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31-1 Early Models of the 31-2 The Spectrum of Atomic 31-3 Bohr’s Model of the 31-3 Bohr’s Model of the Hydrogen Atom

Bohr′s model of Hydrogen Atom

n = 1, 2, 3, …..

Figure 31-7. The First Three Bohr Orbits 31-3 Bohr’s Model of the Hydrogen Atom

The spectrum of Hydrogen

How to create spectrum? • jumps from one orbit to the another: it gives up or absorbs (as a result of energy difference). • Different orbit has different . • The ppghoton is determined b y hf.

Electron Energies include: • Electric . • Kinetic energy. 31-3 Bohr’s Model of the Hydrogen Atom

Assumptions of the Bohr model:

• The electron in a hydrogen atom moves in a circular orbit around the nucleus. • Only certain orbits are allowed, where the angular in the nth allowed orbit is

• in allowed orbits do not radiate photons. Radiation is emitted when an electron changes from one orbit to another, with frequency given by the energy difference: 31-3 Bohr’s Model of the Hydrogen Atom

In order for an electron to move in a circle of radius r at speed v, the electrostatic force must provide the mv22 e required . Hence: ()= k rr2 e2 mv2 =− k 31 3 rn Adding the requirement gives: According Eq.11− 12 L= rmv ,

mrnn v= nh /2π ,

nh vnn ==, 1,2,3,... 31 − 4 2π mrn 31-3 Bohr’s Model of the Hydrogen Atom

Solving for rn and vn from these two equations, the allowed radii are:

h2 rnn==(),(),,,,2 1,2,3,... 31 − 5 n 4π 22mke

The speeds at the allowed radii are:

2π ke2 vn==, 1,2,3,... 31 − 6 n nh Example 1 Find the radius of the Hydrogen at the smallest orbit (Bohr Orbit).

Solution:

h2 r==(), n2 with n 1 n 4π 22mke (6.626× 10−34 ) 2 =×()12 4π 231922192 (9.109×× 10−−kg )(8.988 10 N . m / C )(1.602 × 10 ) =×5.29 10−11 m Examples Find the speed and kinetic energy of the electron in the Second Bohr orbit (n=2).

Solution: 2π ke2 v==,2 with n n nh 2π (8.99× 10922Nm⋅× / C )(1.60 10− 192 ) v = 2 2×× (6.63 10−34 Js . ) =×1.09 10−6 ms / 1) Speed

2π ke2 v==,2 with n n nh 2π (8.99×⋅ 10922Nm / C )(1.60 × 10− 192 ) v = 2 2(663102(6××.63 10−34 Js. ) =×1.09 10−6 ms /

2) Kinetic energy 1 K = mv2 222 1 =×(9.11 10−31kJkgms )(1. 09 × 10 6 / ) 2 =× 5. 4410 10− 19 J 2 31-3 Bohr’s Model of the Hydrogen Atom

The total mechanical energy of an electron in a Bohr orbit is the sum of its kinetic and potential energies. After some algebraic manipulation, and substituting known values of constants, we find for hydrogen atom:

1 EeVn=−(13.6 ⋅ ) , = 1,2,3,... n n2

1 eV = 1.60x10-19 Joule

The lowest energy is called the ground state. Most at room temperature are ihin the ground state. The Origin of Spectral Series in Hydrogen:

Figure 31-9a The Origin of Spectral Series in Hydrogen:

Figure 31-9b The Origin of Spectral Series in Hydrogen: Paschen Series

Figure 31-9c Active Exam 31-2 Absorbing a and what is the wavelength?

Find the wavelength of a photon that will be absorded, if a electron itis to rai se in a hdhydrogen a tom from the n= 3tttth3 state to the n=5 5tt state. Solution:

1) The energy at n=5 state is 11 EeVeVeV=−(13. 66) ⋅ ) =− (13.6) 6 ⋅ ) =− 0.544 5 n225 =−0.544 × 1.6 × 10−−19JJ =− 0.87 × 10 20

2) The energy a t n= 3 s ta te i s 1 E == −(13.6 ⋅eV ) = − 2.42 × 10−19 J 3 52 3) The energy difference is −19 Δ=E5,3 1.55 × 10 J 4) The wavelength is

−19 14 hf=Δ E5,3 =1.55 × 10 J ⇒ f = 2.34 × 10 Hz fcλλ=⇒ =1.28 × 10−6 m Chapter Summary

The total mechanical energy of an electron in a Bohr orbit is the sum of its kinetic and potential energies. For Hydrogen atom, the total energy is

1 EeVn=−(13. 66) ⋅ ), = 1, 2, 3,... n n2