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Chapter 4: Problem 5

Describe Bohr s theory of hydrogen atom. How is it successful in explaining the spectrum of hydrogen atom?

Short Answer

Expert verified
Bohr's theory explains the hydrogen spectrum by quantized energy levels and electron transitions emitting specific wavelengths, accurately predicting spectral lines.

Step by step solution

01

- Introduction to Bohr's Theory

Niels Bohr proposed a model of the hydrogen atom in 1913, which was based on quantized energy levels. According to this model, electrons orbit the nucleus in specific, quantized orbits without radiating energy.
02

- Energy Levels

In Bohr's model, electrons can only occupy certain allowed orbits, each with a fixed energy level. The energy levels are given by the formula: \[ E_n = - \frac{13.6}{n^2} \text{eV} \] where \( n \) is the principal quantum number (n = 1, 2, 3, ...).
03

- Electron Transitions

When an electron transitions between these quantized orbits, energy is absorbed or emitted in the form of photons. The energy of the photon corresponds to the difference between two energy levels: \[ E_{photon} = E_i - E_f \]
04

- Explanation of Hydrogen Spectrum

The spectrum of the hydrogen atom consists of discrete lines, which Bohr's theory explains by these quantized energy differences. When an electron transitions from a higher energy level to a lower one, it emits a photon with an energy corresponding to the difference between these levels. This produces light of a specific wavelength.
05

- Successes of Bohr's Theory

Bohr's theory successfully explains the Rydberg formula for the spectral lines of hydrogen. It accurately predicts the wavelengths of the Lyman, Balmer, and other spectral series of hydrogen.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

quantized energy levels
In Bohr's model of the hydrogen atom, one of the principal concepts is 'quantized energy levels'. This means that electrons can only occupy specific energy levels, or orbits, rather than being anywhere around the nucleus.
Each energy level corresponds to a fixed amount of energy. The energy of these levels is given by the formula: \[ E_n = - \frac{13.6}{n^2} \text{eV} \]
Here:
  • E_n (energy of the level) is measured in electronvolts (eV),
electron transitions
Electron transitions refer to the movement of an electron between these quantized energy levels. This transition is accompanied by either the absorption or emission of energy in the form of photons (particles of light).
When an electron absorbs energy, it moves to a higher energy level (away from the nucleus). Conversely, when it emits energy, it drops to a lower energy level (closer to the nucleus). The energy of the photon absorbed or emitted is exactly equal to the difference between the initial and final energy levels:
\[ E_{photon} = E_i - E_f \]
Key points to remember:
  • The energy difference determines the photon's color
  • Higher transitions mean higher energy and shorter wavelength photons,
hydrogen spectrum
What is the hydrogen spectrum? It is essentially the set of wavelengths or colors of light emitted by electrons in a hydrogen atom as they transition between energy levels. When we talk about the hydrogen spectrum, we’re describing the discrete lines of color that result from these transitions.
Bohr's theory explains the hydrogen spectrum by quantizing these transitions:
  • When an electron drops from a higher energy level to a lower one, it emits a photon of a specific wavelength.
  • This is seen as a bright line in the spectrum,
  • Different series of lines (e.g., Lyman, Balmer) correspond to drops to different lower energy levels.

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Most popular questions from this chapter

The lowest two excited states of hydrogen atom are \(10.2 \mathrm{eV}\) and \(12 \mathrm{eV}\) above the ground state, calculate the wavelength of radiation that could be produced by transition between these states and the ground state.

How much energy is required to raise the hydrogen atom from the ground state \(n=1\) to the first excited state \(n=2\) ? What is the wavelength of the line emitted if the atom returned back to the ground state.

The energy of a Bohr orbit is given by \(-B / n^{2}\), where \(B=2.179 \times 10^{-18} \mathrm{~J}\). Calculate the frequency of radiation, wavelenght and also the wave number, when the electron jumps from the third orbit to the second.

The critical potential of hydrogen is \(13.65 \mathrm{eV}\). Calculate the wavelength of the radiation emitted by hydrogen atom bombarded by an electron of corresponding energy.

Electrons of energies \(10.2 \mathrm{eV}\) and \(12.09 \mathrm{eV}\) can \text { cause radiation to be emitted from hydrogen atoms. Calculate in each case, the principal quantum number of the orbit to whichan electron in the hydrogen atom is raised and the wavelength of radiation if it drops back to the ground state.
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