Question

What are isotone nuclei? In a nucleus average mass of nucleons is \(1.67\times10^{-27}\) Kg. Find the density of nucleus. [Given \(R_0 = 1.2\times10^{-15}\) m.]

Hint:

Isotones share the same neutron number. Use R = R0 A^(1/3) and mass = A m so density = 3m/(4 pi R0^3); the mass number A cancels.

Answer 1

Step 1: Isotone nuclei.
Nuclei that have the same number of neutrons (N) but different atomic numbers (Z) (and hence different mass numbers) are called isotones. Example: \(_{6}^{13}\text{C}\) and \(_{7}^{14}\text{N}\) both have 7 neutrons.
Step 2: Formula for nuclear density.
Nuclear radius: \(R=R_0 A^{1/3}\), where \(A\) is the mass number.
Nuclear mass: \(M=A\,m\), where \(m\) is the average mass of a nucleon.
Nuclear volume: \(V=\dfrac{4}{3}\pi R^3=\dfrac{4}{3}\pi R_0^3 A\).
Density:
\[\rho=\frac{M}{V}=\frac{A\,m}{\frac{4}{3}\pi R_0^3 A}=\frac{3m}{4\pi R_0^3}\]Note that \(A\) cancels, so nuclear density is the same for all nuclei.
Step 3: Substitute the values.
\(m=1.67\times10^{-27}\,\text{kg}\), \(R_0=1.2\times10^{-15}\,\text{m}\).
\(R_0^3=(1.2\times10^{-15})^3=1.728\times10^{-45}\,\text{m}^3\).
\[\rho=\frac{3\times1.67\times10^{-27}}{4\pi\times1.728\times10^{-45}}\]Step 4: Arithmetic.
Numerator \(=3\times1.67\times10^{-27}=5.01\times10^{-27}\).
Denominator \(=4\times3.1416\times1.728\times10^{-45}=21.72\times10^{-45}=2.172\times10^{-44}\).
\[\rho=\frac{5.01\times10^{-27}}{2.172\times10^{-44}}=2.31\times10^{17}\,\text{kg/m}^3\]\[\boxed{\rho\approx 2.3\times10^{17}\ \text{kg/m}^3}\](The nuclear radius constant is \(R_0=1.2\times10^{-15}\,\text{m}\); the value printed as \(10^{-13}\) on the paper is a misprint.)