Ch.7 - Quantum-Mechanical Model of the Atom

Tro - Chemistry: A Molecular Approach 4th Edition

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Tro 4th Edition

Ch.7 - Quantum-Mechanical Model of the Atom

Problem 44

Chapter 7, Problem 44

A heat lamp produces 32.8 watts of power at a wavelength of 6.5 µm. How many photons are emitted per second? (1 watt = 1 J/s)

Verified step by step guidance

Step 1: Understand that the energy of a single photon can be calculated using the equation E = h*c/λ, where E is the energy of the photon, 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 micrometers to meters. 1 µm = 1 x 10^-6 m. So, 6.5 µm = 6.5 x 10^-6 m.

Step 3: Substitute the values into the equation E = h*c/λ to calculate the energy of a single photon.

Step 4: Understand that the total energy emitted per second (power) is given in the problem as 32.8 watts, which is equivalent to 32.8 J/s.

Step 5: Divide the total energy emitted per second by the energy of a single photon to find the number of photons emitted per second. This is because the total energy is the sum of the energies of all the photons emitted.

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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 transferred or converted, 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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