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

The concept of wavelength is fundamental in understanding light and its interaction with matter, such as why the sky appears blue. A wavelength is the distance between successive crests of a wave, and it's inversely proportional to the frequency of the wave. To calculate the wavelength of light, we use the equation
\( c = \lambda u \), where \( c \) represents the speed of light in a vacuum, \( \lambda \) stands for the wavelength, and \( u \) is the frequency. Given the frequency of the blue light from the sky is \( 7.5 \times 10^{14} \text{Hz} \), and the speed of light is a constant at \( 3.0 \times 10^8 \text{m/s} \), by rearranging the equation to \( \lambda = \frac{c}{u} \) we can deduce the wavelength.

When the calculation is performed, we find that the wavelength is around \( 4.0 \times 10^{-7} \text{m} \), equivalent to 400 nanometers (nm), highlighting that the blue light is in the visible spectrum. Remember, to convert meters to nanometers, multiply by \( 10^9 \), as there are one billion nanometers in a meter.

Planck's constant, denoted by \( h \) and valued at \( 6.63 \times 10^{-34} \text{Js} \), is a key pillar in quantum mechanics. It relates the energy of a photon to its frequency, providing a bridge between the classical and quantum worlds. This constant is named after Max Planck, who introduced the idea that energy is quantized, or comes in discrete 'packets' called quanta.

Understanding Planck's constant allows us to comprehend the quantized nature of light and how energy levels of photons are determined. When tackling problems involving photons, the energy or \( E \) of a single photon can be calculated using the relationship \( E = hu \). This fundamental relationship shows that as the frequency of light increases, so does the energy of its photons.

Photons, the particles of light, carry energy which can be determined using the frequency and Planck's constant. The energy of a photon is expressed through the equation \( E = hu \) where \( E \) is energy, \( h \) is Planck's constant, and \( u \) is the frequency of the photon. It is important to understand that the energy associated with a photon is directly proportional to its frequency; higher frequency results in higher energy.

For the blue light scattered by air molecules, with the given frequency of \( 7.5 \times 10^{14} \text{Hz} \), utilizing the equation yields an energy value of approximately \( 5.0 \times 10^{-19} \text{J} \). This energy is a tiny amount, but it's significant when it comes to understanding how light interacts with atoms and molecules, leading to phenomena such as the blue color of the sky.