝑒_𝑔𝑟𝑎𝑚𝑠 = 1.75 x 10^8 g Metric_Tons_Per_Gram_Ra = 7.0 metric tons Conversion_Factor = 1000 kg/metric ton * 1000 g/kg Pitchblende_Grams_Per_Gram_Ra = Metric_Tons_Per_Gram_Ra * Conversion_Factor The amount of radium in grams can be calculated by the following formula: Radium_Grams = 𝑃𝑖𝑡𝑐ℎ𝑏𝑙𝑒𝑛𝑑𝑒_𝑔𝑟𝑎𝑚𝑠 / Pitchblende_Grams_Per_Gram_Ra

Calculate the number of radium atoms

Now that we have the amount of radium in grams, we can use Avogadro's number to find the number of radium atoms. Avogadro's number (Nₐ) = 6.022 x 10²³ atoms/mole First, we should find the molar mass of radium (Ra). Radium has a molar mass of 226 g/mole. Now, calculate the number of moles of radium: Radium_Moles = Radium_Grams / 226 g/mole Finally, use Avogadro's number to find the number of radium atoms: Radium_Atoms = Radium_Moles * Nₐ #Part 2: Radium atoms remaining in the antique watch in 2025#

Calculate the remaining mass of radium

Now, we need to calculate the remaining mass of radium in the antique watch, using the half-life formula. Initial mass of radium in the watch (Radium_initial) = 15 mg Half-Life of radium (T_1/2) = 1.60 x 10^3 years Time elapsed since 1925 (t) = 2025 - 1925 = 100 years Number of half-lives elapsed (n) = t / T_1/2 Remaining mass of radium in the watch (Radium_final) = Radium_initial * (1/2)^n

Calculate the number of remaining radium atoms

Now that we have the remaining mass of radium in the antique watch, we can use Avogadro's number to find the number of remaining radium atoms. First, convert the remaining mass of radium to grams: Radium_final_grams = Radium_final * (1/1000 g/mg) Now, calculate the number of moles of remaining radium: Remaining_Radium_Moles = Radium_final_grams / 226 g/mole Finally, use Avogadro's number to find the number of remaining radium atoms: Remaining_Radium_Atoms = Remaining_Radium_Moles * Nₐ

Key concepts

Understanding Avogadro's Number

In chemistry, Avogadro's number, denoted by \( N_a \), is a fundamental constant that represents the amount of discrete entities, usually atoms or molecules, in one mole of a substance. It is named after the Italian scientist Amedeo Avogadro and has a value of approximately \( 6.022 \times 10^{23} \) entities per mole.

Considering our radium calculation exercise, this number allows us to convert between the mass of a substance and the number of atoms or molecules it contains. When we say that one gram of radium can be isolated from a certain quantity of pitchblende, it's Avogadro's number that provides the bridge between this macroscopic mass and the microscopic number of atoms.

Radioactive Decay Explained

Radioactive decay is a spontaneous process by which an unstable atomic nucleus loses energy by radiation. In the context of the exercise, radium (Ra) undergoes radioactive decay over time, emitting radiation and transforming into different elements in a predictable way. This process occurs naturally and can't be influenced by external conditions.

Scientists can track the decay of radioactive elements through their half-lives—the time required for half of a given amount of radioisotope to decay. Understanding this concept is essential, as it tells us how many radium atoms will remain in the antique watch over time, helping us calculate the remaining radioactive material.

Molar Mass: A Critical Component

Molar mass is the mass of one mole of a substance, typically expressed in grams per mole (g/mol). It is calculated by summing the atomic masses of all atoms in a molecule. For elements like radium, the molar mass is the atomic mass unit found on the periodic table, reflecting the mass of one mole of atoms.

In the radium calculation exercise, knowing the molar mass of radium, which is 226 g/mol, is pivotal. It enables us to convert the grams of radium obtained from pitchblende into moles, a step that's necessary before using Avogadro's number to find the number of atoms.

The Significance of Half-Life

The half-life of an isotope, symbolized as \(T_{1/2}\), is the time required for half of the isotope to undergo radioactive decay. For radium in our exercise, the half-life is significant, as it helps us understand and calculate how much radium will remain in the antique watch after a certain period.

The antique watch manufactured in 1925 began with 15.0 mg of radium, and by using the half-life of radium, we can predict how much of this radioactive substance has decayed and what quantity remains after 100 years. This is a key step in determining the number of radium atoms still present in the watch in 2025.