Question

The blue color in some fireworks occurs when copper(1) chloride is heated to approximately 1500 \(\mathrm{K}\) and emits blue light of wavelength \(4.50 \times 10^{2} \mathrm{nm} .\) How much energy does one photon of this light carry?

Short answer

One photon of the blue light emitted by copper(1) chloride heated to about 1500 K carries approximately \(4.42 \times 10^{-19}\) Joules of energy.

Step-by-step solution

Write down the given information and necessary constants

Given wavelength, λ = \(4.50 \times 10^2\) nm To convert the wavelength to meters, we'll multiply by \(10^{-9}\): λ = \(4.50 \times 10^2 \times 10^{-9}\) m Planck's constant (h) = \(6.63 \times 10^{-34}\) Js Speed of light (c) = \(3.00 \times 10^8\) m/s

Calculate the energy of the photon

The energy (E) of one photon can be calculated using the formula: E = \(\frac{hc}{\lambda}\) Substitute the values for h, c, and λ: E = \(\frac{(6.63 \times 10^{-34}\, \text{Js})(3.00 \times 10^8\, \text{m/s})}{4.50 \times 10^2 \times 10^{-9}\, \text{m}}\)

Solve for energy

Perform the multiplication and division: E ≈ \(4.42 \times 10^{-19}\) J So, one photon of this blue light carries approximately \(4.42 \times 10^{-19}\) Joules of energy.

Key concepts

Planck's Constant

Planck's constant, denoted by the symbol 'h', is a fundamental constant in physics with a value of approximately \(6.63 \times 10^{-34}\) Joule•seconds (J•s). It plays a critical role in quantum mechanics as it relates to the quantization of energy. This is the foundation upon which quantum theory is built; it implies that energy is not continuous and smooth but rather made up of discrete 'packets' called quanta or photons.

When dealing with photons, which are the smallest possible quantities of electromagnetic radiation, Planck's constant helps us calculate their energy. The energy formula has Planck’s constant as one of its key components, illustrating how energy is directly proportional to the frequency of radiation. This is often central to understanding phenomena such as the emission of light from heated copper(1) chloride producing a blue color in fireworks.

To give a practical example, using the constant in calculations enables us to determine the energy carried by a single photon, which is vital in various fields ranging from spectroscopy to the development of new materials and technologies.

Wavelength

Wavelength, symbolized by the Greek letter lambda \(\lambda\), is a property of any wave-like phenomena, including sound waves, light waves, and other electromagnetic radiations. It's defined as the distance between two identical points in consecutive cycles of a wave — for instance, from crest to crest or trough to trough.

With light, the wavelength determines the color we perceive; shorter wavelengths correspond to blue light, while longer wavelengths correspond to red light. In the context of the exercise, the blue light from the heated copper(1) chloride has a wavelength of \(4.50 \times 10^2 \) nanometers (nm). Since energy and wavelength are inversely proportional in photon energy calculations, a shorter wavelength indicates a higher energy photon.

This correlation is essential for understanding the energy involved in various light-based processes. For example, in medical treatments using lasers, selecting the correct wavelength is crucial as it will determine the energy that can be delivered to tissues.

Speed of Light

The speed of light, commonly denoted by 'c', is a universal physical constant significant in many areas of physics. In a vacuum, it has a precise value of \(3.00 \times 10^8\) meters per second (m/s). This figure is not only crucial for the calculation of photon energy but also for our understanding of space and time as woven into the fabric of space-time in Einstein's theory of relativity.

In terms of photon energy calculation, the speed of light ties together wavelength and frequency, joining them into an expression for the photon's energy. The relationship ensures that, given a constant speed of light, photons with higher frequencies (and therefore shorter wavelengths) will have higher energies.

Moreover, the constancy of the speed of light allows for precise calculations when dealing with electromagnetic radiation of any frequency and is imperative in technologies such as GPS, where light-speed calculations are used to determine distances with remarkable accuracy.