Question

If a radioactive isotope of thorium (atomic number \(90,\) mass number 232 ) emits 6 alpha particles and 4 beta particles during the course of radioactive decay, what are the atomic number and mass number of the stable daughter product?

Short answer

The stable daughter product has an atomic number of 82 and a mass number of 208.

Step-by-step solution

Understand Alpha Decay

In alpha decay, the nucleus emits an alpha particle, which contains 2 protons and 2 neutrons. This reduces the atomic number by 2 and the mass number by 4 each time an alpha particle is emitted.

Calculate Changes Due to Alpha Emission

Since 6 alpha particles are emitted: \(\text{change in atomic number} = 6 \times 2 = 12\). The atomic number of thorium decreases by 12: \(90 - 12 = 78\). \(\text{change in mass number} = 6 \times 4 = 24\). The mass number of thorium decreases by 24: \(232 - 24 = 208\).

Understand Beta Decay

In beta decay, a neutron is converted into a proton, increasing the atomic number by 1 for each beta particle emitted, but the mass number remains unchanged.

Calculate Changes Due to Beta Emission

Since 4 beta particles are emitted, the atomic number increases by 4: \(78 + 4 = 82\). There is no change to the mass number; it remains 208.

Determine the Final Atomic Number and Mass Number

After considering the emissions of 6 alpha particles and 4 beta particles, the final atomic number is 82 and the final mass number is 208.

Key concepts

Alpha Decay

Alpha decay is a type of radioactive decay where an unstable nucleus releases an alpha particle. An alpha particle is composed of 2 protons and 2 neutrons. As a result, the emission of an alpha particle decreases an atom's atomic number by 2 and its mass number by 4.
This transformation leads to a significant change in the identity of the element, since the number of protons determines the element. It's like the atom is shedding a bit of weight to become more stable.
Here are the key effects of alpha decay:

• Reduction of atomic number: losing 2 protons changes the element.
• Decrease in mass number: removing 4 particles slightly reduces the atom's overall mass.

In the original problem, the thorium nucleus emitted 6 alpha particles. This means that its atomic number was reduced by 12 (via 2 protons per alpha particle) and its mass number was reduced by 24 (via 4 nucleons per alpha particle).

Beta Decay

Beta decay involves the transformation of a neutron into a proton inside the nucleus. During this process, a beta particle, which is essentially an electron, is emitted.
Unlike alpha decay, beta decay alters only the atomic number, leaving the mass number unchanged.
This happens because although a neutron (neutral) becomes a proton (positive), which increases the charge by one, the mass hardly changes since both particles have similar masses.
The significant outcomes of beta decay are:

• Increase in atomic number: one neutron becomes a proton, so the element becomes one step higher in the periodic table.
• Mass number stays constant: the change in internal nuclear particles does not affect the overall mass.

In the problem, thorium underwent 4 beta decays, thereby increasing its atomic number by 4. Hence, the final atomic number rose from 78 to 82 after accounting for the increase from beta emissions.

Atomic Number

The atomic number of an element is a fundamental property that identifies the element. It corresponds to the number of protons in the nucleus of an atom. As each element has a unique number of protons, the atomic number acts like its fingerprint.
Changes to an atom's atomic number mean the atom transforms into a different element.
In the context of radioactive decay, both alpha and beta decays impact the atomic number. Alpha decay decreases the atomic number by 2 per event, while beta decay increases it by 1 per event.
Here's a summary of the changes:

• Alpha decay: lowers the atomic number, moves down the periodic table.
• Beta decay: raises the atomic number, moves up the periodic table.

For thorium in the exercise, the atomic number changed from 90 to 82, due to 6 alpha decays (decrease by 12) and 4 beta decays (increase by 4). This final transformation identifies the new element as lead.

Mass Number

The mass number is the total count of protons and neutrons in an atom's nucleus. It gives the "weight" of an atom and is crucial in understanding isotopes, which are the same elements having different mass numbers due to different numbers of neutrons.
When radioactive decay occurs, the mass number can change based on the type of decay:

• Alpha decay: decreases the mass number by 4 (due to loss of 2 protons and 2 neutrons).
• Beta decay: leaves the mass number unchanged since only a neutron is turned into a proton.

In the original exercise, thorium starts with a mass number of 232. After losing 24 units through 6 alpha decays, its mass number becomes 208. The beta decay events do not alter the mass number, so it remains 208, identifying the final stable isotope.