Question

The first line of the Balmer series occurs at a wavelength of \(656.3 \mathrm{nm}\). What is the energy difference between the two energy levels involved in the emission that results in this spectral line?

Short answer

The energy difference is approximately \(1.89 \, \text{eV}\).

Step-by-step solution

Understand the Problem

The problem asks us to find the energy difference between two energy levels involved in the emission that results in the Balmer series spectral line at a wavelength of \(656.3 \, \text{nm}\). This involves using the formula for energy of a photon which correlates with its wavelength.

Convert Wavelength to Meters

Since wavelength is often used in meters in physics equations, convert \(656.3 \, \text{nm}\) to meters: \(656.3 \, \text{nm} = 656.3 \times 10^{-9} \, \text{m}\).

Use the Energy-Wavelength Formula

Utilize the formula \(E = \frac{hc}{\lambda}\), where \(E\) is the energy, \(h\) is Planck's constant \(6.626 \times 10^{-34} \, \text{J s}\), \(c\) is the speed of light \(3.00 \times 10^8 \, \text{m/s}\), and \(\lambda\) is the wavelength. Substitute \(\lambda = 656.3 \times 10^{-9} \, \text{m}\) into the equation.

Calculate the Energy

Substitute the numerical values into the formula: \[E = \frac{(6.626 \times 10^{-34} \, \text{J s}) \times (3.00 \times 10^8 \, \text{m/s})}{656.3 \times 10^{-9} \, \text{m}}.\]Evaluate this to find \(E \approx 3.03 \times 10^{-19} \, \text{J}\).

Convert Energy to Electron Volts

The energy usually required in problems involving atomic energy levels is in electron volts (eV). Convert the energy from joules to electron volts using the conversion factor: \(1 \, \text{eV} = 1.602 \times 10^{-19} \, \text{J}\).Thus, \(E = \frac{3.03 \times 10^{-19} \, \text{J}}{1.602 \times 10^{-19} \, \text{J/eV}} \approx 1.89 \, \text{eV}\).

Key concepts

Energy Levels

In the context of atomic physics, energy levels refer to the fixed energies an electron can have within an atom. When an electron transitions between two energy levels, it emits or absorbs a photon. This photon carries energy that corresponds to the difference between these two levels.
In the Balmer series, electrons transition from higher energy levels to the second energy level of the hydrogen atom. The specific energy levels involved in this series produce visible light. This is why certain wavelengths are observed in spectral lines. Understanding energy levels helps explain how electrons move within atoms and why certain wavelengths appear in the emission spectra.

Wavelength

Wavelength is a key characteristic of a wave, including light waves, and is defined as the distance between consecutive points of a wave, such as crests or troughs. It is usually measured in meters, but can also be expressed in nanometers (1 nm = 10^-9 m).
In the Balmer series, each wavelength corresponds to a specific transition of an electron between energy levels in a hydrogen atom. The wavelength of 656.3 nm mentioned in the original exercise is significant as it represents a visible red line in the hydrogen spectra.
The wavelength is inversely related to energy, meaning that as the wavelength increases, the energy associated with the wave decreases and vice versa. This relationship is explored using the formula:

• Energy ( \(E\) ) = \( \frac{hc}{\lambda} \), which links energy and wavelength.

Understanding wavelength helps in connecting the concepts of photons and their energies.

Planck's constant

Planck's constant is a fundamental constant in physics, symbolized by \(h\) , and is crucial in the quantum world. It describes the quantization of energy and the relationship between the energy of a photon and the frequency of its associated electromagnetic wave.
The value for Planck's constant is \(6.626 \times 10^{-34} \) J s, and it appears in equations that define the energy of photons, such as \(E = h \times \) frequency.
In the energy-wavelength formula \(E = \frac{hc}{\lambda} \) , Planck's constant helps determine the energy from a given wavelength. This equation highlights how energy is not continuous but is exchanged in discrete units called quanta.

Photon Energy

Photons are elementary particles that carry electromagnetic energy, such as light. The energy of a photon is directly related to its wavelength and frequency. The shorter the wavelength (or higher the frequency), the greater the photon’s energy.
The formula to calculate photon energy, given its wavelength, is \(E = \frac{hc}{\lambda} \). Here, \(E\) refers to the photon's energy, \(h\) is Planck's constant, \(c\) is the speed of light, and \(\lambda\) is the wavelength of the photon.
By understanding photon energy, one can predict how electrons behave when they absorb or emit photons. This forms the basis for understanding phenomena such as spectral lines in the Balmer series, where specific photons are emitted during electronic transitions, marking the fingerprint of elements in spectra.