Question

(a) In terms of the Bohr theory of the hydrogen atom, what process is occurring when excited hydrogen atoms emit radiant energy of certain wavelengths and only those wavelengths? (b) Does a hydrogen atom "expand" or "contract" as it moves from its ground state to an excited state?

Short answer

(a) In the Bohr model of the hydrogen atom, the process occurring when excited hydrogen atoms emit radiant energy of certain wavelengths is the transition of the electron from a higher energy level to a lower energy level, which releases the difference in energy as radiant energy. Only discrete energy levels and specific energy transitions are possible, so only particular wavelengths of light are emitted. (b) A hydrogen atom "expands" when it moves from its ground state to an excited state. This is because the electron moves to a higher energy level or orbit, which is farther away from the nucleus, occupying a larger space around the nucleus.

Step-by-step solution

(a) Understanding the Bohr Model

The Bohr Model of the hydrogen atom comprises of a nucleus with a single electron moving around it in specific orbits or energy levels. These energy levels have discrete energy values, and this characteristic of discrete energy levels is what leads to the emission of radiant energy with certain wavelengths.

(a) Energy Emission Process

When a hydrogen atom is excited by a source of energy like heat or light, the electron can absorb that energy and jump to a higher energy level. But this excited state is unstable, so the electron will eventually return to its original energy level (or ground state). When the electron moves from a higher energy level to a lower energy level, the difference in their energy levels is emitted as radiant energy.

(a) Relationship between Energy and Wavelength

The emitted radiant energy has a direct relationship to the wavelength of the emitted light. The greater the energy difference between the energy levels, the shorter the wavelength of the emitted light. This relationship is given by the equation \(E = h \cdot f\), where \(E\) is the emitted energy, \(h\) is the Planck's constant, and \(f\) is the frequency of the emitted light. Since only particular energy level transitions are possible in a hydrogen atom, only specific wavelengths of light (corresponding to those energy differences) are emitted.

(b) Ground State and Excited State

The ground state of a hydrogen atom refers to the state when the electron is in the lowest energy level (n=1). An excited state occurs when the electron moves to a higher energy level (n>1) after absorbing energy.

(b) Expansion or Contraction of the hydrogen atom

As the electron moves from the ground state to an excited state, it occupies a higher energy level or orbit, which is farther away from the nucleus. This means that the hydrogen atom effectively "expands" when it moves from the ground state to an excited state because the electron now occupies a larger space around the nucleus.