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 daughter product is lead (Pb) with atomic number 82 and mass number 208.

Step-by-step solution

Understand Alpha and Beta Decay

When a nucleus emits an alpha particle, it loses 2 protons and 2 neutrons, decreasing both its atomic number and mass number. An alpha particle emission decreases the atomic number by 2 and the mass number by 4. In beta decay, a neutron is transformed into a proton, increasing the atomic number by 1 while the mass number remains unchanged.

Calculate Effects of Alpha Decay

Thorium-232 undergoes alpha decay six times. Each alpha decay reduces the atomic number by 2 and the mass number by 4. After six alpha decays: \ Atomic number: \( 90 - 6 \times 2 = 78 \) \ Mass number: \( 232 - 6 \times 4 = 208 \).

Calculate Effects of Beta Decay

Next, we account for the four beta decay processes. Each beta decay increases the atomic number by 1 without changing the mass number. Thus, the updated atomic number is: \ Atomic number: \( 78 + 4 \times 1 = 82 \) \ Mass number remains 208.

Identify the Stable Daughter Product

The final atomic number is 82 and the mass number is 208. An element with an atomic number of 82 is lead (Pb). Therefore, the stable daughter product is lead-208.

Key concepts

Alpha Decay

Alpha decay is a natural process by which certain unstable atomic nuclei emit an "alpha particle" to become more stable. An alpha particle consists of 2 protons and 2 neutrons, similar to a helium nucleus. This decay type significantly alters the original atom. When an atom undergoes alpha decay, its atomic number decreases by 2. This means the number of protons is reduced, changing the element's identity. Additionally, the mass number, which is the total of protons and neutrons, decreases by 4.

• Example: An atom originally with an atomic number of 90 and a mass number of 232 will have these numbers changed to 88 and 228, respectively, after emitting one alpha particle.
• Energy Release: Alpha decay results in the release of energy, which is carried away by the emitted alpha particle, reducing the potential energy of the nucleus.

Beta Decay

Beta decay is another form of radioactive decay critical in the transformation of unstable nuclei. In beta decay, a neutron in the nucleus is transformed into a proton while releasing an electron or beta particle. This process results in an increase in the atomic number by 1, with the mass number remaining unchanged.

• Example: For a nucleus starting with an atomic number of 78, after one beta decay, the atomic number will be 79, while the mass number remains the same.
• Neutrinos: During beta decay, a small, nearly massless particle known as a neutrino is also emitted but does not affect the atomic or mass numbers.

Atomic Number

The atomic number ( Z ) of an element is a fundamental characteristic that defines what element the atom represents. It is equivalent to the number of protons in the nucleus of an atom. The atomic number is critical because it determines the chemical behavior of the element, placing it uniquely in the periodic table.

• Periodic Table: Elements are ordered in the periodic table based on their atomic numbers, starting with hydrogen ( Z = 1 ).
• Changes in Decay: As seen in radioactive decay, the atomic number decreases with alpha decay and increases with beta decay, reflecting the transformation of the element.

Mass Number

The mass number ( A ) of an atom is the total number of protons and neutrons in its nucleus. Unlike the atomic number, the mass number does not uniquely identify an element because different isotopes of the same element have the same atomic number but different mass numbers due to varying neutron counts.

• Example: Carbon can exist as carbon-12 or carbon-14, where the number denotes its mass number.
• Stability: During radioactive decay, the mass number helps calculate how the atom's nucleus changes and stabilizes over time, such as when it loses mass with alpha decay.

Radioactive Isotopes

Radioactive isotopes, or radioisotopes, are unstable variants of elements characterized by their atomic and mass numbers. These isotopes spontaneously emit particles and energy, undergoing radioactive decay to reach a stable state. This property makes them integral in fields like medicine, dating, and energy production.

• Unstable: The imbalance in the number of neutrons relative to protons often leads to instability.
• Applications: Radioisotopes are used in medical diagnostics (e.g., iodine-131 in thyroid treatment), carbon dating, and as tracers in biochemical research.
• Decay Process: Through processes like alpha and beta decay, radioisotopes transition into stable isotopes by adjusting proton-to-neutron ratios.