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Objective:__

1) To use the emission spectrum of hydrogen atom in order to verify the relation between energy levels and the photon wavelength, and

2) to calculate Rydberg's constant, R = 1**.**097x10^{7}m^{-1}**.**

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Equipment:__

A computer with the internet connection, a calculator (The built-in calculator of the computer may be used.), a few sheets of paper, and a pencil

__Theory:__

When an electron in an atom receives some energy by any
means, it moves to a bigger radius orbit whose energy level fits that electron's
energy. Such atom is then said to be in an **excited state**. The excited
state is unstable however, and the electron returns to lower levels by giving
off its excess energy in the form of electromagnetic radiation ( visible light
is a small part of the E&M waves spectrum). **Max Planck**
showed that the **frequency of occurrence ( f )** of a particular **
transition** between two energy levels in an atom depends on the **energy
difference** between those two layers.

E_{n} - E_{m}
= hf

In this formula
*E _{n}*
is the energy of the n-th level,

Possibilities for the occurrence of **electron jump**
from one level to other levels are numerous. It depends on the amount of energy
an electron receives. An electron can get energized when a photon hits it, or
is passed by another more energetic electron that repels it, or by any other
means. The **electron return** can occur in one step or many steps depending
on the amount (s) of energy it loses. In each possibility, the
red arrow shows electron
going to a higher energy level, and the black arrows show possible return
occurrences.

**Hydrogen** is the simplest atom. It has **one proton**
and **one electron**. Click on the
following applet for a better
understanding of the transitions:
http://www.colorado.edu/physics/2000/quantumzone/lines2.html . In this
applet, if you click on a higher orbit than where the electron is orbiting, a
wave signal must be received by the electron (from outside) to give it energy to
go to that higher level. If the electron is already in a higher orbit and
you click on a lower orbit, then the electron loses excess energy and gives off
a wave signal before going to that lower orbit.

Also click on the following link**: **
http://www.walter-fendt.de/ph14e/bohrh.htm and try both options
of "Particle Mode" and "Wave Mode". You can put the mouse on the applet
near or exactly on any circle and change the orbit of the electron to anywhere
you wish; however, there are only discrete orbits whose
each circumference is an integer multiple of a certain wavelength.
It is at those special orbits that the applet shows
principal quantum numbers for the electron on the right side.

**For hydrogen atom**, possible transitions from the
ground state (E_{1}) to 2^{nd} state (E_{2}), 3^{rd}
state (E_{3}), and 4^{th} state (E4) are shown in **Fig. 1**.
The possibilities for **electron return** are also shown. The greater the
energy difference between two states, the more energetic the released photon is
when an excited electron returns to lower orbits. If the return is very
energetic, the wavelength may be too short to fall in the visible range and
cannot be seen in the spectroscope. Some transitions are weak and result in
larger wavelengths in the infrared region that cannot be seen either. However
some intermediate energy transitions fall in the visible range and can be seen

__Grouping of the Transitions:__

**Transitions** made from **higher levels** to the **
first orbit** form the **Lyman Series.**

**Transitions** made from **higher levels** to the **
second orbit** form the **Balmer Series.**

**Transitions** made from **higher levels** to the **
third orbit** form the **Paschen Series.**

**Transitions** made from **higher levels** to the **
fourth orbit** form the **Pfund Series.**

__Emission and
Absorption Spectra__

A hot gas emits light because of
the energy it receives by any means to stay hot. As was mentioned earlier, the
received energy by an atom sends its electrons to higher levels, and in their
returns, the electrons emit light at different wavelengths. The emitted
wavelengths can be observed in a prism spectrometer in the form of a few lines
of different colors. *Each element has its own unique spectral lines
that can be used as an ID for that element.* Such spectrum coming from a

__
__

**For white light** entering a
spectrometer **the spectrum is a continuous band of rainbow colors**. This
continuous band of colors in a spectrometer ranges from violet to red and gives
the following colors**: ** violet, blue, green, yellow, orange, and red. Light
emitted from the Sun contains so many different colors (or electronic
transitions) that its spectrum gives variety of colors changing
gradually from violet to red**. It contains so many different violets, blues,
greens, yellows, oranges, and reds that it appears continuous**.

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Procedure:__

Click on the following link**: **
http://mo-www.harvard.edu/Java/MiniSpectroscopy.html . The "Mini
Spectroscopy" Applet appears. At the very top it has a dropdown window
that allows you chose different hot gases. First
Select the hydrogen gas. The very top picture is what you see of
hot (excited) hydrogen if you view it through a spectrometer. All a
spectrometer does is that it passes the light emitted from an excited gas
through a prism (a triangular piece of glass) causing different colors to
separate. Under the separated colors it show a scale with
nm or Angstrom
graduations. 1nm = 10^{-9}m and 1Angstrom = 10^{-10}m.
This applet is calibrated in
nm. For hydrogen, you should see a red
band at about 650nm, a light blue at about 490nm , a dark
blue at about 440nm, and a violet color at
about 410nm..

- Read the exact values of the wavelengths
from the peaks on the second graph and record them.
If you place the mouse on the second graph an move it, a vertical line appears
and helps you locate the peaks exactly and read the wavelength exactly.
If the top figure does not show you the dark blue and the violet colors
clearly, click on the following applet and you will see them better
**:**http://online.cctt.org/physicslab/content/PhyAPB/lessonnotes/dualnature/discharge/index.html . - Use the formula for Balmer Series above and calculate the Rydberg's constant
**R**by using**each wavelength**you obtained for**hydrogen**atom. - Average the 4 values you obtained for
**R**in the previous step. This is your**measured value**of the constant. - Compare it with the accepted value of
R = 1
**.**097x10^{7}m^{-1}by calculating a %error. - Click on the 2nd link and observe the emission spectra for other atoms. Why other atoms have more transitions (spectral lines) in the visible region? Give this an explanation as part of your conclusion.

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Data:__

__Given: __

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__

** **
R_{accepted}
= 1**.**097x10^{7}m^{-1}

__Measured:__

The hydrogen visible wavelengths are:

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λ_{62} = ? nm, λ_{52}
= ? nm, λ_{42} = ? nm, λ_{32}
= ? nm. ( λ_{62}
means the wavelength corresponding to the transition from n=6 to n=2.)

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Calculations:__

Use the Balmer Series equation to calculate R for each of the measured wavelengths. Next, find the average value of R. This gives the measured value for R.

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Comparison of the Results:__**
**Calculate a %error on R using
the usual %error formula.

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Conclusion:__ To be
explained by students. Also, explain why
heavier elements have more transitions visible to us.

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Discussion:__ To be
explained by students