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 atomic number is 82, and the mass number is 208.

Step-by-step solution

Understanding Alpha and Beta Decay

An alpha particle consists of 2 protons and 2 neutrons. Thus, when an alpha particle is emitted, the atomic number decreases by 2 and the mass number decreases by 4. On the other hand, beta decay involves the conversion of a neutron into a proton, causing the atomic number to increase by 1 while the mass number remains the same.

Calculate Changes from Alpha Decay

Since 6 alpha particles are emitted, the atomic number decreases by \(6 \times 2 = 12\), and the mass number decreases by \(6 \times 4 = 24\). The thorium initially has an atomic number of 90 and a mass number of 232.

Atomic Number after Alpha Decay

Subtract the change in atomic number from the original atomic number: \(90 - 12 = 78\).

Mass Number after Alpha Decay

Subtract the change in mass number from the original mass number: \(232 - 24 = 208\).

Calculate Changes from Beta Decay

For the 4 beta particles, the atomic number increases by \(4 \times 1 = 4\). The mass number remains the same.

Final Atomic Number

Add the increase in atomic number due to beta decay: \(78 + 4 = 82\).

Final Mass Number

The mass number remains unchanged from the result of alpha decay: 208.

Key concepts

Alpha Decay

Alpha decay is a type of radioactive decay where the nucleus of an atom emits an alpha particle. An alpha particle consists of two protons and two neutrons. When an atom undergoes alpha decay, it loses these four nucleons. This causes a change in both the atomic and mass numbers of the element:

• The atomic number decreases by 2 because of the loss of 2 protons.
• The mass number decreases by 4 due to the combined loss of 2 protons and 2 neutrons.

For example, in the case of thorium, which originally has an atomic number of 90 and a mass number of 232, emitting alpha particles results in a reduction in both these numbers. This shift alters the element's identity on the periodic table, moving it to a position 2 places to the left.

Beta Decay

Beta decay is another common form of radioactive decay where a neutron in the nucleus is transformed into a proton and an electron. The electron is emitted from the nucleus, becoming the beta particle. This process entails:

• The atomic number increasing by 1 as a neutron converts to a proton, adding a positive charge without affecting the overall number of nucleons.
• No change in the mass number since the total number of protons and neutrons (nucleons) remains unchanged.

In the context of the thorium isotope in the exercise, beta decay will increase the atomic number, thus shifting its place forward by one on the periodic table. It's important to understand that while beta decay changes the atomic number, the identity of the atom changes but its mass remains constant.

Thorium Isotope

An isotope refers to atoms that have the same number of protons but different numbers of neutrons. Thorium-232 is a specific isotope of thorium characterized by 90 protons (its atomic number) and a total of 232 nucleons (its mass number).

• The isotopes of a given element have similar chemical properties because they have the same electronic structure.
• However, their nuclear properties, such as stability and mode of decay, can vary significantly.

In the exercise, thorium-232 undergoes both alpha and beta decay. The initial element starts with an atomic number of 90 and a mass number of 232, which changes as it emits particles during the decay processes, eventually leading to a new, stable isotope.

Atomic Number

The atomic number is a fundamental characteristic of an element, representing the number of protons in the nucleus of an atom. It defines the element's identity and its position on the periodic table.

• The atomic number is crucial for determining the chemical behavior of an element.
• An increase or decrease in the atomic number changes the identity of the element.

For thorium in the exercise, the starting atomic number is 90. The emission of alpha particles reduces this number, whereas the emission of beta particles increases it. After undergoing 6 alpha decays and 4 beta decays, thorium's atomic number changes from 90 to 82.

Mass Number

The mass number of an isotope refers to the total number of protons and neutrons in its nucleus. This number is essential in identifying different isotopes of an element.

• Unlike atomic numbers, mass numbers are not found on the periodic table.
• It is used often in nuclear chemistry and helps in understanding isotopic changes during decay.

For thorium-232, the initial mass number is 232. During radioactive decay, specifically alpha decay, the mass number decreases. In this exercise, after emitting 6 alpha particles, the mass number decreases by 24 (as each alpha decay reduces it by 4), resulting in a final mass number of 208. Beta decay does not change the mass number, ensuring it remains constant after initial deductions.