2.8: The Bohr Model for the Earth-Sun System
- Page ID
- 154877
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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.