Problem 141 - Chemistry

Question

Owls have good night vision because their eyes can detect a light intensity as low as \(5.0 \times 10^{-13} \mathrm{~W} / \mathrm{m}^{2}\). Calculate the number of photons per second that an owl's eye can detect if its pupil has a diameter of \(9.0 \mathrm{~mm}\) and the light has a wavelength of \(500 \mathrm{nm}\) \((1 \mathrm{~W}=1 \mathrm{~J} / \mathrm{s})\)

Step-by-Step Solution

Answer

The result corresponds to the number calculated in the last step. The exact value depends on the constants and values provided, such as the Planck's constant, the speed of light and the values given in the exercise (intensity, diameter and wavelength)

Step 1: Calculating the area of the owl's pupil

Using the diameter of the owl's eye, calculate the area of the pupil. This is a circular area, so the formula to calculate the area is \(\pi (\frac{d}{2})^2\), where d is the diameter of the circle. The diameter is given as 9 mm, which should be converted to meters (i.e., multiplied by \(10^{-3}\)) to keep the units consistent.

Step 2: Calculating the power of light reaching the owl's eye

Multiply the light intensity (in W/m\(^2\)) which the owl's eye can detect by the area of the pupil. The intensity of light is given as \(5.0 \times 10^{-13}\) W/m\(^2\). This gives the power in watts.

Step 3: Calculating the energy of a single photon

Based on the wavelength of the light, calculate the energy of a single photon. The formula for energy of a photon is \( \frac{hc}{λ}\), where h is the Planck’s constant (\(6.63 \times 10^{-34}\) Js), c is the speed of light (\(3.0 \times 10^8\) m/s), and \(λ\) is the wavelength of light. The wavelength is given in nanometers and should be converted to meters (i.e., multiplied by \(10^{-9}\)).

Step 4: Calculating the number of photons per second

Finally, calculate the number of photons per second. This is done by dividing the power of light reaching the owl's eye by the energy of a single photon. This gives the number of photons per second that an owl's eye can detect.

Key Concepts

Night Vision

Night vision is a fascinating adaptation that some creatures, like owls, have developed to see in low-light conditions. Unlike humans, these animals have eyes that are extremely sensitive to light. This heightened sensitivity allows them to detect very faint light levels. Owls, for instance, can perceive light intensities as low as \(5.0 \times 10^{-13} \mathrm{~W/m^2}\).
Several factors contribute to effective night vision:

• Large pupils: A larger pupil size allows more light to enter the eye, which is crucial in low-light settings. For owls, a 9 mm diameter pupil is quite expansive compared to human pupils.
• Photoreceptor Cells: Specialized cells in owl eyes, called rods, are sensitive to dim light and help in night vision. They lack the color detection of cone cells but excel at capturing light.
Understanding these adaptations highlights why owls are such effective nocturnal hunters.

Light Intensity

Light intensity refers to the amount of light energy hitting a given area over a specific period. It's measured in watts per square meter (W/m extsuperscript{2}). This measurement helps us understand how much light is available for observation. When discussing night vision, particularly, low light intensity levels are crucial.
The owl's ability to detect a minimal light intensity of \(5.0 \times 10^{-13} \, \mathrm{W/m^2}\) underlines its capability of functioning efficiently in near darkness. The formula to determine light power received is: \[ \text{Power of light} = \text{Intensity} \times \text{Area of pupil} \]
This calculation provides the total amount of light energy that enters the eye, enabling the owl to detect and navigate even with minimal visible light.

Energy of a Photon

Each particle of light, or photon, carries energy. The energy of a single photon depends directly on the light's wavelength. This relationship can be expressed using the formula:\[ E = \frac{hc}{\lambda} \] where \( E \) is the energy, \( h \) is Planck’s constant \((6.63 \times 10^{-34} \, \mathrm{Js})\), \( c \) is the speed of light \((3.0 \times 10^8 \, \mathrm{m/s})\), and \( \lambda \) is the wavelength in meters.
In the case of the brown owl scenario, the light wavelength is \(500 \, \mathrm{nm}\). Before making calculations, conversion of wavelength from nanometers to meters is necessary by multiplying by \(10^{-9}\). Understanding this relationship is essential for determining how many photons an owl eyes can detect, based on the available light.

Wavelength Conversion

Wavelength is the key to converting between different energy states for photons. Owls detect light of given wavelengths, such as \(500 \, \mathrm{nm}\), which is common in the visible spectrum. When performing calculations involving photons, it's crucial to convert this wavelength into meters by multiplying by \(10^{-9}\).
This conversion is vital when calculating photon energy and translates into practical applications like determining how many photons reach an owl's eye per second:

• Why Convert? Different units provide clarity in equations and ensure accurate results, especially in scientific contexts where SI units are standard.
• Impact on Calculations: Accurate wavelength conversion influences the energy calculation, which in turn affects the photon count measurement.

Understanding wavelength conversion not only aids in academic exercises but also enhances our grasp of light's interaction with the environment.