Textbook Question
An electron in a hydrogen atom relaxes to the n = 4 level, emitting light of 114 THz. What is the value of n for the level in which the electron originated?
Ch.8 - The Quantum-Mechanical Model of the Atom
Chapter 8, Problem 83
An argon ion laser puts out 5.0 W of continuous power at a wavelength of 532 nm. The diameter of the laser beam is 5.5 mm. If the laser is pointed toward a pinhole with a diameter of 1.2 mm, how many photons travel through the pinhole per second? Assume that the light intensity is equally distributed throughout the entire cross-sectional area of the beam. (1 W = 1 J/s)
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Calculate the energy of a single photon using the formula: \( E = \frac{hc}{\lambda} \), where \( h \) 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 \( 532 \text{ nm} \) converted to meters.
Determine the cross-sectional area of the laser beam using the formula for the area of a circle: \( A = \pi \left(\frac{d}{2}\right)^2 \), where \( d \) is the diameter of the laser beam \( 5.5 \text{ mm} \) converted to meters.
Calculate the cross-sectional area of the pinhole using the same formula for the area of a circle, with \( d \) as the diameter of the pinhole \( 1.2 \text{ mm} \) converted to meters.
Find the fraction of the laser beam's power that passes through the pinhole by dividing the area of the pinhole by the area of the laser beam.
Calculate the number of photons passing through the pinhole per second by multiplying the total power of the laser \( 5.0 \text{ W} \) by the fraction of power passing through the pinhole, and then dividing by the energy of a single photon.
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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, given by 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. For a wavelength of 532 nm, this relationship allows us to calculate the energy of each photon emitted by the laser.
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Photon Energy Formulas
Laser Beam Intensity
The intensity of a laser beam is defined as the power per unit area. In this case, the total power output of the laser (5.0 W) is distributed over the cross-sectional area of the beam, which can be calculated using the formula A = π(d/2)², where d is the diameter of the beam. This intensity helps determine how many photons pass through a given area, such as the pinhole.
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Photon Flux
Photon flux refers to the number of photons passing through a unit area per unit time. It can be calculated by dividing the intensity of the laser beam by the energy of a single photon. By determining the area of the pinhole and multiplying the photon flux by this area, we can find the total number of photons that travel through the pinhole each second.
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Related Practice
Textbook Question
Ultraviolet radiation and radiation of shorter wavelengths can damage biological molecules because these kinds of radiation carry enough energy to break bonds within the molecules. A typical carbon–carbon bond requires 348 kJ/mol to break. What is the longest wavelength of radiation with enough energy to break carbon–carbon bonds?
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Textbook Question
The human eye contains a molecule called 11-cis-retinal that changes shape when struck with light of sufficient energy. The change in shape triggers a series of events that results in an electrical signal being sent to the brain that results in vision. The minimum energy required to change the conformation of 11-cis-retinal within the eye is about 164 kJ/mol. Calculate the longest wavelength visible to the human eye.
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Textbook Question
An X-ray photon of wavelength 0.989 nm strikes a surface. The emitted electron has a kinetic energy of 969 eV. What is the binding energy of the electron in kJ/mol? [KE = 1/2 mv2; 1 electron volt (eV) = 1.602×10–19 J]
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