Show that nuclear density of matter in nuclei is independent of mass number A.

To demonstrate that the nuclear density of matter in atomic nuclei is independent of the mass number (A), we first need to understand what we mean by nuclear density and how it relates to the structure of the nucleus itself.

Defining Nuclear Density

Nuclear density refers to the mass of the nucleus divided by its volume. It is typically expressed in units like grams per cubic centimeter (g/cm³). The mass number, A, is the total number of protons and neutrons in the nucleus, which gives us the total mass of the nucleus since protons and neutrons have similar masses.

Volume of the Nucleus

The volume of a nucleus can be approximated using the formula for the volume of a sphere:

• Volume (V) = (4/3)πr³

Here, r represents the radius of the nucleus. A useful empirical relation for nuclear radii is given by:

• r = r₀A^(1/3)

In this equation, r₀ is a constant (approximately 1.2 to 1.3 femtometers). Substituting this into the volume formula, we get:

• V = (4/3)π(r₀A^(1/3))³ = (4/3)πr₀³A

Calculating Nuclear Density

Now that we have the volume in terms of A, we can find the nuclear density (ρ) using the formula:

• Density (ρ) = Mass (M) / Volume (V)

Since the mass of the nucleus is approximately equal to the mass number A (in atomic mass units), we can write:

• ρ = A / [(4/3)πr₀³A]

Notice that the mass number A cancels out in this equation:

• ρ = 3 / (4πr₀³)

Resulting Independence from Mass Number

This final equation shows that nuclear density is a constant value that depends only on the constant r₀ and fundamental geometric factors. Therefore, it does not vary with A, meaning that nuclear density remains approximately the same for different nuclei, regardless of whether A is 12 (like in carbon) or 238 (like in uranium).

Conclusion

In summary, the nuclear density of matter in atomic nuclei is remarkably consistent across different elements. This characteristic is fundamental to our understanding of nuclear structure and stability, and it highlights the uniformity of the strong nuclear force that holds nucleons together within the nucleus.