Question

A certain radiation has a wavelength of \(574 \mathrm{nm}\). What is the energy, in joules, of (a) one photon; (b) a mole of photons of this radiation?

Short answer

The energy of one photon and a mole of photons for the given radiation are roughly \(3.45 \times 10^{-19} \mathrm{J}\) and \(207.4 \mathrm{kJ/mol}\) respectively.

Step-by-step solution

Convert wavelength to meters

The wavelength is given in nanometres. Convert it to metres as energy calculations require the SI unit of length. There are \(1 \times 10^{-9}\) metres in one nanometre. So, \(574 \mathrm{nm} = 574 \times 10^{-9}\) metres.

Calculate the Energy of One Photon

Use the Planck-Einstein relation ((\(E = \frac{hc}{\lambda}\)) to calculate the energy of one photon. Here, \(h\) is the Planck's constant, \(6.63 \times 10^{-34} \mathrm{Js}\), \(c\) is the speed of light, \(3 \times 10^8 \mathrm{m/s}\), and \(\lambda\) is the wavelength. Plug in the values and solve for \(E\).

Calculate the Energy of a Mole of Photons

After energy of one photon is obtained, the energy of a mole of photons can be found by multiplying the energy of one photon with Avogadro's number (\(6.022 \times 10^{23} \mathrm{mole^{-1}}\)).

Key concepts

Wavelength Conversion

When dealing with photon energy calculations, wavelength must be in meters. This is because the standard unit for length in physics is the meter. In the given problem, the wavelength is provided in nanometers (nm), which is a non-SI unit.
To convert nanometers to meters, we use the fact that 1 nanometer equals \(1 \times 10^{-9}\) meters. So, the wavelength of 574 nm becomes \(574 \times 10^{-9}\) meters.

• 1 nm = \(1 \times 10^{-9}\) m
• 574 nm = \(574 \times 10^{-9}\) m = \(5.74 \times 10^{-7}\) m

This conversion is essential to ensure compatibility in our calculations, especially when dealing with other constants like the speed of light and Planck's constant, which are typically given in meters.

Planck-Einstein Relation

The Planck-Einstein relation is pivotal in determining the energy of a photon. This relation links the energy of a single photon to its wavelength through the formula: \[E = \frac{hc}{\lambda}\] Here:

• \(E\) represents the energy of the photon
• \(h\) is Planck's constant (\(6.63 \times 10^{-34}\) Js)
• \(c\) is the speed of light in a vacuum (\(3 \times 10^8\) m/s)
• \(\lambda\) is the wavelength in meters

By plugging the appropriate values, including our converted wavelength, you can determine the energy of a photon. This step is crucial as it forms the foundation for further calculations, such as finding the energy of more than one photon.

Mole of Photons

Once the energy of a single photon is known, calculating the energy for a mole of photons is straightforward. A mole is a standard unit in chemistry that contains \(6.022 \times 10^{23}\) entities, known as Avogadro's number.
To find the energy of a mole of photons:

• First, calculate the energy of one photon using the Planck-Einstein relation
• Multiply this energy by Avogadro's number

For example, if one photon's energy is \(E\), then a mole of photons will have an energy of \(E \times 6.022 \times 10^{23}\).
This is especially useful in practical scenarios where reactions involve large numbers of photons, allowing us to understand energy transfers at a macroscopic level.