Question

The eyes of a certain member of the reptile family pass a single visual single to the brain when the visual receptors are stuck by photons of wavelength \(662.6 \mathrm{~nm}\). If a total energy of \(3.0 \times 10^{-14} \mathrm{~J}\) is required to trap the signal, what is the minimum number of photons that must strike the receptor? (a) \(1.0 \times 10^{5}\) (b) \(1.0 \times 10^{6}\) (c) 1000 (d) 1

Short answer

The minimum number of photons must be at least 1, considering that photons are indivisible, so none of the answer choices (a), (b), or (c) are correct, making (d) the correct answer.

Step-by-step solution

Determine the energy of a single photon

Calculate the energy of one photon using the equation for the energy of a photon, which is given by Planck's equation: \(E = h u\), where \(E\) is the energy of the photon, \(h\) is Planck's constant \( (6.626 \times 10^{-34} \mathrm{J \cdot s}) \), and \(u\) is the frequency of the light. The frequency can be found using the speed of light \(c\) and the wavelength \(\lambda\): \(u = \frac{c}{\lambda}\).

Calculate the frequency of the photon

Using the speed of light \(c = 3.00 \times 10^8 \mathrm{m/s}\) and the given wavelength \(\lambda = 662.6 \mathrm{nm}\), calculate the frequency \(u\) using the formula: \(u = \frac{c}{\lambda}\). Remember to convert the wavelength from nanometers to meters by dividing by \(10^9\).

Calculate the energy of a single photon

Substitute the values for Planck's constant \(h\) and the calculated frequency \(u\) into the photon energy equation \(E = h u\) to find the energy of a single photon.

Calculate the minimum number of photons required

To find the minimum number of photons required, divide the total energy needed to transmit the signal by the energy of one photon, using the formula: \(N = \frac{E_{\text{total}}}{E_{\text{photon}}}\), where \(N\) is the minimum number of photons, \(E_{\text{total}}\) is the total energy required, and \(E_{\text{photon}}\) is the energy of one photon.

Compare results with the given options

After calculating the minimum number of photons, round up to the nearest whole number if necessary since you can't have a fraction of a photon. Compare the resulting number with the provided answer options to find the right one.

Key concepts

Planck's Equation

Planck's equation is a fundamental principle in quantum mechanics that describes how the energy of a photon is related to its frequency. Mathematically, it is represented as:
\[ E = h u \]
where \(E\) stands for the energy of the photon, \(h\) represents Planck's constant\( (6.626 \times 10^{-34} \mathrm{J \cdot s}) \), and \(u\) (Greek letter nu) denotes the frequency. This equation implies that the energy of a photon is directly proportional to its frequency: the higher the frequency, the higher its energy. In context with the exercise, to find the energy of a photon that activates the visual receptors in a reptile's eye, one would use this crucial equation.

Speed of Light

The speed of light in a vacuum is a constant value and one of the most important in physics. It is generally represented by the symbol \(c\) and has a value of approximately \(3.00 \times 10^{8} \mathrm{m/s}\).
This universal constant is not only fundamental to the theory of relativity but is also crucial when it comes to relating the wavelength of a photon to its frequency. Since light travels at a constant speed, knowing this value allows us to calculate either of these characteristics of light when one is known.

Photon Frequency

Understanding Photon Frequency

In the context of light and other forms of electromagnetic radiation, the frequency refers to the number of wave cycles that pass a point per second. Frequency is measured in hertz (Hz), and it directly correlates with the energy of a photon: a higher frequency means a higher energy and vice versa.
The link between energy and frequency is crucial for various applications, including electronics, astronomy, and even biology, as in the exercise related to reptiles' visual processing.

Wavelength to Frequency Conversion

To convert between wavelength and frequency, the formula combining the two with the speed of light is used:
\[ u = \frac{c}{\lambda} \]
where \(u\) is frequency, \(c\) is the speed of light, and \(\lambda\) is the wavelength. Specifically, as the wavelength increases, the frequency decreases, since the two are inversely proportional.
This relationship is important in the exercise as it allows us to convert the given wavelength of 662.6 nm to the frequency required to calculate photon energy using Planck's equation.

Photoelectric Effect

The photoelectric effect occurs when photons incident upon a material cause it to emit electrons. This phenomenon was crucial in the development of quantum mechanics.
The minimum energy needed by an electron to escape from the material is known as the work function, and the photoelectric effect illustrates that only photons with sufficient energy (equivalent to or greater than the work function) can eject electrons.

Relation to Photon Energy Calculation

Although the photoelectric effect is not directly part of the exercise, understanding how photons can cause the photoelectric effect helps reinforce why the energy of a photon, which depends on its frequency according to Planck's equation, is important in the biological context of the reptile's visual receptors in the original exercise.