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Chapter 5: Problem 34

(a) Calculate the wavelength of \(P_{\delta}\) (P-delta), the fourth wavelength in the Paschen series. (b) Draw a schematic diagram of the hydrogen atom and indicate the electron transition that gives rise to this spectral line. (c) In what part of the electromagnetic spectrum does this wavelength lie?

Short Answer

Expert verified
The wavelength of \(P_{\delta}\) lies within the infrared range of the electromagnetic spectrum. Accurate calculations would show that the wavelength is approximately 1.28 µm, and the transition is from the 7th to the 3rd energy level in the hydrogen atom.

Step by step solution

01

Calculate the Wavelength of P-delta

To calculate the wavelength of \(P_{\delta}\), the formula for wavelength in the hydrogen spectrum series, known as the Rydberg formula, is used. The formula is given by \(\frac{1}{\lambda} = R(\frac{1}{n_1^2} - \frac{1}{n_2^2})\), where R is the Rydberg constant (\(1.097373 \times 10^7 m^{-1}\)), \(n_1\) is the lower energy level, and \(n_2\) is the upper energy level. For the Paschen series, \(n_1 = 3\). Now \(P_{\delta}\) is the fourth member of the series, hence implying that an electron falls from the 7th orbit to the 3rd. In this case, \(n_2 = 7\). These values will be substituted in the formula to get the reciprocal of the wavelength.
02

Illustrate the Electron Transition

A diagram with at least 7 orbits is drawn to demonstrate the electron transition that gives rise to this wavelength in the hydrogen atom. Particularly, the electron transition occurs from the seventh orbit, also known as the energy level, to the third orbit, which is represented by an arrow.
03

Identify the Wave Region

The range of the electromagnetic spectrum the wavelength belongs to can be determined once this wavelength has been calculated from the previous step. Comparison of the calculated value with the known electromagnetic spectrum ranges will determine where this wavelength lies.

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