CAcT HomePage

## Hydrogen Spectra

### Skills to develop

- Explain the spectrum from hydrogen gas.
- Describe Rydberg's theory for the hydrogen spectra.
- Interpret the hydrogen spectrum in terms of the energy states of electrons.

# Hydrogen Spectra

Hydrogen, the simplest but the most abundant element in the universe is
also the most studied element. During the early development of science,
people have been investigating light emitted by a heated tubes of hydrogen
gas. When these lights passes prisms, they saw some lines in the visible
region.
The picture shown here is a brilliant star taken by the
Hubble telescope exploring the universe. This brilliant star may be one
of the largest mass of hydrogen.

##
Hydrogen spectrum in the visible region

When a hydrogen gas is heated or excited by electric means, the visible
spectrum consists of lines. The wavelengths (*wl*) of these
lines are given below:
*wl*** 656 486 434 410 397 **nm
These lines are shown here together with lines emitted by hot gases of Hg and
He. These lines are called the **Balmer Series**, because Balmer saw some
regularity in their wavelength, and he has given a formula to show
the regularity.
####
The Balmer Series

Do you see any regularity in the wavelength?
Well, J.J. Balmer analyzed these lines and he gave the following relationship:
**
***n*^{2}
*wl* = 364.56 -------- nm
*n*^{2} - 2^{2}

for the regularity in terms of integers *n*.
For some integers of *n*, you can confirm the *wl* to be
**
***n* = 3 4 5 6 7 8 9 10...
*wl* = 656 486 434 410 397 389 383 380... nm
red green blue indigo violet (not visible)

As the *n* increase, the lines are getting closer together.
If you plot the lines according to their *wl* on a linear scale,
you will get the apparence of a spectrum as observed by experimentalists.

For this reason, these lines are called the **Balmer series**.

#### The Rydberg Formula

Rydberg inverted both sides of Balmer's formula and gave
**
1 1 1
--- = ***R*_{H} (-- - --)
*wl* 2^{2} *n*^{2}

This is known as the **Rydberg formula**, and *R* is known as the
**Rydberg constant**, its numerical values depends on the units used
**
***R*_{H} = 0.010972 nm^{-1}

= 10972 mm^{-1}

= 109721 cm^{-1}

= 10972130 m^{-1}

This formula shows that if you plot 1/*wl* as a function of
1/*n*^{2}, you will get a straight line. The reciprocal
of *wl* (1/*wl*) is the number of waves per unit length, and
it is called the wavenumber.

####
Other Series

The results given by Balmer and Rydberg for the spectrum in the visible
region of the electromagnetic radiation start with *n* = 3, and the
other integer is 2.
Is there a series with the following formula?

**
1 1 1
--- = ***R*_{H} (-- - --)
*wl* 1^{2} *n*^{2}

If the law of nature is simple and regular, a series should exist, and the
values for *n* and wavenumber (*wn*) should be:
**
***n* = 2 3 4 5 ...
*wl* = 121 102 97 94 nm
*wn* = 82291 97530 102864 105332 ...cm^{-1}

Do you know in what region of the electromagnetic radiation these lines are?
Of course, these lines are in the UV region, and they are not visible,
but they are detected by instruments. These lines form a **Lyman series**.
####
A Generalized Rydberg Formula

Existences of the Lyman series and Balmer's series sugest the existance
of more series, and a generalized formula is suggested.
**
1 1
***wn* = *R*_{H} (--- - ---)
*n*_{f}^{2} *n*_{i}^{2}

Actually, the series with *n*_{f}^{2} = 3, and
*n*_{i}^{2} = 4, 5, 6, 7, ... is called Pashen series.
A SciTech Presentation of Hands-On-Atom provides some fun things to do
regarding the hydrogen spectra. Click it to try out if you wish.

##
Confidence Building Questions

**Calculate the wave numbers of the lines with the longest
wavelength in the Pashen series.**
**Discussion:**

Calculate the wavelength for three lines in this series.
What region are these lines?

**Draw an energy level diagram to show the transition for the
emission of the various series of lines by hydrogen.**
**Discussion:**

A diagram showing the energy levels is shown here.

**
==== ..... very closely spaced lines
---- ***n* = 3
---- *n* = 2
---- *n* = 1

Interpretation of the hydrogen spectrum led to the development
of quantum mechanics.
**
Among all possible photons emitted by hydrogen atoms, what is the shortest
wavelength possible?
**
**Discussion:**

The photo with the highest energy from a hot hydrogen gas is in the Lyman
series. The wavelength is about 91 nm. Confirm this by using the following
equations.

**
1 1
***wn* = *R* (--- - ---)
*n*_{f}^{2} *n*_{i}^{2}

Spectra of Elements shows the spectra of elements in the periodic table.
©cchieh@uwaterloo.ca