What is the wavelength, in \(\mathrm{nm}\), of radiation that has an energy content of \(1.0 \times 10^{3} \mathrm{~kJ} / \mathrm{mol} ?\) In which region of the electromagnetic spectrum is this radiation found?

Understanding Planck's equation is crucial when calculating the wavelength of radiation based on its energy content. Planck's equation, which is a fundamental principle in quantum mechanics, relates the energy of a photon to the frequency of its associated electromagnetic wave. The equation is simply stated as:
\[ E = hu \]
where \(E\) is the energy of the photon, \(h\) is Planck's constant (6.626 x 10^-34 J·s), and \(u\) is the frequency of the electromagnetic wave. But how do we go from the frequency to the wavelength? Since the speed of light \(c\) is equal to the wavelength \(\lambda\) times the frequency \(u\) (\(c = \lambdau\)), we can rearrange Planck's equation to solve for the wavelength:
\[ \lambda = \frac{hc}{E} \]
This formula allows us to calculate the wavelength if we know the energy per photon. When applying Planck's equation, it's essential to use units consistently, converting energy to joules and remembering to convert the wavelength to the desired unit, such as nanometres (nm), for ease of understanding.

The electromagnetic spectrum is a continuum of all electromagnetic waves arranged according to frequency or wavelength. It includes a range of wavelengths from the shortest gamma rays to the longest radio waves. Understanding the electromagnetic spectrum helps us to classify radiation based on its wavelength. Among the various types we have:

• Gamma rays, which have the shortest wavelength and the highest frequencies.
• X-rays, used in medical imaging and security.
• Ultraviolet light, which has wavelengths shorter than visible light and is often associated with the effects of sunburn.
• Visible light, which can be detected by the human eye, consisting of the colors we see in a rainbow, ranging from violet to red.
• Infrared radiation, with wavelengths longer than visible light, felt as heat.
• Microwaves, used for cooking and communication.
• Radio waves, with the longest wavelength, used for broadcasting audio and video signals.

Knowing where various types of radiation fall within the electromagnetic spectrum helps us identify their applications and impacts on health and the environment. For instance, in the exercise above, once we calculate the wavelength of the radiation, we can easily categorize it into the correct region of the spectrum, such as Ultraviolet or Infrared, based on its wavelength value.

The energy per photon is a concept that describes the amount of energy carried by a single photon. Photons are particles of light that carry energy as they travel through space. According to quantum theory, light has both particle-like and wave-like properties, and the energy of each photon is quantized, meaning it can only exist in specific, discrete amounts.

To calculate the energy per photon given the energy content per mole, one must consider Avogadro's number (\(6.022 \times 10^{23}\)), which represents the number of photons (or any particulate entities) in one mole. By dividing the total energy per mole by Avogadro's number, we find the energy content of a single photon:
\[ E_{photon} = \frac{E_{mole}}{N_A} \]
The energy content per mole is often provided in units such as kilojoules per mole (kJ/mol), and for the calculation, it must be converted to joules (J), as Planck's constant is in joules seconds (J·s). This process is a fundamental step in determining the relationship between the energy of a photon and its wavelength and thereby identifying its position on the electromagnetic spectrum. It's worth noting that the higher the energy per photon, the shorter the wavelength of the electromagnetic radiation it represents.