Calculate the nuclear diameters of (a) ⁴He, (b) ⁵⁶Fe, and (c) ²³⁸U.

검증된 단계별 안내

Understand the problem: The nuclear diameter can be estimated using the empirical formula for the radius of a nucleus, which is given by \( R = R_0 A^{1/3} \), where \( R_0 \) is a constant approximately equal to \( 1.2 \times 10^{-15} \, \text{m} \) (1.2 femtometers), and \( A \) is the mass number of the nucleus. The diameter is then \( D = 2R \).

For part (a), ⁴He: The mass number \( A = 4 \). Substitute \( A \) into the formula for the radius: \( R = R_0 A^{1/3} \). Then calculate the diameter using \( D = 2R \).

For part (b), ⁵⁶Fe: The mass number \( A = 56 \). Use the same formula \( R = R_0 A^{1/3} \) to find the radius, and then calculate the diameter \( D = 2R \).

For part (c), ²³⁸U: The mass number \( A = 238 \). Again, use the formula \( R = R_0 A^{1/3} \) to find the radius, and then calculate the diameter \( D = 2R \).

Summarize the results: After substituting the values of \( A \) for each nucleus and performing the calculations, you will have the nuclear diameters for ⁴He, ⁵⁶Fe, and ²³⁸U. Ensure all units are consistent and express the final diameters in meters or femtometers as appropriate.

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주요 개념

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Nuclear Diameter

The nuclear diameter refers to the size of an atomic nucleus, which is typically on the order of femtometers (10^-15 meters). It can be estimated using empirical formulas that relate the number of nucleons (protons and neutrons) to the radius of the nucleus, often expressed as R = R₀A^(1/3), where R₀ is a constant and A is the mass number.

Mass Number

The mass number (A) of an atom is the total number of protons and neutrons in its nucleus. It is a crucial factor in determining the size of the nucleus, as larger mass numbers generally correspond to larger nuclear diameters. For example, helium (⁴He) has a mass number of 4, while iron (⁵⁶Fe) has a mass number of 56.

Empirical Formula for Nuclear Size

The empirical formula for estimating nuclear size, R = R₀A^(1/3), provides a way to calculate the radius of a nucleus based on its mass number. The constant R₀ is approximately 1.2 to 1.3 femtometers. This formula highlights the relationship between the number of nucleons and the overall size of the nucleus, allowing for comparisons across different elements.