Calculate the density of the nucleus of a 26^56 Fe atom, given that the nuclear mass is 9.229 ×

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Calculate the density of the nucleus of a 26^56 Fe atom,...

Calculate the density of the nucleus of a ${ }_{26}^{56} \mathrm{Fe}$ atom, given that the nuclear mass is $9.229 \times 10^{-23} \mathrm{~g}$. From your result, comment on the fact that any nucleus containing more than one proton must have neutrons present as well. (Hint: See Problem $2.106 .)$

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Key Concepts

Nuclear Density

Nuclear density involves calculating the mass of a nucleus divided by its volume. It requires using the relationship that density is mass per unit volume and applying it to the nuclear scale where the nuclear volume is typically approximated assuming a spherical shape. This concept illustrates that despite the small nuclear dimensions, the mass density is extremely high compared to everyday materials.

Nuclear Volume Estimation

Estimating the nuclear volume often relies on the concept that a nucleus can be modeled as a sphere with a radius that scales with the cube root of its mass number. With a typical formula R = r?A^(1/3), where r? is a constant (about 1.2 fm), one can derive a volume using the formula V = (4/3)?R³. This concept is essential for linking the size of the nucleus to its overall mass in density calculations.

Nucleon Composition and Stability

The composition of a nucleus in terms of protons and neutrons is fundamental in nuclear physics. The necessity of including neutrons in any nucleus with more than one proton reflects the role neutrons play in mitigating the electrostatic repulsion among positively charged protons. This enhances nuclear stability by allowing the strong nuclear force, which acts between all nucleons, to bind the nucleus together more effectively.

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