Question

Use Planck's equation (8.3) to determine (a) the energy, in joules per photon, of radiation of frequency \(7.39 \times 10^{15} \mathrm{s}^{-1}\) (b) the energy, in kilojoules per mole, of radiation of frequency \(1.97 \times 10^{14} \mathrm{s}^{-1}\)

Short answer

a) The energy of radiation per photon is around \(4.9 \times 10^{-18}\) Joules. b) The energy of radiation per mole is around \(78.1\) Kilojoules.

Step-by-step solution

Calculating photon energy (part a)

Using the Planck's equation \(E = h\times v\), and substituting Planck's constant \(h\) as \(6.626 \times10^{-34}\) Js and frequency \(v\) as \(7.39 \times 10^{15}\) s^(-1), we can calculate the energy \(E\).

Convert joules to kilojoules (part b)

Still using the Planck's equation, we substitute Planck's constant and frequency \(v\) as \(1.97 \times 10^{14}\) s^(-1) to find the energy per photon. Then we multiply this value by Avogadro's number, \(6.022 \times 10^{23}\) to find the energy per mole, and divide by \(10^{3}\) to convert from Joules to Kilojoules.

Key concepts

Photon Energy

Photon energy refers to the energy carried by a single photon, which is the particle-like unit of light. An essential equation used in calculating photon energy is Planck's equation, given by: \[E = h \times v\] Here, \(E\) is the energy of the photon, \(h\) is Planck's constant \((6.626 \times 10^{-34} \text{Js})\), and \(v\) is the frequency of the photon in hertz (Hz).
Planck's equation is crucial in understanding the relationship between the energy of light and its frequency. It shows that photon energy directly depends on frequency: the higher the frequency, the greater the energy of the photon.
By applying this formula, you can convert the frequency of radiation, given in hertz, to energy in joules per photon. Understanding this concept helps in various fields like quantum physics, chemistry, and even in applications such as photovoltaic energy systems, where the energy from light is used to generate electricity.

Avogadro's Number

Avogadro's number is a fundamental constant that represents the number of constituent particles (usually atoms or molecules) in one mole of a substance. Its value is: \(6.022 \times 10^{23}\) entities per mole. This large number is significant because it allows chemists to count particles by weighing macroscopic amounts of material, which is easier to measure in the lab. When talking about electron or photon energy in chemistry, Avogadro's number is crucial. It helps convert the energy of a single photon (in joules) to the energy contained per mole, which is more useful for practical chemical and physical calculations.
In part b of our exercise, to determine the energy in kilojoules per mole instead of joules per photon, we multiply the energy for one photon by Avogadro's number. This calculation shows how much energy a mole of these photons would contain and is typically expressed in kilojoules when dealing with larger values of energy.

Frequency

Frequency is a term commonly used in physics to describe the number of cycles of a wave that pass a point in one second. The unit of frequency is hertz (Hz), where 1 Hz means one cycle per second. In the context of light and electromagnetic radiation, frequency refers to how often the wave oscillates as it travels through space.
The relationship between frequency, wavelength, and the speed of light is given by the equation: \[c = \lambda \times v\] where \(c\) is the speed of light \((3.00 \times 10^8 \text{ m/s})\), \(\lambda\) is the wavelength, and \(v\) is the frequency.
Understanding frequency is important in the calculation of photon energy using Planck's equation. A change in frequency will directly affect the energy of photons because, as mentioned, higher frequency means higher energy. This concept is fundamental when studying the spectral characteristics of light and analyzing phenomena like the photoelectric effect, where light frequency determines the energy needed to eject electrons from a material.