Question

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

Short answer

The final product has an atomic number of 82 and a mass number of 208, which is lead (Pb).

Step-by-step solution

Understanding Alpha Decay

An alpha particle is composed of 2 protons and 2 neutrons. When an isotope emits an alpha particle, it loses 2 protons and 4 in mass number (due to the loss of 2 protons and 2 neutrons as an alpha particle).

Applying Alpha Decay to Thorium

Since the isotope emits 6 alpha particles, the atomic number will decrease by 6 times 2 (as each alpha particle decreases the atomic number by 2). The original atomic number 90 becomes: 90 - 12 = 78. Similarly, the mass number decreases by 6 times 4: 232 - 24 = 208.

Understanding Beta Decay

A beta particle is an electron, which occurs when a neutron is converted into a proton. This increases the atomic number by 1 each time, while the mass number remains unchanged.

Applying Beta Decay to Thorium

Since 4 beta particles are emitted, increase the atomic number by 4. The atomic number after 6 alpha decays was 78 (from the previous step), now: 78 + 4 = 82. The mass number remains 208 as beta decay doesn't affect it.

Identify the New Element

The atomic number after all decays is 82, which corresponds to lead (Pb) on the periodic table. The mass number is 208.

Key concepts

Alpha Decay

Alpha decay is a type of radioactive decomposition where an atomic nucleus loses an "alpha particle." An alpha particle consists of:

• 2 protons
• 2 neutrons

When an atom undergoes alpha decay, its atomic number decreases by 2, as it loses 2 protons. Likewise, its mass number decreases by 4, due to the 2 neutrons and 2 protons departing. This decay process has a significant impact, transforming the original isotope into a new element, which is:
- Less massive - Positioned two places earlier in the periodic table.
In the original exercise, thorium undergoes alpha decay six times, resulting in significant atomic and mass number reductions.

Beta Decay

In beta decay, a neutron in an atom's nucleus turns into a proton, which changes the balance of subatomic particles. This transformation involves the emission of a beta particle, which is essentially an electron. When beta decay occurs:

• The atomic number increases by 1.
• The mass number remains unchanged.

Each beta decay means the atom has one more proton, requiring a shift to the next element in the periodic table. However, since the mass of an electron is negligible, the overall mass number stays unchanged. Following beta decay, elements often become more stable, due to the shift in neutron-to-proton ratio.

Thorium Transformation

During the radioactive decay process, the thorium isotope undergoes transformations due to the emission of alpha and beta particles. Initially, thorium has:

• Atomic number: 90
• Mass number: 232

With each alpha decay, thorium transforms to a lighter and different element by losing helium atoms. After six such events, the atomic structure changes significantly. Subsequent beta decays further alter its identity, leading to a more profound shift in atomic arrangement.
After the full decay cycle, thorium metamorphoses into a stable element known as lead (Pb), specifically with an atomic number of 82 and a mass number of 208.

Atomic Number Changes

The atomic number of an element reflects the number of protons in its nucleus. Alterations in the atomic number signify an element's shift in the periodic table, as protons define elemental identity. During decay:

• Alpha decay decreases atomic number by 2 per decay.
• Beta decay increases atomic number by 1 per decay.

In the problem of thorium decay, the initial atomic number was 90. After six alpha decays (90 - 12 = 78) and four beta decays (78 + 4 = 82), the atomic number ultimately settles at 82. This reflects thorium's transformation into lead, based on these cumulative changes in proton count.

Mass Number Changes

The mass number is determined by the total count of protons and neutrons in the nucleus of an atom. This number changes notably through the process of alpha decay. During radioactive transformation:

• Alpha decay reduces the mass number by 4 per event.
• Beta decay does not affect the mass number.

For thorium, the starting mass number was 232. After six instances of alpha decay, it decreased by 6 times 4, resulting in a final mass number of 208. Beta decay, focusing on converting neutrons to protons, does not alter mass, so this number remains constant through that phase of decay. Thus, the mass is solely impacted by alpha emissions.