Question

The blue color of the sky results from the scattering of sunlight by molecules in the air. The blue light has a frequency of about \(7.5 \times 10^{14} \mathrm{~Hz}\). (a) Calculate the wavelength (in \(\mathrm{nm}\) ) associated with this radiation, and (b) calculate the energy (in joules) of a single photon associated with this frequency.

Short answer

The wavelength is 400 nm, and the energy is \(4.97 \times 10^{-19} \text{ J}\).

Step-by-step solution

Understand the problem

We need to find two things: (a) the wavelength of blue light with a frequency of \(7.5 \times 10^{14} \, \text{Hz}\), and (b) the energy of a single photon corresponding to this frequency.

Use wave equation to find wavelength

The wave equation relates the speed of light \(c\), frequency \(f\), and wavelength \(\lambda\). The equation is: \(c = \lambda \cdot f\). The speed of light \(c\) is approximately \(3.0 \times 10^{8} \, \text{m/s}\). Solve for wavelength \(\lambda\):\[\lambda = \frac{c}{f} = \frac{3.0 \times 10^{8} \, \text{m/s}}{7.5 \times 10^{14} \, \text{Hz}} = 4.0 \times 10^{-7} \, \text{m}\]. Convert meters to nanometers by multiplying by \(10^9\): \(400 \, \text{nm}\).

Use Planck's equation to find energy

Planck's equation relates energy \(E\) of a photon to its frequency \(f\) with the equation: \(E = h \cdot f\), where \(h\) is Planck's constant, \(6.626 \times 10^{-34} \, \text{Js}\). Substitute the given frequency:\[E = 6.626 \times 10^{-34} \, \text{Js} \cdot 7.5 \times 10^{14} \, \text{Hz} = 4.97 \times 10^{-19} \, \text{J}\].

Present results

The wavelength of the blue light is \(400 \, \text{nm}\), and the energy of a single photon of this light is \(4.97 \times 10^{-19} \, \text{J}\).

Key concepts

Photon Energy

Photon energy is a concept that captures the energy of a photon related to its frequency. Photons are tiny particles of light, each carrying a specific amount of energy determined by its frequency.
To find this energy, we use Planck's equation:

• Planck's equation: \( E = h \cdot f \), where \( E \) is the energy, \( h \) is Planck's constant, and \( f \) is the frequency.
• Planck's constant \( h \) has a value of approximately \( 6.626 \times 10^{-34} \, \text{Js} \).
• Energy, so calculated, indicates how much energy a single photon from a particular light source possesses.

With this, you can understand why different colors of light, which have different frequencies, also have photons with different energies.

Electromagnetic Spectrum

The electromagnetic spectrum is the range of all types of electromagnetic radiation, which includes visible light, microwaves, radio waves, ultraviolet light, and more. It characterizes radiation according to various wavelengths and frequencies.
Each part of the spectrum has distinct properties:

• The visible light range is only a small part of the entire electromagnetic spectrum.
• Different wavelengths correspond to different types of radiation, from the long radio waves to the short gamma rays.

When talking about the blue sky, we are focusing on the visible section where blue light is situated. Blue light has a shorter wavelength and higher frequency compared to red light, explaining why different colors behave differently when interacting with matter.

Light Frequency

Frequency is a key concept when discussing light. It refers to how many waves pass a point in a given time frame and is measured in Hertz (Hz).
For example, the frequency of blue light in the sky is approximately \(7.5 \times 10^{14} \, \text{Hz}\).

• Different sections of the electromagnetic spectrum, including visible light, have their own frequency ranges.
• Frequency helps determine both a light's color and energy.

Understanding frequency is essential for calculations involving photon energy, as well as interpreting the broader electromagnetic spectrum.

Speed of Light

Speed of light in a vacuum is the fastest speed at which light can travel and is a constant, represented by the symbol \(c\). It is approximately \(3.0 \times 10^8 \, \text{m/s}\).
This constant plays a crucial role in various equations within physics, especially in calculating wavelengths and frequencies:

• The relationship involving speed of light is described by the equation: \( c = \lambda \cdot f \).
• This equation allows us to convert between frequency and wavelength values.

Since the speed of light is a constant, it acts as a fundamental bridge connecting these important properties of light.

Planck's Constant

Planck's constant, denoted \(h\), is a fundamental constant in physics that links energy with frequency in photons. Its value is around \(6.626 \times 10^{-34} \, \text{Js} \).
This constant is crucial for calculating the energy carried by photons using Planck's equation.

• Planck's constant is essential in the quantum realm to understand energy transitions.
• It highlights the quantized nature of light, signifying how energy levels are not continuous but discrete.

Understanding Planck's constant helps in comprehending the dual wave-particle nature of light and forms the backbone of quantum mechanics.