Question

The wavelength of the green light in the hydrogen line spectrum is \(434.1 \mathrm{~nm}\). What is the photon energy of the green light emitted by a hydrogen atom?

Short answer

The photon energy of the green light emitted by a hydrogen atom is 2.85 eV.

Step-by-step solution

Convert wavelength to meters

Given that the wavelength of light is 434.1 nm, and knowing that 1 nm equals \(1 \times 10^{-9}\) meters, we can convert the wavelength to meters: \(\lambda = 434.1 nm = 434.1 \times 10^{-9} m\).

Calculate photon energy

Now, plug the value for \(\lambda\), as well as the constants \(h\) and \(c\) into Planck's formula to find the energy of the photon: \(E = \frac{hc}{\lambda} = \frac{(6.63 \times 10^{-34} J.s)(3 \times 10^{8} m/s)}{434.1 \times 10^{-9} m}\).

Calculate and express the result in electron volts

When you do the arithmetic, you get: \(E = 4.57 \times 10^{-19} J\). However, this unit of energy (joules) is quite small for atomic scales, so it's customary to express the result in electron volts (eV). Given that \(1 eV = 1.60 \times 10^{-19} J\), divide the energy by the value of 1 eV to express it in eV: \(E = \frac{4.57 \times 10^{-19} J}{1.60 \times 10^{-19} J/eV} = 2.85 eV\).

Key concepts

Hydrogen Line Spectrum

When observing the hydrogen atom, one of the most prominent features is its line spectrum. This spectrum is created when electrons in hydrogen atoms transition between energy levels. Each transition results in the emission or absorption of light at specific wavelengths, producing a series of discrete colored lines known as the spectral lines.

For hydrogen, these lines can be seen in different regions of the electromagnetic spectrum, including the visible range. For instance, the Balmer series consists of lines visible to the human eye. The wavelength given in the exercise, 434.1 nm, typically corresponds to one of the lines in the Balmer series. This particular green light emission occurs when an electron transitions from a higher energy level to the second energy level in the hydrogen atom.

The visibility of these lines not only confirms the quantized nature of electron energies in atoms but also provides a powerful tool for analyzing the atomic structure and understanding the interaction of light and matter.

Planck's Formula

The energy of a photon can be calculated using Planck's formula, which describes the relationship between the energy of a photon (\(E\)) and its wavelength (\( \text{\lambda} \formatting\))). The formula is expressed as:
\[E = \frac{hc}{\text{\lambda}}\]
where \(h\) is Planck's constant (approximately \(6.63 \times 10^{-34} J.s\)) and \(c\) is the speed of light in a vacuum (about \(3 \times 10^{8} m/s\)).

By this relationship, we can see that the energy of a photon is inversely proportional to its wavelength; that is, shorter wavelengths correspond to higher energies. This becomes particularly important in understanding the transition of electrons in atoms: the larger the energy difference between the levels, the shorter the wavelength of the emitted photon. The use of Planck's formula enables us to calculate the energy associated with a specific wavelength of light, like the green light specified in our textbook exercise.

Electron Volts

An electron volt (eV) is a unit of energy commonly used in the fields of atomic and particle physics. It is defined as the amount of kinetic energy gained by a single electron when it accelerates through an electric potential difference of one volt. It's a much more convenient unit for these small-scale phenomena than the joule, which is why scientists often convert energy measurements into electron volts when talking about atomic energy levels.

The relationship between electron volts and joules is given by the conversion factor:
\[1 eV = 1.60 \times 10^{-19} J\]
This unit allows for a more intuitive grasp of the energy scale involved in photon emission and absorption, such as the energy of the green light emitted by a hydrogen atom in our exercise. Expressing energy in electron volts simplifies comparisons and calculations when working with subatomic particles and quantum mechanical processes.