Question

The Solar Constant measured by Earth satellites is roughly \(1400 . \mathrm{W} / \mathrm{m}^{2}\). Though the Sun emits light of different wavelengths, the peak of the wavelength spectrum is at \(500 . \mathrm{nm}\) a) Find the corresponding photon frequency. b) Find the corresponding photon energy. c) Find the number flux of photons arriving at Earth, assuming that all light emitted by the Sun has the same peak wavelength.

Short answer

Question: Calculate the photon frequency, energy, and number flux of photons arriving at Earth if the Solar Constant is 1400 W/m² and the peak wavelength emitted by the Sun is 500 nm. Solution: a) The corresponding photon frequency is \(6.00 x 10^{14} Hz\). b) The corresponding photon energy is \(3.98 x 10^{-19} J\). c) The number flux of photons arriving at Earth is \(3.52 x 10^{21} \frac{photons}{m^2\cdot s}\).

Step-by-step solution

Find the Corresponding Photon Frequency

To find the frequency of the light at the peak wavelength, we can use the equation \(v = \frac{c}{\lambda}\). We know the peak wavelength is 500 nm (\(5.00 x 10^{-7} m\)) and the speed of light is \(3.00 x 10^8 \frac{m}{s}\). \(v = \frac{3.00 x 10^8 \frac{m}{s}}{5.00 x 10^{-7} m}\) \(v = 6.00 x 10^{14} Hz\) The corresponding photon frequency is \(6.00 x 10^{14} Hz\).

Find the Corresponding Photon Energy

To find the energy of the photons with the frequency we've calculated, we can use the equation \(E = h \cdot v\). We know the photon frequency is \(6.00 x 10^{14} Hz\) and Planck's constant is \(6.63 x 10^{-34} Js\). \(E = (6.63 x 10^{-34} Js)(6.00 x 10^{14} Hz)\) \(E = 3.98 x 10^{-19} J\) The corresponding photon energy is \(3.98 x 10^{-19} J\).

Find the Number Flux of Photons Arriving at Earth

To find the number flux of photons arriving at Earth, we can use the equation \(\text{Flux} = \frac{\text{Solar Constant}}{E}\). We know the Solar Constant is 1400 \(\frac{W}{m^2}\) and the photon energy is \(3.98 x 10^{-19} J\). Flux = \(\frac{1400 \frac{W}{m^2}}{3.98 x 10^{-19} J}\) Flux = \(3.52 x 10^{21} \frac{photons}{m^2\cdot s}\) The number flux of photons arriving at Earth is \(3.52 x 10^{21} \frac{photons}{m^2\cdot s}\).

Key concepts

Photon Frequency

Understanding photon frequency is essential because it determines the color and energy of light. The frequency of a photon refers to the number of times the wave peaks pass a point in one second, measured in hertz (Hz). The higher the frequency, the more energy the photon has. In our exercise, the frequency of a photon at the peak wavelength emitted by the Sun (500 nm) is found using the equation
\( v = \frac{c}{\lambda} \)
where \(v\) is the frequency, \(c\) is the speed of light (approximately \(3.00 \times 10^8 \frac{m}{s}\)), and \(\lambda\) is the wavelength. When the wavelength of 500 nm (\(5.00 \times 10^{-7} m\)) is plugged into the equation, we determine the photon frequency to be \(6.00 \times 10^{14} Hz\), a value occurring in the visible spectrum which humans perceive as light.

Photon Energy

Photon energy is directly related to the frequency; the higher the frequency, the greater the energy. The equation that expresses this relationship is
\( E = h \cdot v \)
where \(E\) represents the photon energy, \(h\) is Planck's constant (\(6.63 \times 10^{-34} J\cdot s\)), and \(v\) is the frequency. Using our calculated frequency of \(6.00 \times 10^{14} Hz\), we find the energy of each photon to be \(3.98 \times 10^{-19} J\). This energy is indicative of a photon in the visible spectrum and is fundamental to understanding how sunlight interacts with objects, such as solar panels or the human retina.

Number Flux of Photons

The number flux of photons describes how many photons pass through a certain area every second. It's a convenient way to measure the intensity of sunlight reaching Earth. To find it, we divide the Solar Constant, which represents the radiant energy received from the Sun per unit area, by the energy per photon.
\( Flux = \frac{\text{Solar Constant}}{E} \)
With the Solar Constant at \(1400 \frac{W}{m^2}\) and the photon energy previously found, we calculate the number flux to be \(3.52 \times 10^{21} \frac{photons}{m^2\cdot s}\). This flux helps us estimate the rate at which photons are reaching and potentially providing energy to the Earth's surface.

Planck's Constant

Planck's constant (\(h\)) is a fundamental piece in quantum mechanics, inferring that energy transfer occurs in discrete amounts called quanta. Its current accepted value is \(6.63 \times 10^{-34} J\cdot s\), and it shows up in several essential quantum equations, including those for photon energy and the photoelectric effect. Planck’s constant reflects the granular nature of energy at microscopic scales, which debunks the classical notion of energy as a continuous variable. The tiny number represents the scale at which quantum effects become noticeable. It also illustrates that light, although often thought of as a wave, can also behave as a particle (photon), with discrete energy related to its frequency.