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Ch.8 - The Quantum-Mechanical Model of the Atom
Tro - Chemistry: A Molecular Approach 6th Edition
Tro6th EditionChemistry: A Molecular ApproachISBN: 9780137832217Not the one you use?Change textbook
All textbooksTro 6th EditionCh.8 - The Quantum-Mechanical Model of the AtomProblem 43
Chapter 8, Problem 43

Calculate the energy of a photon of each wavelength and state what part of the electromagnetic spectrum is associated with the wavelength.
a. 105 nm
b. 715 nm
c. 2.52 cm

Verified step by step guidance
1
insert step 1: Understand the relationship between energy and wavelength using the formula: E = \(\frac{hc}{\lambda}\), where E is the energy of the photon, h is Planck's constant (6.626 \(\times\) 10^{-34} \(\text{ J s}\)), c is the speed of light (3.00 \(\times\) 10^8 \(\text{ m/s}\)), and \(\lambda\) is the wavelength.
insert step 2: Convert the given wavelengths into meters for consistency in units. For example, 105 nm = 105 \(\times\) 10^{-9} m, 715 nm = 715 \(\times\) 10^{-9} m, and 2.52 cm = 2.52 \(\times\) 10^{-2} m.
insert step 3: Substitute the converted wavelength values into the energy formula for each case to calculate the energy of the photon.
insert step 4: Identify the part of the electromagnetic spectrum each wavelength corresponds to. For instance, 105 nm is in the ultraviolet range, 715 nm is in the visible range (red light), and 2.52 cm is in the microwave range.
insert step 5: Summarize the findings by stating the energy calculated for each wavelength and the corresponding part of the electromagnetic spectrum.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Photon Energy Calculation

The energy of a photon can be calculated using the equation E = hc/λ, where E is the energy, h is Planck's constant (6.626 x 10^-34 J·s), c is the speed of light (3.00 x 10^8 m/s), and λ is the wavelength in meters. This relationship shows that energy is inversely proportional to wavelength; shorter wavelengths correspond to higher energy photons.

Electromagnetic Spectrum

The electromagnetic spectrum encompasses all types of electromagnetic radiation, arranged by wavelength or frequency. It includes gamma rays, X-rays, ultraviolet light, visible light, infrared radiation, microwaves, and radio waves. Each region of the spectrum has distinct properties and applications, with visible light being the only part detectable by the human eye.

Wavelength Conversion

To calculate photon energy accurately, wavelengths must be in meters. For example, to convert nanometers (nm) to meters, divide by 1 billion (1 nm = 1 x 10^-9 m), and for centimeters (cm), divide by 100 (1 cm = 0.01 m). This conversion is essential for applying the photon energy formula correctly and identifying the corresponding part of the electromagnetic spectrum.
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Conversion Factors
Related Practice
Textbook Question

A heat lamp produces 17.7 watts of power at a wavelength of 6.5 μm. How many photons are emitted per second? (1 watt = 1 J/s)

Textbook Question

List these types of electromagnetic radiation in order of (ii) increasing energy per photon.

a. visible light

b. radio waves

c. X-ray

d. ultraviolet radiation

Textbook Question

A laser pulse with wavelength 532 nm contains 1.85 mJ of energy. How many photons are in the laser pulse?

Textbook Question

List these types of electromagnetic radiation in order of (i) increasing frequency and (ii) decreasing energy per photon.

a. gamma rays

b. ultraviolet radiation

c. infrared radiation

d. microwaves

Textbook Question

The nearest star to our sun is Proxima Centauri, at a distance of 4.3 light-years from the sun. A light-year is the distance that light travels in one year (365 days). How far away, in km, is Proxima Centauri from the sun?

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Textbook Question

Calculate the energy of a photon of each frequency and state what part of the electromagnetic spectrum is associated with the frequency.

a. 15.77×1017 Hz

b. 1.18×1014 Hz

c. 3.00×1020 Hz