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Chapter 6: Problem 13

The wavenumber \(\bar{\lambda}\) is the number of waves that exist over a specified distance, very often \(1 \mathrm{~cm}\). The wavenumber can easily be calculated by taking the reciprocal of the wavelength. Give typical wavenumbers for (a) X-rays \((\lambda=1 \mathrm{nm})\) (b) visible light \((\lambda=500 \mathrm{nm})\) (c) microwaves $(\lambda=1 \mathrm{~mm})$.

Short Answer

Expert verified
The typical wavenumbers for each type of wave are: (a) X-rays: \(\bar{\lambda}_{\mathrm{X-rays}} = 10^7 \mathrm{~cm}^{-1}\) (b) visible light: \(\bar{\lambda}_{\mathrm{visible}} = 2 \times 10^4 \mathrm{~cm}^{-1}\) (c) microwaves: \(\bar{\lambda}_{\mathrm{microwaves}} = 10 \mathrm{~cm}^{-1}\)

Step by step solution

01

(Step 1: Convert Wavelengths to Centimeters)

Before we find the wavenumbers, we need to make sure all wavelengths are in the same unit (centimeters). Since we are given the wavelengths in nanometers and millimeters, we can easily convert these values to centimeters: - For X-rays: \(1 \mathrm{nm} = 10^{-7}\mathrm{~cm}\) - For visible light: \(500 \mathrm{nm} = 5 \times 10^{-5}\mathrm{~cm}\) - For microwaves: \(1 \mathrm{~mm} = 0.1\mathrm{~cm}\)
02

(Step 2: Calculate the Wavenumber for X-Rays)

Now, we can find the wavenumber of X-rays by taking the reciprocal of the wavelength (in centimeters): $$ \bar{\lambda}_{\mathrm{X-rays}} = \frac{1}{10^{-7}\mathrm{~cm}} = 10^7 \mathrm{~cm}^{-1} $$
03

(Step 3: Calculate the Wavenumber for Visible Light)

Similarly, we can find the wavenumber of visible light by taking the reciprocal of the wavelength (in centimeters): $$ \bar{\lambda}_{\mathrm{visible}} = \frac{1}{5 \times 10^{-5}\mathrm{~cm}} = 2 \times 10^4 \mathrm{~cm}^{-1} $$
04

(Step 4: Calculate the Wavenumber for Microwaves)

Finally, we can find the wavenumber of microwaves by taking the reciprocal of the wavelength (in centimeters): $$ \bar{\lambda}_{\mathrm{microwaves}} = \frac{1}{0.1\mathrm{~cm}} = 10 \mathrm{~cm}^{-1} $$
05

(Step 5: State the Results)

We have found the typical wavenumbers for each type of wave: (a) X-rays: \(\bar{\lambda}_{\mathrm{X-rays}} = 10^7 \mathrm{~cm}^{-1}\) (b) visible light: \(\bar{\lambda}_{\mathrm{visible}} = 2 \times 10^4 \mathrm{~cm}^{-1}\) (c) microwaves: \(\bar{\lambda}_{\mathrm{microwaves}} = 10 \mathrm{~cm}^{-1}\)

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Most popular questions from this chapter

A diode laser emits at a wavelength of \(987 \mathrm{nm} .\) (a) In what portion of the electromagnetic spectrum is this radiation found? (b) All of its output energy is absorbed in a detector that measures a total energy of $0.52 \mathrm{~J}\( over a period of \)32 \mathrm{~s}$. How many photons per second are being emitted by the laser?

(a) A green laser pointer emits light with a wavelength of \(532 \mathrm{nm}\). What is the frequency of this light? (b) What is the energy of one of these photons? (c) The laser pointer emits light because electrons in the material are excited (by a battery) from their ground state to an upper excited state. When the electrons return to the ground state, they lose the excess energy in the form of \(532-\mathrm{nm}\) photons. What is the energy gap between the ground state and excited state in the laser material?

Indicate whether energy is emitted or absorbed when the following electronic transitions occur in hydrogen: (a) from \(n=2\) to \(n=3,(\mathbf{b})\) from an orbit of radius 0.529 to one of radius \(0.476 \mathrm{nm},(\mathbf{c})\) from the \(n=9\) to the \(n=6\) state.

Give the values for \(n, l,\) and \(m_{l}\) for \((\mathbf{a})\) each orbital in the \(3 p\) subshell, (b) each orbital in the \(4 f\) subshell.

Consider a fictitious one-dimensional system with one electron. The wave function for the electron, drawn below, is \(\psi(x)=\sin x\) from \(x=0\) to $x=2 \pi .\( (a) Sketch the probability density, \)\psi^{2}(x),\( from \)x=0\( to \)x=2 \pi .(\mathbf{b})\( At what value or values of \)x$ will there be the greatest probability of finding the electron? (c) What is the probability that the electron will be found at \(x=\pi ?\) What is such a point in a wave function called? [Section 6.5\(]\)
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