Question

The particle nature of light was first proposed by Albert Einstein, who suggested that light could be described as a stream of particles called photons. A photon of wavelength \(\lambda\) has an energy (E) given by the equation: \(E=h c / \lambda\), where \(E\) is the energy of the photon in \(\mathrm{J}, h\) is Planck's constant \(\left(6.626 \times 10^{-34} \mathrm{~J} \cdot \mathrm{s}\right)\), and \(c\) is the speed of light \(\left(3.00 \times 10^{8} \mathrm{~m} / \mathrm{s}\right)\). Calculate the energy of \(1 \mathrm{~mol}\) of photons with a wavelength of \(632 \mathrm{~nm}\).

Short answer

The energy of 1 mole of photons with a wavelength of 632 nm is calculated by determining the energy of a single photon using the given equation and constants, and then multiplying by Avogadro's number.

Step-by-step solution

Understanding the problem and gathering constants

This problem requests the calculation of the energy of 1 mole of photons with a specific wavelength. We are provided that the energy of a single photon is calculated using the equation E = h * c / λ, where E is the energy of the photon, h is Planck's constant (6.626 x 10^-34 J*s), and c is the speed of light (3.00 x 10^8 m/s). Our goal is to use this equation to find the energy of one photon and then to convert that energy to the energy of a mole of photons.

Calculate the energy of a single photon

First, we convert the given wavelength from nanometers to meters because the speed of light (c) is given in meters per second. Wavelength is given as 632 nm, which is 632 x 10^-9 meters. We can then substitute values for h, c and λ into the energy equation: E = h * c / λ = (6.626 x 10^-34 J*s) * (3.00 x 10^8 m/s) / (632 x 10^-9 m).

Convert the Energy of one photon to Energy of 1 mole of photons

Once we have the energy of one photon, we can convert it to the energy of one mole of photon using Avogadro's number (6.022 x 10^23 photons/mol). The total energy of one mole of photons is the energy of one photon multiplied by Avogadro's number.

Perform the calculation

Implement the calculation from step 2 and 3 to get the total energy for a mole of photons. The calculation will be: E_mole = E_photon * Avogadro's number.

Key concepts

Photon Energy Calculation

When we talk about the particle nature of light, we refer to the concept that light can behave as both a wave and a stream of particles known as photons. To calculate the energy (\( E \) ) of a single photon, we can use the formula \( E = \frac{hc}{\text{\lambda}} \) where \text{\lambda} represents the photon's wavelength, \(h\) is Planck's constant, and \(c\) is the speed of light. This equation beautifully ties together the wave and particle descriptions of light, allowing us to calculate how much energy is contained within a single quantum - or packet - of light.

For instance, if the wavelength of the photon is provided, we first convert it to meters if it's given in any other unit like nanometers. Once we have our wavelength in the correct unit, we simply plug the values into this equation to find the energy of one photon. Understanding this process is crucial as it lays the groundwork for various applications in quantum mechanics and photochemistry. It's interesting to note that while this calculation is for a single photon, in practice we often need to consider a substantial number of photons, like a mole as used in chemistry, to appreciate the effects at a macroscopic level.

Planck's Constant

Planck's constant (\(h\)) is a fundamental constant that plays a central role in quantum mechanics. Its value is approximately \(6.626 \times 10^{-34} J\cdot s\), and it relates the energy carried by a photon to its frequency; a sort of atomic-scale conversion factor between the world of the very small (photons and atoms) and the more familiar world of energy as we experience it. Named after Max Planck, the physicist who proposed it, this constant has profound implications for our understanding of the quantum world.

\(h\) is the key to quantifying energy at the microscopic level, allowing us to calculate how energy is quantized, or 'packaged', in discrete units rather than being continuous. Essentially, it helps us understand that energy transfer in the realm of atoms and subatomic particles comes in little 'bumps' of fixed sizes, rather than smoothly and continuously.

Speed of Light

The speed of light (\(c\)) is one of the most well-known physical constants, and it is the speed at which all massless particles and waves, including light, travel in a vacuum. It is approximately \(3.00 \times 10^{8} m/s\). The significance of this value lies in its universality; it is a cosmic speed limit that is the same in all frames of reference, regardless of the observer's relative movement.

\(c\) does not just apply to light, but also to the entire electromagnetic spectrum from radio waves to gamma rays. In our calculations for photon energy, the speed of light connects the properties of the photon as a particle (like its energy) to its wave characteristics (such as frequency and wavelength). This interrelation beautifully embodies how light's dual nature is represented in its speed - something that remains consistent no matter how light behaves.

Avogadro's Number

Avogadro's number is a constant that represents the number of constituent particles, usually atoms or molecules, that are contained in one mole, which is a standard unit of measurement in chemistry. It is approximately equal to \(6.022 \times 10^{23}\) entities per mole. This incredibly large number allows chemists to count out the number of molecules or atoms using mass, which is practical and measurable.

In the context of photon energy, Avogadro's number enables us to scale up from the microscopic energy of a single photon to the macroscopic energy of a mole of photons. When dealing with quantities on a macroscopic scale, which is standard in chemistry, it becomes essential to relate the microscopic quantities to amounts that we can work with more practically. Therefore, by using Avogadro's number, students can easily switch back and forth between the energy of a single photon and the energy of a mole of photons, making calculations much more meaningful in a real-world context.