Question

The energy difference between the ground state of an atom and its excited state is \(3 \times 10^{-19} \mathrm{~J}\). What is the wavelength of the photon required for this transition? (a) \(6.6 \times 10^{-34} \mathrm{~m}\) (b) \(3 \times 10^{-8} \mathrm{~m}\) (c) \(1.8 \times 10^{-7} \mathrm{~m}\) (d) \(6.6 \times 10^{-7} \mathrm{~m}\)

Short answer

The wavelength of the photon required for the transition is approximately \(6.6 \times 10^{-7} \mathrm{~m}\), which corresponds to option (d).

Step-by-step solution

Understand the relationship between energy and wavelength

The energy (\(E\)) of a photon is related to its wavelength (\(\lambda\)) by the equation \(E = \frac{hc}{\lambda}\), where \(h\) is Planck's constant (\(6.626 \times 10^{-34} \,\text{Js}\)) and \(c\) is the speed of light in a vacuum (\(3 \times 10^8 \,\text{m/s}\)).

Calculate the wavelength

Use the given energy to find the wavelength with the previous equation: \(\lambda = \frac{hc}{E}\). Substitute the values \(h = 6.626 \times 10^{-34} \,\text{Js} \), \(c = 3 \times 10^8 \,\text{m/s}\), and \(E = 3 \times 10^{-19} \,\text{J}\) into the equation.

Solve for the wavelength

The calculation will be \(\lambda = \frac{6.626 \times 10^{-34} \,\text{Js} \cdot 3 \times 10^8 \,\text{m/s}}{3 \times 10^{-19} \,\text{J}}\). Simplifying this will give the wavelength \(\lambda\) of the photon.

Perform the calculation

\(\lambda = \frac{6.626 \times 10^{-34} \times 3 \times 10^8}{3 \times 10^{-19}} = 6.626 \times 10^{-7} \,\text{m}\)

Determine the correct option

The calculated wavelength corresponds to the option (d) \(6.6 \times 10^{-7} \,\text{m}\), which is the closest value to the calculated wavelength.

Key concepts

Planck's Constant

One of the fundamental constants in physics is Planck's constant, denoted by the symbol 'h'. This constant is exceptionally crucial in the realm of quantum mechanics as it relates to the quantization of energy. The value of Planck's constant is approximately equal to \(6.626 \times 10^{-34}\text{ Js}\) (joule-seconds).

What it essentially means is that energy exists in discrete packets, or 'quanta', as opposed to being continuous. Specifically, it tells us how much energy is carried by photons or particles of light. The constant represents the proportionality factor that connects the energy of a photon to the frequency of its associated electromagnetic wave. This relationship is famously encapsulated by the equation \(E = hf\text{,}\) where \(E\text{}\) is the energy of the photon, \(h\text{}\) is Planck's constant, and \(f\text{}\) is the frequency of the photon. This concept underscores the particle-like properties of light, a key principle of quantum mechanics.

Photon Energy Calculation

Quantifying the energy of a photon is essential to comprehend various phenomena in quantum physics and chemistry, particularly those involving electronic transitions in atoms or in photosensitive materials.

The equation used to calculate this energy is \(E = \frac{hc}{\text{}}\text{,}\) where \(E\text{ }\) denotes the energy of the photon, \(h\text{ }\) is Planck's constant, \(c\text{ }\) represents the speed of light in a vacuum (approximately \(3 \times 10^8\text{ m/s}\text{),}\) and \(\text{ }\) corresponds to the wavelength of the photon.

It is imperative to note that the longer the wavelength of a photon, the less energy it contains. Conversely, a shorter wavelength signifies a higher energy photon. This inverse relationship is pivotal to understanding the behavior of photons in various contexts, such as light absorption, emission spectra, and even the functioning of solar panels. Essentially, the energy of the photon determines the possible interactions it can have with other particles or fields, playing a key role in applications ranging from medical imaging to solar energy collection.

Wavelength of a Photon

The wavelength of a photon is a measure of the distance over which the photon's electromagnetic wave repeats itself. It is intimately connected to the photon's energy through the equation \(E = \frac{hc}{\text{}}\text{,}\) where \(\text{ }\) is the wavelength, further emphasizing an essential quantum mechanical principle that light exhibits both particle and wave characteristics.

To visualize it, think of the wavelength as the distance between two successive crests or troughs of the wave associated with a photon. It is typically measured in meters but can vary immensely, from the very short wavelengths of gamma rays to the extremely long wavelengths of radio waves. Photons with different wavelengths make up the electromagnetic spectrum, encompassing all varieties of light, many of which are not visible to the human eye.

Different wavelengths of light have different applications and behaviors. For instance, ultraviolet light can initiate chemical reactions, while infrared light is often related to heat. Therefore, the wavelength of a photon is not just a distance measure; it reveals a wealth of information about the photon itself, including its potential interactions with matter. It is a cornerstone concept for students and professionals working with lasers, optical fibers, and other technologies relying on light manipulation.