Ch.8 - The Quantum-Mechanical Model of the Atom

Tro - Chemistry: A Molecular Approach 6th Edition

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Tro 6th Edition

Ch.8 - The Quantum-Mechanical Model of the Atom

Problem 45

Chapter 8, Problem 45

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

Verified step by step guidance

Convert the wavelength from nanometers to meters by using the conversion factor: 1 nm = 1 x 10^{-9} m.

Use the formula for energy of a photon: E = \(\frac{hc}{\lambda}\), where h is Planck's constant (6.626 x 10^{-34} J\(\cdot\) s), c is the speed of light (3.00 x 10^8 m/s), and \(\lambda\) is the wavelength in meters.

Calculate the energy of a single photon using the converted wavelength.

Convert the total energy of the laser pulse from millijoules to joules by using the conversion factor: 1 mJ = 1 x 10^{-3} J.

Divide the total energy of the laser pulse by the energy of a single photon to find the number of photons in the laser pulse.

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Key Concepts

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

Photon Energy

The energy of a single 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 shorter wavelengths correspond to higher energy photons.

Wavelength and Frequency

Wavelength (λ) and frequency (ν) are inversely related through the equation c = λν, where c is the speed of light. This means that as the wavelength of light decreases, its frequency increases, which in turn affects the energy of the photons. Understanding this relationship is crucial for calculating the number of photons in a given energy pulse.

Calculating Number of Photons

To find the number of photons in a laser pulse, divide the total energy of the pulse by the energy of a single photon. This is expressed mathematically as N = E_total / E_photon, where N is the number of photons, E_total is the total energy of the pulse, and E_photon is the energy of one photon calculated from its wavelength.

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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

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

Textbook Question

Determine the energy of 1 mol of photons for each kind of light. (Assume three significant figures.) a. infrared radiation (1500 nm) b. visible light (500 nm) c. ultraviolet radiation (150 nm)

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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

How much energy is contained in 1 mol of each? a. X-ray photons with a wavelength of 0.135 nm b. γ-ray photons with a wavelength of 2.15×10^–5 nm

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