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% ---------------------Bohr_Atom.m ---------------------------
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% 				Atomic Spectra and the Bohr Atom
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%				Chemistry 455
%				Professor James B. Callis
%				Summer Quarter 1997
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% Press any key to continue
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% 		The Discovery of the Electron
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% J. J. Thompson analyzed the cathode rays of a Crooke's tube
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% Measured the charge to mass ratio:
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% q/m =1.7588e11 coulombs/kg (modern value)
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% Deduced the mass to be ~ 6e(-31) kg - tiny.
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% Millikan oil droplet experiment refined mass and charge to 
% be 9.109e(-31) kg and 1.60219e(-19) C.
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% The Discovery of the Nucleus
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% Ernest Rutherford analyzed the scattering of alpha particles 
% from thin foils.
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% Results could only be explained by a tiny dense nucleus.
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% The end of the "plum pudding" model.
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% Proposed the planetary model of the atom, with an electron 
% orbiting about the nucleus.
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% Based on the similar form of gravitational and coulombic
% forces.
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% Problems With the Planetary Model
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% The classical theory of Electromagnetism predicts that a 
% charged body in cyclic motion will radiate energy. 
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% Thus, the elctron is predicted to lose energy by radiating
% and spiral into the nucleus. 
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% Emission spectra of excited atoms are predicted to have a 
% continuous distribution of frequencies. 
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% Real atoms, in contrast, radiate energy at a series of
% discrete wavelengths.  
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% The Empirical Analysis of the Visible Spectrum by Balmer
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% Balmer noted that the visible lines were found at the 
% following wavelengths:
% nu_bar = 15,233, 20,565, 23,033, 24,373 and 25,182 cm-1
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% where the wavelengths are expressed as inverse wavelengths
% or wavenumbers
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% Balmer attempted to associate with each wavelength an 
% integer. 
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% He found that there was a linear relationship between the
% square of the assigned integer and the frequency. 
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% This relationship is known as the Balmer formula.
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% MATLAB Can Help You Discover the Balmer Formula
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% First we assemble the known wavelengths into a vector, nu-bar:
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nu_bar = [15233, 20565, 23033, 24373, 25182];
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% Then we assign a quantum number to each member of the array:
num = 3:7;
% Now we make a plot of nu-bar vs num
plot(num, nu_bar) 
xlabel('num')
ylabel('Emission Frequency, wavenumbers')
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% Clearly this Doesn't Yield a Linear Relationship
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% let's try plotting the inverse integers:
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num_inv=1./num;
% Now we make a plot of nu-bar vs num
plot(num_inv, nu_bar) 
xlabel('Inverse of Interger Number')
ylabel('Emission Frequency, wavenumbers')
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% Clearly There Is Still Not a Linear Relationship
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% let's try plotting the inverse integers squared:
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num_inv_sq= (1./num).^2;
% Now we make a plot of nu-bar vs num
plot(num_inv_sq, nu_bar) 
xlabel('Inverse of Integer Number Squared')
ylabel('Emission Frequency, wavenumbers')
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% The Above Works Much Better
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% let's try a different set of inverse integers squared:
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num = 4:8;
num_inv_sq= (1./num).^2;
% Now we make a plot of nu-bar vs num_inv_sq
plot(num_inv_sq, nu_bar) 
xlabel('Inverse of Integer Number Squared')
ylabel('Emission Frequency, wavenumbers')
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% Summary
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% We have seen that the Balmer formula fits the atomic emission
% spectrum of hydrogen better than any of the other models 
% we tried. Moreover, it appears that the constants, 109680 and 
% intial num = 3 seem to give an uniquely good linear relation-
% ship. 
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% Thus, we have used MATLAB as an effective tool to rule out
% various functional forms. This is one of the major jobs of 
% scientists: to disprove hypothesis. For now we will accept
% the Balmer formula as a provisional model. 
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% Limitations of the Balmer Formula and Its Generalization, the 
% Rydberg Formula
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% The problem with these formulae (models) is that they are 
% purely empirical, and there is no underlying explanation 
% of the macroscopic observations in terms of the microscopic 
% behavior of the atoms. 
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% Also, these models offer no explanation of the Rydberg constant
% in terms of other, more fundamental constants of nature.
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% Finally, there is no explanation of why the relationships
% involve integers. 
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% The Bohr Atom
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% Assumptions:
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% The atom consists of an electron orbiting about the nucleus.
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% The total energy of the system is the sum of the kinetic 
% and potential energies.
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% The potential energy arises from the Coulombic attraction of 
% electron and nucleus.
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% The angular momentum of the electron is quantized.
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% With these assumptions, Bohr derived an expression for the
% energy of the one-electron atomic system.
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% It was dependent on the inverse of the square of the quantum 
% number governing the quantization of the angular momentum.
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% It rationalized the line spectra of one-electron atoms and 
% explained the Rydberg constant in terms of more fundamental
% physical constants of nature. 
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% Shortcommings of the Bohr Atom
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% It is incapable of explaining the spectra of atoms and ions
% with more than one electron. 
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% There is no explanation of why the angular momentum should
% be quantized.
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% End of Lecture 2
% Press any key to clear screen and end session.
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