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

Problem 47

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Chapter 8, Problem 47

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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Step 1: Recall the energy of a photon can be calculated using the equation: E = h * c / 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.

Step 2: Convert the wavelength from nanometers to meters. 1 nm = 1 x 10^-9 m. So, 1500 nm = 1500 x 10^-9 m = 1.5 x 10^-6 m.

Step 3: Substitute the values into the equation to calculate the energy of one photon: E = (6.626 x 10^-34 J*s) * (3.00 x 10^8 m/s) / (1.5 x 10^-6 m).

Step 4: To find the energy of 1 mol of photons, multiply the energy of one photon by Avogadro's number (6.022 x 10^23). So, E(mol) = E * (6.022 x 10^23).

Step 5: The result from step 4 is the energy of 1 mol of infrared radiation photons.

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Key Concepts

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

Photon Energy

Photon energy is the energy carried by a single photon, which can be calculated using the equation E = hc/λ, where E is energy, h is Planck's constant, c is the speed of light, and λ is the wavelength of the light. This relationship shows that energy is inversely proportional to wavelength; shorter wavelengths correspond to higher energy photons.

Wavelength and Frequency

Wavelength and frequency are fundamental properties of electromagnetic radiation. Wavelength (λ) is the distance between successive peaks of a wave, while frequency (ν) is the number of wave cycles that pass a point per second. They are related by the equation c = λν, where c is the speed of light, indicating that as wavelength increases, frequency decreases.

Moles and Avogadro's Number

A mole is a unit in chemistry that represents 6.022 x 10²³ entities, such as atoms or photons. When calculating the energy of 1 mol of photons, it is essential to multiply the energy of a single photon by Avogadro's number to find the total energy for that amount of photons, allowing for practical calculations in chemical contexts.

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Related Practice

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

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

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

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

Sketch the interference pattern that results from the diffraction of electrons passing through two closely spaced slits.

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Textbook Question

The resolution limit of a microscope is roughly equal to the wavelength of light used in producing the image. Electron microscopes use an electron beam (in place of photons) to produce much higher resolution images, about 0.20 nm in modern instruments. Assuming that the resolution of an electron microscope is equal to the de Broglie wavelength of the electrons used, to what speed must the electrons be accelerated to obtain a resolution of 0.20 nm?

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