Download Hydrogen Spectrum

Hydrogen Spectrum
I.
OBJECTIVES.
II.
In this lab you will learn how to use a spectrometer in order to analyze the light emitted
by a light source. In particular you will complete the following tasks:
a. Examine the continuous spectrum emitted by an incandescent lamp.
b. Determine the wavelengths emitted by a hydrogen lamp.
c. Determine the cutoff wavelength of two optical filters.
II. EQUIPMENT.
Grating spectrometer, Hydrogen lamp, incandescent lamp, optical filters, optical jack.
III. INTRODUCTION.
a. Dual nature of light
Light can be described in two different ways using either classical physics or modern
(quantum) physics.
Classical physics describes light as a wave that can propagate through vacuum as well as
through matter. In vacuum light propagates with speed c  3 108 m/s . Associated with


every light wave is an oscillating electric and magnetic field vector (symbols E and B ,
respectively). The resulting wave is called electromagnetic (EM). Such a wave is shown


in fig.1. The electric field E is perpendicular to the magnetic field B , and both vectors
are perpendicular to the propagation direction of the wave. The electromagnetic wave
shown in fig.1 is periodic in space i.e. it consists of a repeating pattern (sinusoidal). The
distance between two adjacent wave crests (or wave troughs) is called the wavelength
(symbol  ). If we observe the EM wave at a particular point along the x-axis the electric


and magnetic fields E and B oscillate rapidly in time with a frequency f, defined as the
number of oscillations per second (unit: Hz). The wavelength and the frequency of an
EM wave are connected via the equation:
f c
(eqs.1)
EM waves have an enormous range of wavelengths and frequencies. This is illustrated in
fig.2. Depending on the wavelength EM waves encompass radio waves of all types (AM,
FM, TV, radar), infra-red (IR) light, visible light, ultraviolet (UV) light, x-rays, and
gamma-rays. Please note that visible light is only a very narrow section of the entire EMwave spectrum with wavelengths from 400 nm (violet) to 700 nm (red).
Note: 1 nm  109 m . This picture was proposed by James Clerk Maxwell.
Quantum physics describes light as a stream of particles called photons. These have
zero mass and travel in vacuum with speed c  3 108 m/s . Each photon carries an
energy given by the expression:
hc
E  hf 
(eqs.2)

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Here h is a constant known as Planck’s constant whose value is equal to:
h  6.63  10 34 J  s . When a photon is absorbed by an atom the photon disappears (the
formal term is annihilated) and all its energy is transferred to the atom. This model for
the description of light has been proposed by Albert Einstein.
The two pictures given above (classic and quantum) are complimentary to each
other, something like the two different faces of the same coin. Some effects such light
interference and light polarization are explained by the classic picture. Others, such as
the emission spectrum of Hydrogen and the photoelectric effect can only be explained
using the quantum picture.
b. Hydrogen atom
The main features of the emission spectra of hydrogen can be understood using the Bohr
model which makes the following assumptions:
1. The electron moves around the nucleus on circular orbits which a stable. The electron
does not loose energy by radiating electromagnetic waves as predicted by Maxwell’s
equations. These orbits have radii r1 , r2 , r3 , ... and corresponding energies E1 , E2 , E3 , ... .
2. An electron moving in an orbit of energy Ei can make a transition to an orbit of lower
energy E f
E
i
 E f  and conserve energy by emitting a photon of wavelength
 according to the equation:
hc
Ei  E f 
(eqs.3)

Here h is Planck’s constant and c is the speed of light in vacuum. h  6.63  1034 J  s
and c  3 108 m/s.
3. Each orbit of radius rn and energy En has an angular momentum mvrn which is given
by the equation:
Ln  n
h
2
where n  1, 2,3, ...
(eqs.4)
Based on the above assumptions the Bohr model predicts the following expression for the
energy of each electron orbit:
En  
R
, n  1, 2,3, ...
n2
(eqs.5)
Here R  2.173264  1018 J is known as Rydberg’s constant.
The radius of the electron orbit is given by the equation:
2
rn  ao n2 ,
n  1, 2,3, ...
(eqs.6)
Here ao  5.3  1011 m and is known as the “Bohr radius” of the hydrogen atom.
Equations 4, 5, and 6 tell express the fact that parameters such as the angular momentum,
energy , and orbit radius cannot take any value but can have only certain discreet values.
This effect is known as “quantization”.
A schematic diagram of the energies of the hydrogen orbits is shown in fig.3.
The possible transitions are grouped together according to the final state. In this
experiment you will record the wavelengths that correspond to the Balmer series which
includes transitions from the n  3, 4,5,... orbits to the n  2 orbit. The wavelengths of
the Balmer series fall in the visible part of the electromagnetic spectrum. The color of
light associated with the  3  2  transition is red, while that associated with the  4  2 
is green. In this experiment you will measure the wavelengths of the photons associated
with the  3  2  and  4  2  transitions in the Balmer series and compare the
experimental values with the calculated values predicted by the Bohr model.
hc
 1 1 
 En  E2  R  2  2  , n  3, 4,5,...
n
2 n 
(eqs.7)
IV. EXPERIMENTAL METHOD.
In this experiment you will determine the wavelengths of the light emitted by a hydrogen
lamp. For this task you will use the spectrometer shown in fig.4. This instrument uses a
diffraction grating to separate the various wavelengths of light. We will study this
optical component in the next experiment. The light to by analyzed by the spectrometer
enters through a slit at point A (see fig.4). The user observes the various wavelengths
through eyepiece B. A reflector C is used to illuminate the instrument’s internal
wavelength scale. The spectrometer is mounted on post D and is supported by tripod E.
The spectrometer wavelength scale reads in nanometers ( 1 nm  109 m )
V. PROCEDURE.
V1. Continuous emission spectrum of an incandescent lamp. Place the incandescent
lamp on an optical jack. Adjust the jack height so that the lamp is at the same height as
the entrance slit (point A in fig.4) of the spectrometer. The spectrum is continuous with
colors varying in color from the violet to the red. View the spectrum through the
eyepiece (point B in fig.4)
Record on the data sheet the wavelengths violet
Record on the data sheet the longest wavelength red
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V2. Place the hydrogen lamp at the entrance slit. The spectrum now is discrete and
consists of 4 separate sharp lines. These are known collectively as the Hydrogen Balmer
lines. Record the data sheet the wavelength as well as the color for each line.
V3. Place the red filter (#29) between the incandescent lam and the entrance slit A of the
spectrometer as shown in fig.5. This is known as a low pass filter in the sense that it
absorbs the shorter wavelengths and transmits the longer wavelengths of the incident
light. For filter #29 only the red part of the lamp’s spectrum is transmitted through the
filter. Record on the data sheet the shortest wavelength of the transmitted light 29 .
This is called the cut-off wavelength of the filter.
V4. Repeat the procedure described in section V7 for the orange filter (#15). For filter
#15 only the red and yellow part of the lamp’s spectrum is transmitted through the filter.
Record on the data sheet the shortest wavelength of the transmitted light 15 .
VI. FOR THE REPORT.
VI-1. List the wavelengths violet and red measured in section V1.
VI-2. List the four wavelengths of the hydrogen Balmer lines as well as their colors,
determined in section V2.
VI-3. List the cut-off wavelength 29 for the red filter (#29) measured in section V7.
VI-4. List the cut-off wavelength 15 for the orange filter (#15) measured in section V8.
VII QUESTIONS.
VII-1. Based on what you learned in this lab suggest a method to verify the presence of
hydrogen in the atmosphere of a distant star without sending a spacecraft.
VII-2. From equation 7 the calculated wavelengths of the Balmer lines are given by:
hc
calc 
 1 1 
R 2  2 
2 n 
Compare the experimental wavelengths of the Balmer lines measured in V2 with the
exp  calc
calculated values by calculating the quantity
 100
calc
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Fig. 1: A snapshot of an
electromagnetic wave propagating
along the x-axis (Halliday, Resnick and
Walker, Fundamentals of Physics)
Fig.2: Wavelengths and frequencies of the electromagnetic spectrum. (Halliday, Resnick,
and Walker, “Fundamentals of Physics”
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E
E5
E4
E3
Paschen Series
E2
Balmer Series
Lyman Series
Fig.3: Transitions among orbits in the hydrogen atom
B
A
E1
Incandescent
light source
C
A
D
Filter
Spectrometer
B
E
Observer
Fig.5: Experimental setup
used in section V3 and V4
Fig.4: Picture of the grating
spectrometer used in this
experiment
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