When atoms are excited, such as by passing an electric current through a gas, they absorb energy and later release it as light. This emitted light is not a continuous rainbow but consists of distinct, sharp lines of color, known as a line spectrum. Each line corresponds to a specific, unique wavelength of light, which acts like a fingerprint for the element that produced it. The hydrogen atom, the simplest element, produces a particularly clear and predictable spectrum of light. Scientists have categorized these distinct patterns of light into different spectral series, and one of these groups, the Balmer series, holds a special place in understanding atomic structure.
Understanding Atomic Emission and Energy Levels
The appearance of specific lines in a spectrum provides direct evidence that electrons within an atom can only possess certain, fixed amounts of energy. These permitted energy states are referred to as discrete energy levels, which can be visualized as shells surrounding the nucleus. An electron typically resides in the lowest possible energy level, known as the ground state, but it can jump to a higher, excited state if the atom absorbs energy.
An atom in an excited state is unstable, and the electron will quickly fall back down to a lower energy level. As the electron transitions from a high-energy level to a lower one, it must release the excess energy it possesses. This energy is emitted in the form of a single packet of light, called a photon. The energy of the emitted photon is exactly equal to the difference in energy (\(\Delta E\)) between the two levels the electron jumped between.
This energy difference is directly related to the wavelength (\(\lambda\)) of the emitted light by the fundamental equation \(E=hc/\lambda\), where \(h\) is Planck’s constant and \(c\) is the speed of light. A large energy drop produces a high-energy, short-wavelength photon, often in the ultraviolet region. Conversely, a small energy drop results in a low-energy, long-wavelength photon, typically found in the infrared or far-red part of the spectrum. These energy levels are mathematically identified by a whole number, the principal quantum number, denoted by the letter \(n\), where \(n=1\) is the lowest level closest to the nucleus, \(n=2\) is the next level, and so on.
Defining the Balmer Series
The Balmer series is a specific collection of spectral lines resulting from electron transitions in the hydrogen atom. This series is uniquely defined by the condition that the electron’s final destination, the lower energy level, is always the second energy level, or \(n=2\). The electron can begin its descent from any higher level, meaning its initial energy level (\(n_{initial}\)) must be \(n=3\), \(n=4\), \(n=5\), and beyond.
The Balmer series was the first hydrogen spectral series discovered because its emissions include four lines visible to the human eye. These visible lines appear as red, blue-green, blue, and violet, ranging from approximately \(656 \text{ nm}\) down to \(410 \text{ nm}\). Other hydrogen series, like the Lyman (ending at \(n=1\)) or Paschen (ending at \(n=3\)) series, produce photons outside the visible range, primarily in the ultraviolet and infrared regions.
Calculating the Longest Wavelength
To determine the longest possible wavelength emitted in the Balmer series, we must identify the transition involving the smallest change in energy (\(\Delta E\)). Since wavelength and energy are inversely related, the lowest energy photon will possess the greatest wavelength. All Balmer transitions end at the \(n=2\) energy level, so the smallest energy difference occurs when the electron starts from the level immediately above \(n=2\). This transition is the drop from \(n_{initial}=3\) down to \(n_{final}=2\), known as the H-alpha line, which causes the characteristic red glow in hydrogen gas clouds.
Applying the Rydberg Formula
The exact wavelength is calculated using the Rydberg formula, a mathematical generalization that describes all hydrogen spectral lines: \(1/\lambda = R_H (1/n_1^2 – 1/n_2^2)\). In this formula, \(R_H\) is the Rydberg constant for hydrogen (\(1.097 \times 10^7 \text{ m}^{-1}\)), \(n_1\) is the final level (2 for Balmer), and \(n_2\) is the initial level (3 for the longest wavelength). Substituting these values yields the inverse of the longest wavelength: \(1/\lambda = 1.097 \times 10^7 \text{ m}^{-1} (1/2^2 – 1/3^2)\).
Calculating the Result
The term in the parentheses simplifies to \((1/4 – 1/9)\), which is \(5/36\). Performing the calculation, \(1/\lambda = 1.097 \times 10^7 \text{ m}^{-1} \times (5/36)\), results in an inverse wavelength of approximately \(1,523,611 \text{ m}^{-1}\). Taking the reciprocal to find the wavelength, \(\lambda\), gives a value of \(6.564 \times 10^{-7} \text{ meters}\). Therefore, the longest possible wavelength emitted in the Balmer series is approximately \(656.4 \text{ nanometers}\) (nm).