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Home , Discrete spectrum

and Atomic Structure

A spectrometer is an instrument for measuring the of light. The pattern of wavelengths recorded is the called a . Spectral analysis provided a powerful tool to study the structure of Spectroscopy and Atomic Structure We have seen that heated objects emit a radiation. Gases in a discharge tube absorb and emit discrete wavelengths (discrete spectral lines). Every gas emits and absorbs a unique discrete spectrum. The absorption and emission spectra differ: Every absorbed is emitted but not every emitted wavelength is absorbed (???) Spectral Lines of Hydrogen

Balmer Series: Balmer looked at the visible spectral lines for hydrogen and empirically deduced a formula:

# n2 & ! = C , m = 2, n = 3,4,5....) 2 $% n2 " m2 '(

!(n = )) = C2 = 364.56nm

C2 is called the convergence limit. The discrete lines converge to this limit. Spectral Lines of Hydrogen

Other series of spectral lines in hydrogen were found for different values of m: Lyman Series Paschen Series # n2 & # n2 & ! = C , n = 2,3,4....) ! = C , n = 4,5,6....) 1 $% n2 " 12 '( 3 $% n2 " 32 '(

Bracket Series Pfund Series # n2 & # n2 & ! = C , n = 4,5,6....) ! = C , n = 6,7,8....) 4 $% n2 " 42 '( 5 $% n2 " 52 '(

General Formula: (describes every observed of hydrogen)

1 # 1 1 & = R% 2 " 2 ( , m > n ! $ n m ' 1 / R = 91.18nm R: Rydberg constant Spectroscopy and Atomic Structure

Classical model: • Rutherford had demonstrated that atoms have a small, positively charges nucleus. • Negatively charge electrons orbit the nucleus. • According to Maxwell’s equations, an oscillating charge emits radiation. The orbiting electron should emit a continuous spectrum. • As the electrons radiate, they lose energy, and their orbits decay until they fall into the nucleus! Bohr Model of the

Bohr’s postulates: • Electrons orbit the nucleus and feel the Coulomb force (as in the Rutherford model). • Only certain orbits are stable. An electron in these orbits does not radiate energy. It is said to be in a stationary state of energy. • An electron jumps between stationary states by absorbing or emitting a photon of energy equal to the energy difference between the two stationary states:

Ei ! E f = hf

• The of an electron in a stable orbit (stationary state) is quantized in integer multiples of h/2!.

nh L = = n! 2! Bohr Model of the Atom Bohr model of the : Coulomb force on the electron: ke2 mv2 ke2 F = = ma = v2 = r 2 r mr If the electron is moving in a circular orbit, then from quantization of angular momentum, n2 2 L = mvr = n! 2 ! v = 2 2 m r Equating the two expressions for v we obtained quantized radii of the allowed stable orbits:

2 2 2 n ! 2 ! rn = 2 = n a0 , a0 = 2 = 0.0529nm mke mke

a0 is the Bohr radius (orbit corresponding to n=1). This radius agrees with the experimentally measured radius of the hydrogen atom! Bohr Model of the Atom Bohr model of the hydrogen atom: Coulomb force on the electron: ke2 mv2 ke2 F = = ma = v2 = r 2 r mr Total energy of the electron = kinetic energy + potential energy 1 ke2 1 ke2 ke2 ke2 E = K + V = mv2 ! = m ! = ! 2 r 2 mr r 2r Inserting the quantized orbital radii, 2 rn = n a0

2 2 ke E1 ke E = ! 2 = ! 2 , E1 = 2 = 13.6eV 2a0n n 2a01

E1: lowest energy ground state( corresponding to n=1). Energy states with quantum number n > 1: excited states. Bohr Model of the Atom Bohr model of the hydrogen atom:

2 2 ke E1 ke E = ! 2 = ! 2 , E1 = 2 = 13.6eV 2a0n n 2a01 The negative sign indicates that the electron is bound to the nucleus due to Coulombs force.

The energy required for the electron in the ground state to just escape the attraction of the proton is the ionization energy:

Eion = E! " E1 = 13.6eV

This agrees with the measured ionization energy for hydrogen! Bohr Model of the Atom Bohr model of the hydrogen atom:

2 2 ke E1 ke E = ! 2 = ! 2 , E1 = 2 = 13.6eV 2a0n n 2a01 Frequency of light absorbed or emitted when the electron jumps between energy levels: " E E % E " 1 1 % hf = E ! E = ! 1 + 1 f = 1 ! m n #$ m2 n2 &' h #$ n2 m2 &'

1 f E1 # 1 1 & = = % 2 " 2 ( , ! c hc $ n m ' 1 1 = 91.18nm = E1 hc R This formula agrees exactly with the general Balmer formula! Bohr Model of the Atom Bohr model of the hydrogen atom: Energy level diagram

1 f E1 # 1 1 & = = % 2 " 2 ( ! c hc $ n m '

• When an electron jumps to the ground energy state it cannot jump down any further. It will remain in the ground state.

• If an electron is in the ground state. Transitions from 1 to n are observed but transitions from 2 to n will not be observed in the absorption spectrum. However, transitions from n to 2 and n to 1 will be observed in the . Hence the difference between absorption and emission spectra. Bohr Model of the Atom Bohr model of the hydrogen atom:

2 2 ke E1 ke E = ! 2 = ! 2 , E1 = 2 = 13.6eV 2a0n n 2a01 Bohr’s model works for other hydrogen-like atoms in which all but one electron has been removed. For a single electron orbiting a nucleus of charge Ze,

2 2 2 2 ke Z E1 Z ke E = ! 2 = ! 2 , E1 = 2 = 13.6eV 2a0n n 2a01

n2a r = 0 n Z For multi-electron atoms, the orbits are more complex, so Bohr’s simple model with circular orbits does not apply. Bohr Model of the Atom Bohr model of the hydrogen atom: Find the energy quantum number, speed and energy of a 500nm diameter hydrogen atom.

r d / 2 250 nm n = n = = = 68.75 a a 0.0529 nm 0 0

mvrn = n! n! ! " 1% v 2.19 ( 106 m/s ! v = = = 1 = = 3.2 ( 104 m/s mr ma #$ n&' 69 69 n 0

E1 E1 13.60 eV E69 = ! 2 = ! 2 = ! 2 = !0.0029 eV n 69 69 ( ) ( ) Bohr Model of the Atom Bohr model of the hydrogen atom: If the energy of a particular transition in the Helium Paschen series is 2.644 eV, find the corresponding transition, i.e. initial and final n values.

# & 2 1 1 !E = E f " Ei = Z E1 % 2 " 2 ( $ n f ni ' 1 1 !E 2 = 2 " 2 ni n f Z E1 "1/2 # 1 !E & ni = % 2 " 2 ( $ n f Z E1 '

where n f = 3 for Paschen Series, Z = 2 for He "1/2 # 1 2.644 eV & n = % " ( = 4 i 32 22 13.6 eV $ ( )' Bohr Model of the Atom

Find the ionization energy, and wavelength for the nf = 4 spectral lines of Be3+.

# & hc 2 1 1 E f ! Ei = = Z E1 % 2 ! 2 ( " $ n f ni '

n f = 4 for Brackett, ni = ), Z = 4 # 1 1 & E = 42 13.6 eV ! = 13.6eV ) ( )$% 42 )2 '( hc 1240 eV nm " = = = 91.2nm ) E 13.6 eV c 3* 108 m / s f = = = 3.29 * 1015 s!1 ) " 91.2 10!9 m ) * Correspondence Principle

The correspondence principle formulated by Bohr states that quantum physics should correspond to classical physics in the regime where classical theory is known to be valid.

Roughly speaking the classical regime occurs when the quantum numbers of a system are very large.

In general classical physics does not simply emerge from in the way that Newtonian dynamics emerges from relativistic dynamics at small velocities.

Connecting the classical limit to quantum theory is a tricky business and the subject of much research even today. Frank-Hertz Experiment In 1914, Franck and Hertz directly measured the energy quantization of atoms via the inelastic scattering of electrons.

Measure current of electron beam (I) vs. accelerating grid voltage (V) inside a glass tube filled with mercury gas Frank-Hertz Experiment Results: A series of peaks and dips are seen in the current as a function of accelerating voltage

Explanation: • As V is increased electron energy increases. • Electrons can reach the collector if they overcome the retarding potential of 1.5V. At this point current starts to increase. • As V is increased further, electrons gain enough energy to excite the mercury atoms to their first excited state and hence lose energy. • These electrons cannot overcome the retarding potential and the current dips. Frank-Hertz Experiment

• As V is increased further, the electrons again reach an energy that excites the mercury atoms and hence lose energy again. • This repeated process causes the rise and dips of current.

• Each dip corresponds to a transition between quantized energy levels in mercury

The energy of the electrons at which the dips occur correspond to the first excitation energy of mercury (4.9eV)