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Chapter 8: Problem 16

The Lyman series of the hydrogen spectrum can be represented by the equation $$\nu=3.2881 \times 10^{15} \mathrm{s}^{-1}\left(\frac{1}{1^{2}}-\frac{1}{n^{2}}\right)(\text { where } n=2,3, \ldots)$$ (a) Calculate the maximum and minimum wavelength lines, in nanometers, in this series. (b) What value of \(n\) corresponds to a spectral line at 95.0 nm? (c) Is there a line at \(108.5 \mathrm{nm} ?\) Explain.

Short Answer

Expert verified
For part (a), the maximum wavelength is approximately 121.5 nm, and minimum value is 0. The value of \(n\) in part (b) that corresponds to a spectral line at 95.0 nm is 3. For part (c), the calculated value of \(n\) is an integer, so there is a line at 108.5 nm.

Step by step solution

01

Calculate Maximum and Minimum Wavelength

To achieve this, remember that frequency and wavelength are inversely related (ν = c/λ), where c is the speed of light with a known value of \(3.0 × 10^8 \, m/s\). Also consider that nanometer is \(10^{-9}\,m\). Let's first calculate the maximum wavelength which corresponds to the smallest frequency, that is \(n = 2.\) Plug in \(n = 2\) into the frequency equation and obtain frequency. Now find the wavelength by plugging the obtained frequency into the wavelength equation and solving it for max wavelength. Now, to calculate the minimum wavelength which corresponds to the maximum frequency when \(n\) approaches infinity. Calculate the wavelength using the frequency obtained when \(n\) is quite large.
02

Find the Value of n that Corresponds to Wavelength of 95.0 nm

Given the wavelength, find frequency by use frequency-wavelength relationship. Then substitute the obtained frequency in given frequency-equation and obscure \(n\). Solve the equation to get the value of \(n\).
03

Is There a Line at 108.5 nm?

To check if 108.5 nm is a valid wavelength in this series, firstly convert the wavelength to frequency using frequency-wavelength relation. Then, put the frequency in given frequency-equation and solve for \(n\). You will find real number and if it is a natural number then there is a spectral line at 108.5 nm.

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Most popular questions from this chapter

For the Bohr hydrogen atom determine (a) the radius of the orbit \(n=4\) (b) whether there is an orbit having a radius of \(4.00 \AA\) (c) the energy level corresponding to \(n=8\) (d) whether there is an energy level at \(-2.5 \times 10^{-17} \mathrm{J}\)

A line is detected in the hydrogen spectrum at \(1880 \mathrm{nm}\). Is this line in the Balmer series? Explain.

Between which two orbits of the Bohr hydrogen atom must an electron fall to produce light of wavelength \(1876 \mathrm{nm} ?\)

When atoms in excited states collide with unexcited atoms they can transfer their excitation energy to those atoms. The most efficient energy transfer occurs when the excitation energy matches the energy of an excited state in the unexcited atom. Assuming that we have a collection of excited hydrogen atoms in the \(2 s^{1}\) excited state, are there any transitions of \(\mathrm{He}^{+}\) that could be most efficiently excited by the hydrogen atoms?

Briefly describe each of the following ideas or phenomena: (a) atomic (line) spectrum; (b) photoelectric effect; (c) matter wave; (d) Heisenberg uncertainty principle; (e) electron spin; (f) Pauli exclusion principle; (g) Hund's rule; (h) orbital diagram; (i) electron charge density; (j) radial electron density.
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