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Chapter 38: Problem 6

What is the photon energy for yellow light from a highway sodium lamp at a wavelength of \(589 \mathrm{~nm}\) ?

Short Answer

Expert verified
The photon energy is approximately \(3.37 \times 10^{-19} \text{ Joules}\).

Step by step solution

01

Understand the Formula

The energy of a photon can be calculated using the formula \( E = \frac{hc}{\lambda} \), where \( E \) is the energy of the photon, \( h \) is Planck's constant \(6.626 \times 10^{-34} \text{ J}\cdot\text{s}\), \( c \) is the speed of light \(3 \times 10^8 \text{ m/s}\), and \( \lambda \) is the wavelength of the light.
02

Convert Wavelength

Since 1 nm = \(10^{-9}\) m, convert the wavelength from nanometers to meters: \(\lambda = 589 \text{ nm} = 589 \times 10^{-9} \text{ m}\).
03

Plug Values into the Formula

Substitute the known values into the energy formula: \[E = \frac{(6.626 \times 10^{-34} \text{ J}\cdot\text{s})(3 \times 10^8 \text{ m/s})}{589 \times 10^{-9} \text{ m} }\]
04

Calculate Energy

Perform the calculation: \[E = \frac{(6.626 \times 10^{-34})(3 \times 10^8)}{589 \times 10^{-9}} = 3.37 \times 10^{-19} \text{ Joules}\]
05

Finalize the Energy Calculation

The energy of a photon of yellow light at a wavelength of 589 nm is approximately \(3.37 \times 10^{-19} \text{ Joules}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Wavelength
The wavelength of light is a crucial property that defines its color and energy. In simple terms, a wavelength is the distance between two consecutive peaks of a wave.
In the context of light waves, which are electromagnetic waves, this distance helps dictate the type of light we perceive. Shorter wavelengths correspond to higher energy and often fall into the blue or ultraviolet range, while longer wavelengths have lower energy and can appear red.
  • The symbol for wavelength is usually represented as \( \lambda \), and it is measured in meters. However, since light waves are very small, it is common to use units like nanometers (nm), where 1 nm = \(10^{-9}\) meters.
  • Visible light, which human eyes can see, typically ranges from about 400 nm (violet) to 700 nm (red).
  • In the exercise, yellow light has a wavelength of 589 nm, placing it in the middle of the visible spectrum.
Understanding wavelength is key to exploring light鈥檚 properties, including its energy.
Planck's Constant
Planck's constant is a fundamental constant that plays a pivotal role in quantum mechanics. It relates the energy of a photon with the frequency of its electromagnetic wave.
This constant is denoted by \( h \), and has a value of approximately \( 6.626 \times 10^{-34} \, \text{J} \cdot \text{s} \).
  • Planck's constant serves as a bridge between the macroscopic and the quantum world. It shows how quantum particles like photons carry discrete amounts of energy instead of a continuous range.
  • Within the photon energy formula, \( E = \frac{hc}{\lambda} \), using Planck's constant allows us to relate the energy to the wavelength of light. This link is fundamental to understanding phenomena like photoelectric effects, where light can eject electrons from a material.
  • Every change in light's wavelength or frequency is directly tied to changes in the energy, all due to Planck's constant鈥攁 testament to its significance in physics.
Recognizing how this constant impacts calculations can demystify many quantum theories and equations.
Speed of Light
The speed of light is a universal constant representing how fast light travels through a vacuum. It is one of the cornerstones of physics, impacting our understanding of the universe.
Light travels at a speed of approximately \( 3 \times 10^8 \) meters per second (m/s).
  • The use of the speed of light, symbolized by \( c \), in calculations highlights its critical importance. It ties together space and time, as seen in Einstein's theory of relativity.
  • In the context of photon energy, it is part of the equation \( E = \frac{hc}{\lambda} \), where the speed of light allows the calculation of energy for photons, given their wavelength. Thus, it becomes a determinant factor in how much energy a light wave can carry.
  • This constant underscores the limit of speed in the universe, affirming that nothing can travel faster than light in a vacuum.
In practice, understanding the speed of light helps us grasp concepts of time, distance in space travel, and even the fundamental limits of physics itself.

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Most popular questions from this chapter

(a) The smallest amount of energy needed to eject an electron from metallic sodium is \(2.28 \mathrm{eV}\). Does sodium show a photoelectric effect for red light, with \(\lambda=680 \mathrm{~nm} ?\) (That is, does the light cause electron emission?) (b) What is the cutoff wavelength for photoelectric emission from sodium? (c) To what color does that wavelength correspond?

You wish to pick an element for a photocell that will operate via the photoelectric effect with visible light. Which of the following are suitable (work functions are in parentheses): tantalum \((4.2\) \(\mathrm{eV}\) ), tungsten \((4.5 \mathrm{eV})\), aluminum \((4.2 \mathrm{eV})\), barium \((2.5 \mathrm{eV})\), lithium \((2.3 \mathrm{eV}) ?\)

Monochromatic light (that is, light of a single wavelength) is to be absorbed by a sheet of photographic film and thus recorded on the film. Photon absorption will occur if the photon energy equals or exceeds \(0.6 \mathrm{eV}\), the smallest amount of energy needed to dissociate an AgBr molecule in the film. (a) What is the greatest wavelength of light that can be recorded by the film? (b) In what region of the electromagnetic spectrum is this wavelength located?

What (a) frequency, (b) photon energy, and (c) photon momentum magnitude (in \(\mathrm{keV} / \mathrm{c}\) ) are associated with \(\mathrm{x}\) rays having wavelength \(35.0 \mathrm{pm}\) ?

An orbiting satellite can become charged by the photoelectric effect when sunlight ejects electrons from its outer surface. Satellites must be designed to minimize such charging because it can ruin the sensitive microelectronics. Suppose a satellite is coated with platinum, a metal with a very large work function \((\Phi=5.32 \mathrm{eV})\). Find the longest wavelength of incident sunlight that can eject an electron from the platinum.
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