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 46

Chapter 8, Problem 46

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)

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Identify the given values: power (P) = 17.7 watts, wavelength (\(\lambda\)) = 6.5 \(\mu\)m.

Convert the wavelength from micrometers to meters: \(\lambda = 6.5 \times 10^{-6}\) m.

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

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

Determine the number of photons emitted per second by dividing the total power by the energy of a single photon: \(\text{Number of photons} = \frac{P}{E}\).

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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 photon is determined by its wavelength and can be calculated using the equation E = hc/λ, where E is 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.

Power and Energy Relationship

Power is defined as the rate at which energy is emitted or transferred, measured in watts (1 watt = 1 joule/second). In this context, the power of the heat lamp indicates how much energy is emitted per second, which is crucial for determining the number of photons emitted based on their individual energy.

Calculating Photon Emission

To find the number of photons emitted per second, divide the total power output of the lamp by the energy of a single photon. This calculation involves first determining the energy of a photon using its wavelength, and then using the formula: Number of photons = Power / Energy per photon, which provides the total count of photons emitted each second.

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Gamma Emission

Related Practice

Textbook Question

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

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

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

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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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