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##### general description

- Introduce the concept of absorption and emission line spectra and describe the Balmer equation to describe the visible lines of atomic hydrogen.

The first to recognize that white light is made up of the colors of the rainbow was Isaac Newton, who in 1666 passed sunlight through a narrow slit and then through a prism to project the spectrum of colors onto a wall. This effect was not only observed in the sky, of course, but earlier attempts by Descartes and others to explain it suggested that white light becomes colored when it is refracted, and the color depends on the angle of refraction. Newton clarified the situation by using a second prism to reconstitute white light, making the idea that white light was composed of separate colors much more plausible. He then took a monochromatic component of the spectrum produced by one prism and passed it through a second prism to find that no other colors were produced. That is, light of a single color does not change color when it is refracted. He concluded that white light is made up of all the colors of the rainbow and that when passing through a prism, these different colors are refracted at slightly different angles, separating them in the observed spectrum.

## Atomlinienspektren

The spectrum of hydrogen*Atom*, which became crucial to the first understanding of atomic structure more than half a century later, was first observed in 1853 by Anders Ångström in Uppsala, Sweden. His work was translated into English in 1855. Ångstrom, the son of a home minister, was a reticent man disinterested in the social life that revolved around court. Consequently, many years passed before his achievements were recognized at home and abroad (most of his results were published in Swedish).

Most of what is known about atomic (and molecular) structure and mechanics has been derived from spectroscopy. Figure 1.4.1 shows two different types of spectra. A solid or gas that glows at high pressure can produce a continuous spectrum (for example, blackbody radiation is continuous). An emission spectrum can be generated by a low pressure gas excited by heat or by collisions with electrons. An absorption spectrum is produced when light from a continuous source penetrates a cooler gas, which consists of a series of dark lines characteristic of the gas's composition.

##### Fraunhofer lines

In 1802, in England, William Wollaston discovered that the solar spectrum had small gaps: there were many thin dark lines in the rainbow of colors. These were much more systematically examined by Joseph von Fraunhofer beginning in 1814. He increased the dispersion by using more than one prism. He found an "almost uncountable number" of lines. He labeled the strongest dark lines A, B, C, D, etc. Between 1814 and 1823, Frauenhofer discovered some 600 dark lines in the high-resolution solar spectrum and labeled the main features with the letters A through K and the fainter lines with other letters (Table 1.4.1). Modern observations of sunlight can detect many thousands of lines. It is now clear that these lines are caused by absorption by the Sun's outer layers.

Designation | Element | Wave length (nm) |
---|---|---|

j | o_{2} | 898.765 |

Z | o_{2} | 822.696 |

A | o_{2} | 759.370 |

B | o_{2} | 686.719 |

C | H | 656.281 |

A | o_{2} | 627.661 |

D_{1} | Of | 589.592 |

D_{2} | Of | 588.995 |

D_{3}o d | Es | 587.5618 |

Fraunhofer lines are typical spectral absorption lines. These dark lines are created whenever there is a cold gas between a broadband photon source and the detector. In this case, a decrease in the intensity of the light is observed at the frequency of the incident photon, since the photons are absorbed and re-emitted in random directions, mostly in different directions than the original one. This results in a*absorption line*, because the narrow frequency band of light originally traveling towards the detector has been converted to heat or re-emitted in other directions.

On the other hand, if the detector sees photons emitted directly from a hot gas, then in general the detector sees photons emitted in a narrow frequency range by quantum emission processes in atoms in the hot gas, resulting in a*emission line*. Visible on the Sun are Fraunhofer lines of gas in the outer regions of the Sun that are too cold to directly produce emission lines from the elements they represent.

Bunsen, Kirkhoff, and others discovered that gases heated to incandescence emit light in a range of sharp wavelengths. Emitted light analyzed by a spectrometer (or even a simple prism) appears as infinitely narrow bands of color. these calls*line spectra*are characteristic of the atomic composition of the gas. The line spectra of various elements are shown in Figure 1.4.3.

## Die Balmer Hydrogen-Serie

Obviously, if a pattern can be discerned in the spectral lines of a given atom (relative to the mixture that the Fraunhofer lines represent), this could be a clue to the internal structure of the atom. Maybe you could build a model. Since the 1860s, great efforts have been made to analyze spectral data. The big discovery was made by Johann Balmer, a math and Latin teacher at a girls' school in Basel, Switzerland. Balmer had never worked with physics before and made the big discovery about it when he was in his sixties.

Balmer decided that the atom most likely to show simple spectral patterns was the lightest atom, hydrogen. Ångstrom measured the four visible spectral lines at wavelengths of 656.21, 486.07, 434.01, and 410.12 nm (Figure 1.4.4). Balmer focused precisely on these four numbers and discovered that they were represented by the phenomenological formula:

\[\lambda = b \left( \dfrac{n_2^2}{n_2^2 -4} \right) \label{1.4.1} \]

where \(b\) = 364.56 nm and \(n_2 = 3, 4, 5, 6\).

The first four wavelengths of the equation \(\ref{1.4.1}\) (with \(n_2\) = 3, 4, 5, 6) were in excellent agreement with Ångstrom's experimental lines (Table 1.4. 2). Balmer predicted that there are other lines in the ultraviolet that correspond to \(n_2 \ge 7\), and in fact some of them have already been observed without Balmer's knowledge.

\(n_2\) | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|

\(\lambda\) | 656 | 486 | 434 | 410 | 397 | 389 | 383 | 380 |

Kor | Putrefaction | investigator | blue | Indigo | remove | no visible | no visible | no visible |

The integer \(n_2\) in the Balmer series theoretically extends to infinity and the series represents monotonically increasing energy (and frequency) of absorption lines with increasing values of \(n_2\). Furthermore, the energy difference between consecutive lines decreases as \(n_2\) increases (1.4.4). This behavior converges towards the highest possible energy as shown in Example 1.4.1. If you plot the lines according to their \(\lambda\) on a linear scale, the spectrum looks like Figure 1.4.4; These lines are called**Serie Balmer**.

The general Balmer formula (equation \(\ref{1.4.1}\)) can be rewritten relative to the inverse wavelength commonly called*wave number*(\(\widetilde{\now}\)).

\[ \begin{align} \widetilde{\nu} &= \dfrac{1}{ \lambda} \\[4pt] &=R_H \left( \dfrac{1}{4} -\dfrac{1}{ n_2^2}\direita) \label{1.4.2} \end{align} \]

where \(n_2 = 3, 4, 5, 6\) and \(R_H\) is the Rydberg constant (discussed in the next section) equal to 109.737 cm^{-1}.

Furthermore, he conjectured that the 4 could be replaced by 9, 16, 25,... and it turned out to be true, but these lines, more in the infrared, were not discovered until the beginning of the 20th century, along with the ultraviolet lines. .

##### The wave number as a unit of frequency

The relationship between the wavelength and the frequency of electromagnetic radiation is

\[\lambda \nu= c \number \]

In the SI system of units, the wavelength (\(\lambda\)) is measured in meters (m), and since wavelengths are generally very small, the nanometer (nm) is often used, which is \(10^{-9 }\ ;m\). The frequency (\(\nu\)) in the SI system is measured in reciprocal seconds 1/s − which is called Hertz (after the discovery of the photoelectronic effect) and is represented by Hz.

It is common to use the reciprocal of the wavelength in centimeters as a measure of the frequency of radiation. This unit is called the wave number and is represented by (\(\widetilde{\nu}\)) and is defined by

\[ \begin{align*} \widetilde{\nu} &= \dfrac{1}{ \lambda} \\[4pt] &= \dfrac{\nu}{c} \end{align*} \nonumber \ ]

The wavenumber is a convenient unit in spectroscopy because it is directly proportional to energy.

\[ \begin{align*} E &= \dfrac{hc}{\lambda} \nonumber \\[4pt] &= hc \times \dfrac{1}{\lambda} \nonumber \\[4pt] &= hc\widetilde{\nu} \label{energy} \\[4pt] &\propto \widetilde{\nu} \end{align*} \]

##### Example 1.4.1: Balmer Series

Calculate the longest and shortest emitted wavelengths (in nm) in the Balmer series of the emission spectrum of the hydrogen atom.

**Solution**

The behavior of the Balmer equation (equation \(\ref{1.4.1}\) and Table 1.4.2) results in the value of \(n_2\), which has the longest wavelength (that is, the longest) (\(\lambda \)) is the smallest possible value of \(n_2\), which for this series is (\(n_2\)=3). That leads to

\[ \begin{align*} \lambda_{más larga} &= (364,56 \;nm) \left( \dfrac{9}{9 -4} \right) \\[4pt] &= (364,56 \;nm) \left( 1.8 \right) \\[4pt] &= 656.2\; nm \end{align*} \nonúmero \]

This is also called the \(H_{\alpha}\) line of atomic hydrogen and is bright red (Figure \(\PageIndex{3a}\)).

For the smallest wavelength, it should be noted that the smallest wavelength (higher energy) results in the limit of the largest (\(n_2\)):

\[ \lambda_{minor} = \lim_{n_2\rightarrow\infty}(364.56\;nm)\left(\dfrac{n_2^2}{n_2^2-4}\right)\nonumber\]

This can be solved byhospital rule, or alternatively, the limit can be expressed through the equally useful energy expression (equation \ref{1.4.2}) and is easily solved:

\[ \begin{align*} \widetilde{\nu}_{maior} &= \lim_{n_2 \rightarrow \infty} R_H \left( \dfrac{1}{4} -\dfrac{1}{n_2^ 2}\direita) \\[4pt] &= \lim_{n_2 \rightarrow \infty} R_H \left( \dfrac{1}{4}\right) \\[4pt] &= 27,434 \;cm^{- 1} \end{align*} \sinnúmero\]

Since \( \dfrac{1}{\widetilde{\nu}}= \lambda\) is converted to cm units, this results in 364 nm as the shortest possible wavelength for the Balmer series.

The Balmer series is particularly useful in astronomy because, due to the abundance of hydrogen in the universe, Balmer lines appear in many stellar objects and are therefore frequently seen and relatively strong compared to other elements.

## employees and tasks

miguel hunter(Professor Vigas,Physics department,University of Virginia)

(Video) A Level Chemistry Revision "The Mass Spectrometer"