An unstable nucleus undergoes alpha decay with the release of 5.52 MeV of energy. The combined mass of the parent and daughter nuclei is 452 u. What was the mass number of the parent nucleus?

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Understand the concept of alpha decay: In alpha decay, a parent nucleus emits an alpha particle (which consists of 2 protons and 2 neutrons, with a mass number of 4 and a charge of +2). The mass number of the parent nucleus decreases by 4, and the atomic number decreases by 2.

Use the energy-mass equivalence principle: The energy released during the decay (5.52 MeV) corresponds to a small loss in mass of the system. Use Einstein's equation, \( E = \Delta m c^2 \), to calculate the mass defect \( \Delta m \). Rearrange the equation to find \( \Delta m = \frac{E}{c^2} \).

Convert the energy released into mass units: Since 1 u of mass corresponds to 931.5 MeV of energy, the mass defect can be calculated as \( \Delta m = \frac{5.52 \text{ MeV}}{931.5 \text{ MeV/u}} \).

Determine the mass of the parent nucleus: The combined mass of the parent and daughter nuclei is given as 452 u. Subtract the mass defect \( \Delta m \) from this value to find the mass of the parent nucleus.

Find the mass number of the parent nucleus: The mass number is approximately the integer value of the mass of the parent nucleus (since the mass number represents the total number of protons and neutrons, which are whole numbers).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Alpha Decay

Alpha decay is a type of radioactive decay in which an unstable nucleus emits an alpha particle, consisting of two protons and two neutrons. This process reduces the mass number of the parent nucleus by four and the atomic number by two, resulting in the formation of a new element. The energy released during this decay is a result of the mass difference between the parent and daughter nuclei, as described by Einstein's mass-energy equivalence principle.

Mass-Energy Equivalence

Mass-energy equivalence, encapsulated in Einstein's equation E=mc², states that mass can be converted into energy and vice versa. In the context of nuclear reactions, the energy released during decay processes can be calculated from the mass defect, which is the difference between the mass of the parent nucleus and the total mass of the decay products. This principle is crucial for understanding the energy dynamics in nuclear decay, including alpha decay.

Mass Number

The mass number of an atomic nucleus is the total count of protons and neutrons it contains. It is an important identifier for isotopes of elements and plays a key role in nuclear reactions. In the case of alpha decay, knowing the mass number of the parent nucleus allows us to determine the mass number of the daughter nucleus and understand the changes in the atomic structure resulting from the decay process.