The figure below represents part of the emission spectrum...

The figure below represents part of the emission spectrum for a one-electron ion in the gas phase. All the lines result from electronic transitions from excited states to the $n=3$ state. (See Exercise 160.) a. What electronic transitions correspond to lines $A$ and $B ?$ b. If the wavelength of line $B$ is $142.5 \mathrm{nm},$ calculate the wavelength of line $A$.

The figure below represents part of the emission spectrum for a one-electron ion in the gas phase. All the lines result from electronic transitions from excited states to the $n=3$ state. (See Exercise 160.)
a. What electronic transitions correspond to lines $A$ and $B ?$
b. If the wavelength of line $B$ is $142.5 \mathrm{nm},$ calculate the wavelength of line $A$.

Key Concepts

Hydrogen-like Ions

Hydrogen-like ions are atomic systems that contain only one electron orbiting the nucleus, similar to the hydrogen atom. Their simplicity allows for accurate application of theoretical models, such as the Rydberg formula, to predict wavelengths of emitted or absorbed light during electronic transitions.

Rydberg Formula

The Rydberg formula is an equation used to predict the wavelengths of light resulting from electronic transitions in hydrogen-like atoms and ions. It relates the wavelength of emitted or absorbed light to the principal quantum numbers of the initial and final energy levels, utilizing fundamental physical constants.

Electronic Transitions

Electronic transitions occur when an electron moves between discrete energy levels in an atom or ion. The energy difference between the levels is emitted as a photon, and the wavelength of the photon is determined by this energy gap, which is a fundamental concept in understanding atomic spectra.

Emission Spectrum

An emission spectrum is the set of wavelengths of light emitted by an excited atom or molecule as electrons transition between quantized energy levels. Each line in the spectrum corresponds to a specific electronic transition, making it a powerful tool for identifying elements and studying atomic structure.

Quantized Energy Levels

Quantized energy levels refer to the discrete values of energy that electrons can possess within an atom or ion. This quantization is a result of the wave nature of electrons and explains why atoms emit or absorb light at specific wavelengths, rather than a continuous spectrum.

Transcript

00:01 In this exercise, we are given the figure, i've given an abbreviated figure from the book here, that represents the emission spectrum of a one electron ion in a gas phase.

00:13 All the lines result from electronic transitions from excited states to the n equals three state.

00:21 So we're being asked to figure out what electric transition corresponds to lines a and b, and then also going to figure out if the wavelength of line b is 142 .5 nanometers, figure out the wavelength of line a.

00:38 Okay, so first thing we need to do is figure out our equation.

00:43 So we're going to be working with the equation e or energy equals negative 2 .178 times 10 to the negative 18th.

00:57 Times z squared over n f squared minus z squared over n i squared.

01:08 Now we don't know what z is or what e is.

01:12 However, we do know that the wavelength of b is 142 .5 nanometers.

01:28 So we can actually figure out our energy using e equals planks constant times the speed of light divided by our wavelength.

01:40 So in this case, we're going to have e equals 6 .626 times 10 to the negative 34th times 3 .0 times 10 .000 times 3 .0 times 10 to 0 .4th .5 .0.

01:59 The 8th, all divided by 142 .5 times 10 to the negative 9th meters.

02:09 We do have to put it into meters so that it goes with the speed of light.

02:13 And that gives us an energy of 1 .39 times 10 to the negative 18th joules...

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