Which stable nuclei have a diameter of 7.46 fm?

검증된 단계별 안내

Step 1: Understand the relationship between the diameter of a nucleus and its mass number (A). The diameter of a nucleus is related to its radius, which can be approximated using the formula:

R = R

₀⋅A1/3, where

R

₀ is a constant approximately equal to 1.2 fm.

Step 2: Relate the diameter to the radius. The diameter is twice the radius, so

d = 2 \cdot R

. For a diameter of 7.46 fm, the radius is

R = 7.46 / 2 = 3.73

fm.

Step 3: Solve for the mass number (A) using the formula for the radius. Rearrange the formula to find

A = (R / R

₀)3. Substitute

R = 3.73

fm and

R

₀=1.2 fm.

Step 4: Calculate the approximate value of

A

. This will give the mass number of the nucleus, which corresponds to the number of protons and neutrons in the nucleus.

Step 5: Identify stable nuclei with the calculated mass number. Use a table of nuclides to find stable isotopes with the corresponding mass number. Stable nuclei are those that do not undergo radioactive decay under normal conditions.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Nuclear Stability

Nuclear stability refers to the ability of a nucleus to remain intact without undergoing radioactive decay. Stable nuclei have a balanced ratio of protons to neutrons, which helps to overcome the repulsive forces between protons due to their positive charge. Understanding the factors that contribute to nuclear stability is essential for identifying which nuclei are stable at specific sizes.

Nuclear Diameter

The diameter of a nucleus is a measure of its size, typically expressed in femtometers (fm), where 1 fm equals 10^-15 meters. The size of a nucleus is influenced by the number of nucleons (protons and neutrons) it contains, with larger nuclei generally having larger diameters. Knowing the typical diameters of stable nuclei helps in identifying specific nuclei based on their size.

Isotopes and Nuclei

Isotopes are variants of a chemical element that have the same number of protons but different numbers of neutrons, resulting in different mass numbers. Some isotopes are stable, while others are radioactive. Identifying stable isotopes with a specific diameter, such as 7.46 fm, requires knowledge of the isotopes of elements and their respective nuclear properties.