Question

When a tube containing hydrogen atoms is energized by passing several thousand volts of electricity into the tube, the hydrogen emits light that, when passed through a prism, resolves into the "bright line" spectrum shown in Fig. \(11.10 .\) Why do hydrogen atoms emit bright lines of specific wavelengths rather than a continuous spectrum?

Short answer

Hydrogen atoms emit bright lines of specific wavelengths rather than a continuous spectrum because of their quantized energy levels and allowed electron transitions between these energy levels. When hydrogen atoms are energized, their electrons transition between specific energy levels, releasing energy in the form of light with distinct wavelengths corresponding to the energy differences between the levels. This results in a line spectrum with distinct wavelengths, rather than a continuous spectrum.

Step-by-step solution

Understanding energy levels in hydrogen

In a hydrogen atom, the single electron occupies different energy levels. These energy levels are quantized, meaning that the electron can only occupy specific levels of energy. The difference between the energy levels determines the wavelength of light emitted or absorbed when an electron transitions between the levels.

Electron transitions between energy levels

When hydrogen atoms are energized with electricity, their electrons are excited and can jump to higher energy levels. However, these excited electrons tend to return to their original, lower energy levels, as it is a more stable state. When the electron returns to a lower energy level, the energy difference between the two energy levels is released in the form of light, or a photon.

Calculating the wavelength of emitted light

The energy of the emitted photon corresponds to the energy difference between the two energy levels involved in the electron transition. The energy of the emitted photon, E, can be related to its wavelength, λ, using the Planck's equation: \(E = h \cdot f\) where \(h\) is Planck's constant (\(6.63 \times 10^{-34} Js\)) and \(f\) is the frequency of the emitted light. The frequency, f, is related to the light's wavelength through the speed of light, c: \(f = \frac{c}{\lambda}\) Therefore, we can write the energy of the emitted photon as: \(E = \frac{h \cdot c}{\lambda}\)

Explaining the line spectrum

Since the energy levels are quantized, only specific energy differences are allowed, which in turn means that only specific wavelengths of light can be emitted. Therefore, a line spectrum with distinct wavelengths of light is observed when hydrogen atoms emit light, rather than a continuous spectrum. Each bright line in the spectrum corresponds to a particular transition between energy levels in hydrogen atoms. In conclusion, hydrogen atoms emit bright lines of specific wavelengths rather than continuous spectra due to the quantized nature of their energy levels and the corresponding allowed electron transitions between these energy levels.

Key concepts

Energy Levels in Hydrogen

Imagine a hydrogen atom as a tiny, energetic solar system, with its core nucleus acting like the sun and the lone electron as a planet orbiting at certain distances. This electron, however, doesn't follow the free-will paths we might expect in space; instead, it's restricted to specific orbits or energy levels. These levels are not continuous; like steps on a ladder, the electron can only stand firmly on one step at a time, not in between. This is because energy levels in a hydrogen atom are quantized. The electron can't simply choose any orbit; it must adhere to a distinct set of energy rungs.

To sustain an orbit further from the nucleus, the electron requires more energy, much like you need more strength to climb higher on a ladder. When energized with electricity, electrons get a surge of energy, allowing them to ascend to higher orbits. However, they can't stay there for long and will eventually release energy to fall back to a lower, more comfortable rung, emitting light in the process. This emission leads to the creation of the hydrogen atom's hallmark, its spectrum.

Electron Transitions

Electron transitions in a hydrogen atom are like dance steps executed across the energy levels. When we introduce a high voltage, we are essentially inviting the electron to dance, causing it to take a leap to a higher energy level. However, our electron is a creature of habit and prefers its original, low-energy state; thus, in no time, it 'steps' back down. These transitions are not random; they must adhere to specific quantized rules of energy change.

Each step down releases a specific amount of energy in the form of a photon, which is a tiny particle of light. The amount of energy released determines the color of the light and, therefore, its position in the spectrum. Just like a dancer hits only specific spots on the floor, an electron only emits light at certain wavelengths, which is why we see the beautiful line spectrum, not a blur of continuous color.

Quantized Energy Levels

The concept of quantized energy levels is a revolutionary idea that changed how scientists view the atomic world. It suggests that an electron's energy is not a sliding scale but a set of fixed rungs on the energy ladder, each representing an allowed energy state. These are the only energy levels at which the electron can exist. It's not a gradient, but more like a series of discrete steps; either you're on one step or another, not in between.

This 'quantization' of energy levels leads directly to the line spectrum's existence, as it restricts the electron's energy changes to specific values. This is the heart of why we don't see a smear of colors in hydrogen's emission spectrum. Instead, we encounter neat lines each corresponding to the jump or fall from one quantized energy level to another, painting a very distinctive picture of the hydrogen atom's energetic possibilities.

Planck's Equation

At the turn of the 20th century, Max Planck ushered in the quantum revolution with an equation that ties the energy of light to its frequency. Planck's equation is elegantly simple: \(E = h \cdot f\), where \(E\) represents the energy of the photons, \(h\) is Planck's constant, and \(f\) is the frequency of the emitted light. This equation implies that light's energy is quantized, just like the hydrogen atom's energy levels.

Moreover, when we connect the frequency of light to its wavelength through the expression \(f = \frac{c}{\lambda}\), we gain a powerful tool to calculate the exact wavelengths of the light emitted during electron transitions. So, not only does Planck's equation prove that energy is quantized, but it also enables us to predict the spectrum of light emitted by atoms with an accuracy that was previously impossible. Thus, whenever an electron falls to a lower energy level, it emits a photon with a very specific energy and, consequently, a very specific wavelength, filling in another piece in the fascinating spectral puzzle of hydrogen.