Question

The nuclei of large atoms, such as uranium, with 92 protons, can be modeled as spherically symmetric spheres of charge. What is the radius of the uranium nucleus?

Answer 1

The radius of a nucleus can be estimated using the empirical formula
`\( r = r_0 A^(1/3) \)`, where
`\( r_0 \)` is a constant (about 1.2 femtometers), and ( A ) is the mass number of the nucleus. For uranium
`(\( A = 238 \))`, the calculation yields
`\( r \approx 7.4 \)` femtometers. This formula assumes a spherical distribution of charge within the nucleus, providing a simple model for estimating nuclear sizes.

Explanation:

The radius of a nucleus, particularly that of large atoms like uranium with 92 protons, can be approximated using the empirical formula
`\( r = r_0 A^(1/3) \)`, where
`\( r_0 \)` is a constant (around 1.2 femtometers) and ( A ) represents the mass number of the nucleus. For uranium, with a mass number of 238, this calculation results in a nucleus radius of approximately 7.4 femtometers. This model assumes a spherically symmetric distribution of charge within the nucleus, treating it as a compact, dense sphere.

In this context, the constant
`\( r_0 \)` serves as a scaling factor, reflecting the average distance between nucleons within the nucleus. The exponent ( 1/3 ) accounts for the volume dependence on the mass number, indicating that as the number of nucleons increases, the volume occupied by the nucleus grows, leading to a larger radius.

This simplified model is rooted in the liquid drop model of the nucleus, which considers the nucleus as a droplet of incompressible nuclear fluid. While it provides a straightforward means of estimating nuclear sizes, it's essential to note that more sophisticated models, such as those incorporating shell structure and quantum effects, exist for a more comprehensive understanding of nuclear properties.