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Chapter 7: Problem 19

What type of relationship (direct or inverse) exists between wavelength, frequency, and photon energy? What does a photon energy unit of a joule equal?

Short Answer

Expert verified
An inverse relationship exists between wavelength and frequency, and between wavelength and photon energy, while a direct relationship exists between frequency and photon energy. Photon energy is measured in joules (J), with smaller wavelengths corresponding to higher energy photons and vice versa.

Step by step solution

01

1. Key Concepts and Equations

First, let's recall the key concepts and equations relevant to the problem: - Wavelength: It is the distance between two equivalent points in a continuous wave and is represented by the symbol '位', measured in meters (m). - Frequency: It is the number of oscillations of a wave per unit time, represented by the symbol '谓' (nu), and its unit is Hertz (Hz). - Photon Energy: It is the energy carried by a single photon, represented by the symbol 'E', and its unit is Joules (J). The relationship between wavelength, frequency, and the speed of light (c) is given by the equation: \(c = 位谓\) Where \(c\) is the speed of light in a vacuum, approximately equal to \(3.0 \times 10^8\) meters per second (m/s). The energy of a photon is given by Planck's equation: \(E = h谓\) Where \(h\) is Planck's constant, approximately equal to \(6.63 \times 10^{-34}\) Joule-seconds (J.s).
02

2. Relationship between Wavelength and Frequency

Using the equation \(c = 位谓\), we can determine the relationship between wavelength and frequency by isolating one variable: If we solve for frequency (谓), we get: \(谓 = \frac{c}{位}\) From this equation, we can see that frequency is inversely proportional to wavelength. As the wavelength increases, the frequency decreases, and vice versa.
03

3. Relationship between Frequency and Photon Energy

Now, let's examine the relationship between frequency and photon energy using Planck's equation: \(E = h谓\) This equation shows that photon energy is directly proportional to frequency. As the frequency increases, the photon energy also increases, and vice versa.
04

4. Relationship between Wavelength and Photon Energy

We have found the relationship between wavelength and frequency (inverse) and between frequency and photon energy (direct). To find the relationship between wavelength and photon energy, we can combine the two equations \(c = 位谓\) and \(E = h谓\): \(E = h谓\) \(E = h\frac{c}{位}\) Here, we can observe that the photon energy is inversely proportional to the wavelength. As the wavelength increases, the photon energy decreases, and vice versa.
05

5. The Photon Energy Unit - Joule

The unit of photon energy is the joule (J). A single joule is equivalent to \(1\) kg.m虏/s虏 which is also the unit of work done by a force of one Newton when it moves an object by one meter in the direction of the force. This unit is used to measure the energy of a photon, and in our context, it is important to remember that a smaller wavelength corresponds to a higher energy photon in joules, and vice versa. To conclude, there is an inverse relationship between wavelength and frequency, between wavelength and photon energy, and a direct relationship between frequency and photon energy. Photon energy is measured in joules.

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Most popular questions from this chapter

Human color vision is "produced" by the nervous system based on how three different cone receptors interact with photons of light in the eye. These three different types of cones interact with photons of different frequency light, as indicated in the following chart: $$ \begin{array}{|lc|} \hline \text { Cone Type } & \begin{array}{c} \text { Range of Light } \\ \text { Frequency Detected } \end{array} \\ \hline \mathrm{S} & 6.00-7.49 \times 10^{14} \mathrm{~s}^{-1} \\ \mathrm{M} & 4.76-6.62 \times 10^{14} \mathrm{~s}^{-1} \\ \mathrm{~L} & 4.28-6.00 \times 10^{14} \mathrm{~s}^{-1} \\ \hline \end{array} $$ What wavelength ranges (and corresponding colors) do the three types of cones detect?

From the information below, identify element \(\mathrm{X}\). a. The wavelength of the radio waves sent by an FM station broadcasting at \(97.1 \mathrm{MHz}\) is \(30.0\) million \(\left(3.00 \times 10^{7}\right)\) times greater than the wavelength corresponding to the energy difference between a particular excited state of the hydrogen atom and the ground state. b. Let \(V\) represent the principal quantum number for the valence shell of element \(X\). If an electron in the hydrogen atom falls from shell \(V\) to the inner shell corresponding to the excited state mentioned above in part a, the wavelength of light emitted is the same as the wavelength of an electron moving at a speed of \(570 . \mathrm{m} / \mathrm{s}\) c. The number of unpaired electrons for element \(\mathrm{X}\) in the ground state is the same as the maximum number of electrons in an atom that can have the quantum number designations \(n=2\), \(m_{\ell}=-1\), and \(m_{s}=-\frac{1}{2}\) d. Let \(A\) equal the charge of the stable ion that would form when the undiscovered element 120 forms ionic compounds. This value of \(A\) also represents the angular momentum quantum number for the subshell containing the unpaired electron(s) for element \(\mathrm{X}\).

The successive ionization energies for an unknown element are \(I_{1}=896 \mathrm{~kJ} / \mathrm{mol}\) \(I_{2}=1752 \mathrm{~kJ} / \mathrm{mol}\) \(I_{3}=14,807 \mathrm{~kJ} / \mathrm{mol}\) \(I_{4}=17,948 \mathrm{~kJ} / \mathrm{mol}\) To which family in the periodic table does the unknown element most likely belong?

The wave function for the \(2 p_{z}\) orbital in the hydrogen atom is $$ \psi_{2 p_{z}}=\frac{1}{4 \sqrt{2 \pi}}\left(\frac{Z}{a_{0}}\right)^{3 / 2} \sigma \mathrm{e}^{-\sigma / 2} \cos \theta $$ where \(a_{0}\) is the value for the radius of the first Bohr orbit in meters \(\left(5.29 \times 10^{-11}\right), \sigma\) is \(Z\left(r / a_{0}\right), r\) is the value for the distance from the nucleus in meters, and \(\theta\) is an angle. Calculate the value of \(\psi_{2 p_{p}}^{2}\) at \(r=a_{0}\) for \(\theta=0^{\circ}\left(z\right.\) axis) and for \(\theta=90^{\circ}(x y\) plane).

Assume that we are in another universe with different physical laws. Electrons in this universe are described by four quantum numbers with meanings similar to those we use. We will call these quantum numbers \(p, q, r\), and \(s .\) The rules for these quantum numbers are as follows: \(p=1,2,3,4,5, \ldots\) \(q\) takes on positive odd integers and \(q \leq p\) \(r\) takes on all even integer values from \(-q\) to \(+q\). (Zero is considered an even number.) \(s=+\frac{1}{2}\) or \(-\frac{1}{2}\) a. Sketch what the first four periods of the periodic table will look like in this universe. b. Wh?t are the atomic numbers of the first four elements you would expect to be least reactive? c. Give an example, using elements in the forst four rows, of ionic compounds with the formulas \(\mathrm{XY}, \mathrm{XY}_{2}, \mathrm{X}_{2} \mathrm{Y}, \mathrm{XY}_{3}\), and \(\mathrm{X}_{2} \mathrm{Y}_{3}\) d. How many electrons can have \(p=4, q=3 ?\) e. How many electrons can have \(p=3, q=0, r=0 ?\) f. How many electrons can have \(p=6\) ?
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