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 is 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

Understanding Alpha and Beta Decay

In alpha decay, the nucleus emits an alpha particle, which consists of 2 protons and 2 neutrons. This means the atomic number decreases by 2 and the mass number decreases by 4. In beta decay, a neutron is converted into a proton, meaning the atomic number increases by 1 while the mass number remains unchanged.

Calculating Changes from Alpha Emissions

Each alpha particle emission decreases the atomic number by 2 and the mass number by 4. Since 6 alpha particles were emitted: \[ \text{Atomic Number Change} = 6 \times (-2) = -12 \]\[ \text{Mass Number Change} = 6 \times (-4) = -24 \]The new atomic number will be \( 90 - 12 = 78 \) and mass number will be \( 232 - 24 = 208 \).

Calculating Changes from Beta Emissions

Each beta particle emission increases the atomic number by 1, and has no impact on the mass number. Since 4 beta particles were emitted:\[ \text{Atomic Number Change} = 4 \times 1 = 4 \]The atomic number after the beta emissions will be \( 78 + 4 = 82 \). The mass number remains 208, since beta decay does not affect it.

Determining the Daughter Product

After accounting for both the alpha and beta decays, the final atomic number is 82 and the final mass number is 208. Therefore, the stable daughter product is the element with these atomic and mass numbers.

Key concepts

Alpha Particles

Alpha particles play a crucial role in the process of radioactive decay. Composed of 2 protons and 2 neutrons, they are essentially helium nuclei. During alpha decay, a radioactive isotope emits one of these particles, leading to changes in the element's structure.

• The atomic number of the original element decreases by 2.
• The mass number decreases by 4, as both protons and neutrons contribute to mass.

For example, if a thorium isotope (atomic number 90, mass number 232) emits an alpha particle, the new atomic number will be 88 and the mass number will be 228. This significant shift means the element is changed entirely, and if multiple alpha particles are emitted, as in the case of the thorium isotope emitting six alpha particles, the changes are compounded.

Beta Particles

Beta particles are a fascinating aspect of radioactive decay. Unlike alpha particles, beta particles represent more subtle changes in the atomic structure. A beta particle is either an electron or a positron emitted from the nucleus when a neutron transforms into a proton or vice versa.

• Each beta decay increases the atomic number by 1.
• The mass number remains unchanged, as no mass is lost or gained in the process.

In our example, the thorium isotope undergoes 4 beta decays, each adding 1 to the atomic number. Starting from an atomic number of 78 after alpha decays, the emission of 4 beta particles results in a final atomic number of 82, while the mass number remains at 208.

Daughter Product

The daughter product is the result of a radioactive element after undergoing decay processes. It is stable or at least less radioactive than the parent isotope. In our thorium example, after emitting 6 alpha particles and 4 beta particles, we determine the daughter product.

• The final atomic number results from adjusting for both types of decay; 82 in this case.
• The final mass number is determined mainly by alpha decay, here ending at 208.

This set of atomic and mass numbers corresponds to lead, which is the stable daughter product derived from the original thorium isotope. Understanding how emissions affect atomic structure is key to identifying daughter products.

Atomic Number

The atomic number is the count of protons in an atom's nucleus, defining the element's identity. Any change in this number means a change in the element itself. In radioactive decay, both alpha and beta emissions influence the atomic number in different ways.

• Alpha decay decreases the atomic number by 2 with each emission, due to the loss of 2 protons.
• Beta decay increases the atomic number by 1, as a neutron turns into a proton.

In our example, thorium loses 12 protons from alpha emissions (90 to 78), but gains 4 back through beta decay (78 to 82). Therefore, the atomic number of the final product is 82, indicating a change in the elemental composition.

Mass Number

The mass number represents the total number of protons and neutrons in an atom. It is crucial for determining isotopes of elements but differs from atomic weight. During radioactive decay, changes in mass number mostly occur during alpha particle emissions.

• Alpha decay reduces the mass number by 4 with each emission.
• Beta decay typically does not affect the mass number, as the transformation involves particles of equivalent mass.

For our thorium isotope, the mass number starts at 232. With six alpha particles emitted, the mass number reduces by 24, resulting in a new mass number of 208. Therefore, the mass of the thorium significantly decreases, which is essential for differentiating the daughter product from the parent element.