Question

(a) Consider the following nuclides: ${}^{12}_{6}\text{C}$, ${}^{198}_{80}\text{Hg}$, ${}^{14}_{6}\text{C}$, ${}^{197}_{79}\text{Au}$. Group them into isotopes and isotones.

(b) How does the size of a nucleus depend on its mass number $A$? Hence prove that the density of nucleus is a constant, independent of $A$, for all nuclei.

Hint:

Isotopes share Z, isotones share N = A - Z. Use R = R0 A^(1/3) and note that A cancels in mass over volume.

Answer 1

Step 1: Understanding the Concept:
Isotopes have the same atomic number \( Z \) but different mass numbers \( A \). Isotones have the same number of neutrons \( N = A - Z \). Nuclear size depends on how many nucleons are packed in.

Step 2: Part (a), find \( Z \) and \( N \) of each nuclide:
\( {}^{12}_{6}\text{C} \): \( Z = 6,\ N = 12 - 6 = 6 \).
\( {}^{14}_{6}\text{C} \): \( Z = 6,\ N = 14 - 6 = 8 \).
\( {}^{198}_{80}\text{Hg} \): \( Z = 80,\ N = 198 - 80 = 118 \).
\( {}^{197}_{79}\text{Au} \): \( Z = 79,\ N = 197 - 79 = 118 \).
So the isotopes (same \( Z = 6 \)) are \( {}^{12}_{6}\text{C} \) and \( {}^{14}_{6}\text{C} \). The isotones (same \( N = 118 \)) are \( {}^{198}_{80}\text{Hg} \) and \( {}^{197}_{79}\text{Au} \).

Step 3: Part (b), size of the nucleus:
Experiments on scattering show that the nuclear radius grows with the cube root of the mass number:
\[ R = R_0 A^{1/3} \]
Here \( R_0 \approx 1.2 \times 10^{-15}\ \text{m} = 1.2\ \text{fm} \) is a constant.

Step 4: Density of the nucleus:
Each nucleon has a mass close to \( m \), the atomic mass unit (\( 1.66 \times 10^{-27}\ \text{kg} \)). So the mass of the nucleus is \( M = Am \).
The volume is \( V = \tfrac{4}{3}\pi R^3 = \tfrac{4}{3}\pi R_0^3 A \).
\[ \rho = \dfrac{M}{V} = \dfrac{Am}{\tfrac{4}{3}\pi R_0^3 A} = \dfrac{3m}{4\pi R_0^3} \]
The mass number \( A \) cancels. So \( \rho \) depends only on \( m \) and \( R_0 \), which are constants.
Putting in numbers, \( \rho \approx 2.3 \times 10^{17}\ \text{kg m}^{-3} \), the same for all nuclei.

Final Answer:
Isotopes: C-12 and C-14. Isotones: Hg-198 and Au-197. The radius varies as \( A^{1/3} \), and the density \( \rho = 3m/(4\pi R_0^3) \) is independent of \( A \). \[ \boxed{R = R_0A^{1/3},\quad \rho = \dfrac{3m}{4\pi R_0^3} \approx 2.3\times10^{17}\ \text{kg m}^{-3}} \]