Question 43.1: The Volume and Density of a Nucleus Consider a nucleus of ma...
The Volume and Density of a Nucleus
Consider a nucleus of mass number A, containing protons and neutrons, each with mass approximately equal to m.
(A) Find an approximate expression for the mass of the nucleus.
(B) Find an expression for the volume of this nucleus in terms of A.
(C) Find a numerical value for the density of this nucleus.
(A) Conceptualize Imagine the nucleus to be a collection of protons and neutrons (Fig. 43.2). The mass number A counts both protons and neutrons.
Categorize Assume A is large enough that we can model the nucleus as spherical.
Analyze Because the masses of protons and neutrons are each approximated as m, the mass of the nucleus is approximately Am.
(B) Assume the nucleus is spherical and use Equation 43.1:
r=a A^{1 / 3} (43.1)
(1) V_{\text {nucleus }}=\frac{4}{3} \pi r^3=\frac{4}{3} \pi a^3 A
(C) Use Equation 1.1 and substitute Equation (1):
\rho \equiv \frac{m}{V} (1.1)
\rho =\frac{m_{\text {nucleus }}}{V_{\text {nucleus }}}=\frac{A m}{\frac{4}{3} \pi a^3 A}=\frac{3 m}{4 \pi a^8}Substitute numerical values:
\rho=\frac{3\left(1.67 \times 10^{-27} kg\right)}{4 \pi\left(1.2 \times 10^{-15} m\right)^3}=2.3 \times 10^{17} kg /m^3Finalize The nuclear density is approximately 2.3 \times 10^{14} times the density of water \left(\rho_{\text {water }}=1.0 \times 10^3 kg/m^3\right).
