2.8: The Bohr Model for the Earth-Sun System

Last updated
Save as PDF

Page ID: 154877

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

Data for the earth-sun system assuming a circular earth orbit:

\[ \begin{matrix} \text{Mass of the earth:} & Me = 5.974 (10)^{24} \text{kg} & \text{Mass of the sun:} & Ms = 1.989 (10)^{30} \text{kg} \\ \text{Earth orbit radius:} & r = 1.496 (10)^{11} m & \text{Gravitational constant:} & G = 6.674 (10)^{-11} \frac{ \text{N m}^2}{ \text{kg}^2} \\ \text{Planck's constant:} & h = 6.62608 (10)^{-34} \text{J s} \end{matrix} \nonumber \]

Assuming the earth executes a circular orbit of radius r about the sun and has a deBroglie wavelength given by h/mv, yields a quantum mechanical kinetic energy for the earth which is the first term in the total energy expression below. The potential energy of the earth-sun interaction is well-known and is the second term in the total energy expression.

\[ \begin{matrix} E = \frac{n^2 h^2}{8 \pi^2 Me r^2} - \frac{G Me Ms}{r} & \text{where n = 1, 2, 3, ...} \end{matrix} \nonumber \]

Setting the first derivative of the energy with respect to r equal to zero, yields the allowed values of r in terms of the quantum number, n.

\[ \begin{matrix} \frac{d}{dr} \left( \frac{G Me Ms}{r} - \frac{G Me Ms}{r} \right) = 0 & \text{has solution(s)} & \frac{1}{4} n^2 \frac{h^2}{G \left[ Me^2 \left( Ms \pi^2 \right) \right]} \end{matrix} \nonumber \]

Substitution of this value of r in the total energy expression yields the energy of the earth-sun system as a function of the quantum number, n, Planck's constant, the gravitational constant, and the masses of the earth and the sun.

\[ \begin{matrix} E = \frac{n^2 h^2}{8 \pi^2 Me r^2} - \frac{G Me Ms}{r} & \text{by substitution, yields} & E = \frac{-2}{n^2 h^2} \pi^2 Me^3 G^2 Ms^2 \end{matrix} \nonumber \]

Given the radius of the earth's orbit listed above, calcuate the earth's quantum number.

\[ \begin{matrix} r = \frac{1}{4} n^2 \frac{h^2}{G \left[ Me^2 \left( Ms \pi^2 \right) \right]} & \text{has solution(s)} & \begin{pmatrix} \frac{-2}{h} \sqrt{G} Me \sqrt{Ms} \pi \sqrt{r} \\ \frac{2}{h} \sqrt{G} Me \sqrt{Ms} \pi \sqrt{r} \end{pmatrix} = \begin{pmatrix} -2.524 \times 10^{74} \\ 2.524 \times 10^{74} \end{pmatrix} \end{matrix} \nonumber \]

The positive root n is used to calculate the energy of the earth-sun system.

\[ \begin{matrix} n = 2.524 (10)^{74} & E = \frac{-2}{n^2 h^2} \pi^2 Me^3 G^2 Ms^2 & E = -2.65 \times 10^{33} \text{J} \end{matrix} \nonumber \]

According to the virial theorem the classical expression for the energy of the earth-sun system with earth orbit radius r is half the potential energy. Note that this gives a value which is in agreement with the Bohr model for the earth-sun system. Is this a legitimate example of the correspondence principle?

\[ \begin{matrix} E = - \frac{G Me Ms}{2r} & E = -2.65 \times 10^{33} \text{J} \end{matrix} \nonumber \]

*Johnson and Pedersen, Problems and Solutions in Quantum Chemistry and Physics, pages 26-27.