Question

To measure the wavelength of lines in atomic emission and absorption spectra, one uses a spectroscope. Two lines in the yellow emission spectra of hot Na metal, the so-called sodium-D lines, are used to calibrate this instrument. If the wavelength of one of these lines is \(5890 \AA\), find the energy of the electronic transition associated with this line, \(\mathrm{h}=6.62 \times 10^{-27}\) erg-sec.

Short answer

The energy of the electronic transition associated with the given wavelength is approximately \(3.36 \times 10^{-12}\) erg.

Step-by-step solution

Write down the known values

We know the following values: - Wavelength, λ = \(5890 \AA\) (angstroms) - Planck's constant, h = \(6.62 \times 10^{-27}\) erg·sec - Speed of light, c = \(2.998 \times 10^{10}\) cm/sec (commonly used speed of light in cm/sec)

Convert wavelength from angstroms to centimeters

We need to convert the wavelength from angstroms to centimeters to make the units consistent. 1 angstrom = \(10^{-8}\) cm So, λ = \(5890 \AA \times 10^{-8} cm/\AA = 5.89 \times 10^{-5}\) cm.

Use the speed of light equation to find the frequency

The speed of light equation is given by: \[c = \nu \times \lambda\] where \(\nu\) is the frequency and \(\lambda\) is the wavelength. We can rearrange the equation to solve for the frequency: \[\nu = c / \lambda\] Now we can plug in the known values: \[\nu = (2.998 \times 10^{10} \: cm/sec) / (5.89 \times 10^{-5} \: cm) = 5.08 \times 10^{14}\: sec^{-1}\]

Use Planck's equation to find the energy of the electronic transition

Planck's equation is given by: \[E = h \times \nu\] where E is the energy of the electronic transition, h is Planck's constant, and \(\nu\) is the frequency. Now we can plug in the known values: \[E = (6.62 \times 10^{-27} \: erg \cdot sec) \times (5.08 \times 10^{14} \: sec^{-1}) = 3.36 \times 10^{-12}\: erg\] The energy of the electronic transition associated with the given wavelength is approximately \(3.36 \times 10^{-12}\) erg.

Key concepts

Spectroscope

A spectroscope is an instrument utilized to measure and observe the spectrum of light. This device is essential in the field of spectroscopy, wherein scientists investigate the interactions of electromagnetic radiation with matter. The spectroscope typically consists of a slit through which light enters, a series of lenses or mirrors that collimate the light, and a prism or a diffraction grating that disperses the light into its constituent wavelengths, creating a spectrum. By analyzing the spectrum, researchers can identify the different wavelengths present and deduce information about the substance emitting or absorbing the light, such as its composition, temperature, and motion.

Using a spectroscope to analyze the emission spectra of elements enables the calibration of the instrument with known spectral lines, such as the sodium-D lines in the exercise. This process is crucial for precision measurements in a variety of scientific fields, from chemistry to astronomy.

Sodium-D Lines

The sodium-D lines are characteristic spectral lines that appear at specific wavelengths in the emission spectrum of sodium. They are actually doublets, closely spaced lines that are caused by electronic transitions in the sodium atom. When sodium atoms are heated, they emit light at two almost identical yellow wavelengths, commonly denoted as D₁ and D₂. These lines are famously used to calibrate spectroscopes because they have precisely known wavelengths, which are approximately 5890 and 5896 angstroms, respectively.

The importance of the sodium-D lines in applications such as calibrating instruments or studying celestial objects makes them a classic example in textbooks and educational sources.

Planck's Constant

Planck's constant is a fundamental physical constant denoted by the symbol 'h' and is of paramount importance in quantum mechanics. Max Planck introduced this constant in 1900 to describe the quantization of energy. Planck's constant relates the energy of a photon to its frequency, stating that energy is absorbed or emitted in discrete units called quanta. Its value is approximately \(6.62607015 \times 10^{-34}\) joules per second or, as used in the exercise, \(6.62 \times 10^{-27}\) erg-seconds.

Planck's constant enables us to calculate the energy associated with electronic transitions in atoms. Whenever an electron jumps from one energy level to another, the energy difference corresponds to the emission or absorption of a photon with energy given by \(E = h \times u\), where \(E\) is the energy and \(u\) is the frequency of the photon.

Speed of Light

The speed of light, commonly represented by 'c', is a fundamental constant of nature that signifies the maximum speed at which all energy, matter, and information in the universe can travel. In a vacuum, the speed of light is exactly \(299,792,458\) meters per second or about \(3.00 \times 10^{8}\) meters per second for simpler calculations. In the context of the exercise, it's provided as \(2.998 \times 10^{10}\) centimeters per second to match the units used.

The speed of light is vital in formulas that connect the frequency and wavelength of electromagnetic radiation, such as \(c = u \times \lambda\), where \(u\) and \(\lambda\) represent frequency and wavelength, respectively. This relationship allows scientists to infer properties of light and its interactions with matter, which is critically important in fields of study like astrophysics, optics, and quantum mechanics.

Electronic Transition Energy

Electronic transition energy refers to the energy involved when an electron in an atom jumps from one energy level to another. This is a fundamental concept in quantum mechanics, as electrons can only occupy specific energy levels within an atom. Transitions between these levels can either absorb energy (moving to a higher energy level) or release energy (falling to a lower energy level).

The energy associated with such an electronic transition is given by the Planck-Einstein relation \(E = h \times u\), where 'E' represents the energy, 'h' is Planck's constant, and '\(u\)' is the frequency of the emitted or absorbed photon. The frequency can also be related to the wavelength \(\lambda\) through the speed of light 'c', using the relationship \(c = u \times \lambda\). In spectroscopy, the measurement of the energy of electronic transitions provides crucial insights into the structure of an atom and the behavior of its electrons.