How much energy is contained in 1 mol of each? a. X-ray photons with a wavelength of \(0.135 \mathrm{nm}\) b. \(\gamma\) -ray photons with a wavelength of \(2.15 \times 10^{-8} \mathrm{nm}\)

Planck's constant, denoted by the symbol h, is a fundamental quantity in quantum physics which connects the energy of a photon with its frequency. The value of Planck's constant is approximately \(6.626 \times 10^{-34}\, \text{J} \cdot \text{s}\). Understanding this constant is crucial as it appears in various equations that describe the behavior of particles at a quantum level. For example, the energy (E) of a single photon can be calculated using the equation \(E = h u\), where u is the frequency of the photon.

To make it simpler, think of Planck's constant as a bridge between the world we can see and the behavior of particles in the invisible world of atoms and photons. It's a bit like a universal conversion factor that allows us to calculate how something as abstract as the frequency of light relates to something as concrete as energy. This concept is significant when calculating the energy content of photons, as seen in the textbook exercise.

Avogadro's number is another fundamental constant in chemistry and physics, denoted by N_A or simply A. It is defined as the number of constituent particles, usually atoms or molecules, that are contained in one mole of a substance. Its approximate value is \(6.022 \times 10^{23}\) particles per mole. This large number is hard to visualize, but it's standardized for use in calculations involving atomic-scale substances.

Consider Avogadro's number as an accountant for atoms and molecules. When working with chemical substances in the lab, we don't measure out individual atoms or molecules—we work with moles that contain Avogadro's number of them. This makes it possible to scale up from the individual photons or atoms to amounts that are meaningful in everyday chemical contexts, which is what we aim for in exercises like calculating the energy of 1 mol of photons.

The wavelength-energy relationship is a critical concept in understanding the behavior of photons. The energy of a photon is inversely proportional to its wavelength, meaning that photons with shorter wavelengths have higher energy. The relationship can be expressed using the formula \(E = \frac{hc}{\lambda}\), where E is the energy of the photon, h is Planck's constant, c is the speed of light, and \lambda is the wavelength of the photon.

The phrase 'shorter wavelength equals higher energy' can be likened to how a short, tight guitar string vibrates faster and creates a higher-pitched sound compared to a longer, looser string. In a similar fashion, photons with shorter wavelengths pack more energy than those with longer wavelengths. This concept is central when answering questions like those in the textbook exercise, where we determine the energy of photons based on their wavelength.

The speed of light, denoted by c, is a fundamental physical constant representing the speed at which light travels in a vacuum. The approximate value of the speed of light is \(3.00 \times 10^8\, \text{m/s}\). It is not just the speed of light, but the maximum speed at which all energy, matter, and information in the universe can travel.

Imagine if you were driving at the speed of light—you could circle the Earth nearly 7.5 times in just one second! Although this is a striking image, in photon energy calculations, we use the speed of light as a vital part of determining a photon's energy. Recognizing and understanding the value of c is an essential component for accurately calculating the energy of photons as demonstrated in the energy formula used in the textbook solution.