Question

What will be the energy of one photon of radiation whose frequency is \(5 \times 10^{14} \mathrm{~Hz}\) ? (a) \(199.51 \mathrm{~kJ} \mathrm{~mol}^{-1}\) (b) \(3.3 \times 10^{-19} \mathrm{~J}\) (c) \(6.626 \times 10^{-34} \mathrm{Js}\) (d) \(2.31 \times 10^{5} \mathrm{~J}\)

Short answer

The energy of one photon with a frequency of is .

Step-by-step solution

Identify the Known Quantities

The frequency of the radiation () is given as.

Recall the Energy-Photon Relationship

Energy of a photon can be calculated using Planck's equation, which is where is Planck's constant () and is the frequency.

Substitute the Known Values

Substitute the given frequency and the value of Planck's constant into the equation to calculate the energy of one photon.

Calculate the Energy of the Photon

Perform the calculation using the substituted values to find the energy of the photon in joules.

Key concepts

Planck's Equation

Planck's equation is foundational to understanding how energy is quantized in units known as photons. According to quantum mechanics, light can be thought of as being made up of these units or 'packets' of energy, with each photon's energy directly proportional to its frequency. The equation is elegantly simple:
\[ E = h u \]
where \( E \) denotes the energy of a single photon, \( h \) represents Planck's constant (approximately \(6.626 \times 10^{-34} \mathrm{J\cdot s}\)), and \( u \) is the frequency of the photon. Understanding and applying this equation is essential when delving into the properties of photons in various applications, such as in determining the energy output in lasers or the color emitted by an LED.
In our exercise, knowing the frequency of a photon allows us to use Planck's equation to determine its energy. This equation is a testament to the quantized nature of energy - a revelation that led to the birth of quantum mechanics.

Photon Frequency

The frequency of a photon is a measure of how many wave cycles pass a particular point each second. It's like counting the number of waves that hit the shore every minute, but on an incredibly rapid and tiny scale. Frequencies are measured in hertz \( (Hz) \), and for light waves, they are typically very high -- in the orders of terahertz and beyond.
Frequency is related not only to the energy of photons, as per Planck's equation, but also to the color of light we perceive; higher frequencies are associated with colors like blue and violet, while lower frequencies are associated with red. The frequency given in our exercise is \(5 \times 10^{14} Hz\), which falls within the visible spectrum and would appear as a greenish color to our eyes. By associating the concept of frequency with visible color, it becomes easier to grasp the otherwise abstract notion of 'how fast' a photon vibrates.

Quantum Mechanics

Quantum mechanics is the branch of physics that deals with the behavior of particles on the atomic and subatomic levels, where the classical laws of physics like Newton's laws of motion don't apply. The theory suggests that energy, just like the photon's, is not continuous but instead comes in discrete units or 'quanta'. This was a radical departure from the thinking of the time and required a new way to mathematically describe the phenomena observed in experiments involving tiny particles like electrons and photons.
Quantum mechanics not only explains the quantization of energy but also introduces the concept of wave-particle duality, where particles exhibit both wave-like and particle-like characteristics. The exercise we are looking at is an example of quantum principles at work, using Planck's equation to calculate the discrete energy packets of photons given their frequency. Understanding the quantum behavior of particles helps us to design new technologies such as semiconductors and quantum computers.