Question

An einstein (E) is a unit of measurement equal to Avogadro's number \(\left(6.02 \cdot 10^{23}\right)\) of photons. How much energy is contained in \(1 \mathrm{E}\) of violet light \((\lambda=400, \mathrm{nm}) ?\)

Short answer

Answer: The energy contained in 1 Einstein of violet light is approximately \(2.99 \cdot 10^5 \mathrm{J}\).

Step-by-step solution

Convert the wavelength to energy using Planck's formula

To find the energy of a single photon of violet light, we can use Planck's formula: \(E = \frac{hc}{\lambda}\), where \(E\) is the energy, \(h\) is the Planck's constant (approximately \(6.626 \cdot 10^{-34} \mathrm{J} \cdot \mathrm{s}\)), \(c\) is the speed of light \((3.00 \cdot 10^{8} \mathrm{m/s})\), and \(\lambda\) is the wavelength. However, the wavelength is given in nanometers (nm) and we must first convert it to meters (m) by dividing by \(10^9\): \(\lambda = 400 \mathrm{nm} \cdot \frac{1 \mathrm{m}}{10^{9} \mathrm{nm}} = 4.00 \cdot 10^{-7} \mathrm{m}\).

Calculate the energy of a single photon

Now we can substitute the values for \(h\), \(c\), and \(\lambda\) into Planck's formula to find the energy of a single photon of violet light: \(E = \frac{(6.626 \cdot 10^{-34} \mathrm{J} \cdot \mathrm{s})(3.00 \cdot 10^{8} \mathrm{m/s})}{4.00 \cdot 10^{-7} \mathrm{m}} = 4.97 \cdot 10^{-19} \mathrm{J}\)

Multiply the energy of a single photon by the number of photons in 1 Einstein

Finally, to find the energy contained in 1 Einstein of violet light, we need to multiply the energy of a single photon (\(4.97 \cdot 10^{-19} \mathrm{J}\)) by the number of photons in 1 Einstein (Avogadro's number, \(\approx 6.02 \cdot 10^{23}\) photons): Energy = \((4.97 \cdot 10^{-19} \mathrm{J}) \cdot (6.02 \cdot 10^{23}) \approx 2.99 \cdot 10^5 \mathrm{J}\) The energy contained in 1 Einstein of violet light is approximately \(2.99 \cdot 10^5 \mathrm{J}\).

Key concepts

Einstein Unit

Introduced by the renowned physicist Albert Einstein, the 'einstein unit' isn't as widely known as the other concepts that emerged from his work. However, it's quite useful in the context of photochemistry and quantum physics. Essentially, an einstein is a mole of photons - that's where Avogadro's number comes into play. We tie it to the amount of substance, just like the way a dozen refers to 12 of anything, an einstein refers to approximately 6.02 x 10^23 photons.

Phyiscists use this unit to quantify large amounts of photon-related entities, such as when measuring the flux in photosynthesis. It's particularly handy because it correlates the macroscopic world that we can measure, such as a mole of a substance, to the microscopic quantum world of particles like photons.

Photon Energy

When we discuss photon energy, we're delving into the tiny packets of energy that make up light. The energy of a single photon is inversely proportional to its wavelength - the longer the wavelength, the lower the energy, and vice versa. Now, Planck's formula, represented as
\(E = \frac{hc}{\lambda}\)

, is pivotal because it links energy (E) with the wavelength (\(\lambda\)) of the photon. In Planck's equation, 'h' stands for Planck's constant, a fundamental constant that describes the size of these energy packets, while 'c' is the unchanging speed of light.

Understanding photon energy is crucial not just in academic physics problems but also in real-world applications like solar power generation, where we convert light into electricity, or in medical applications such as laser surgery where specific photon energies are utilized to interact in precise ways with tissue.

Avogadro's Number

Avogadro's number, approximately 6.02 x 10^23, is a fundamental constant in chemistry and physics. It represents the number of atoms or molecules (or in the case of our einstein unit, photons) in one mole of a substance. This mammoth number is named after Amedeo Avogadro, an Italian scientist, and scales down the interactions between atoms and molecules to a macroscopic level that we can effectively measure and understand.

When we talk about an einstein of light, we're essentially referring to Avogadro's number of photons. It's like saying one mole of photons. This allows chemists and physicists to quantify the amount of a substance involved in a chemical reaction or physical process in a very precise way.

Convert Wavelength to Energy

Converting the wavelength of light to energy is a fundamental practice in physics, helping us understand phenomena like the color of objects, the workings of lasers, and the harvesting of solar energy. The procedure typically involves using Planck's formula, which requires you to convert the wavelength from commonly used units like nanometers into meters, the standard unit in physics for measuring distance.

Once the wavelength is expressed in meters, we can easily substitute it and other constants into Planck's equation to calculate the photon's energy. This process exemplifies how closely intertwined different areas of physics are, with concepts like Avogadro's number assisting in calculating energies on a larger, almost tangible scale - such as the energy in an einstein of light.