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.

To find the wavelength of radiation emitted by a hydrogen atom, we start by understanding how energy and wavelength are related. This relationship is described by the formula: \[ E = \frac{hc}{\text{wavelength}} \] Here:

• E is the energy of the photon (in joules).
• c is the speed of light (around \(3.00 \times 10^8 \text{ m/s})\).
• h is Planck's constant \(6.626 \times 10^{-34} \text{ J s})\).

Note that when an electron impacts a hydrogen atom with energy equivalent to the critical potential energy, it emits radiation with a specific wavelength. This emission of light relates directly back to the principle that energy has an inverse relationship with wavelength. This core concept helps us find the wavelength once the energy is known.

In physics, energy can be measured in different units. Electron volts (eV) is a common unit when discussing atomic and subatomic particles. However, for calculations involving Planck's constant and the speed of light, we use joules (J). We convert electron volts to joules using the relationship:\[1 \text{ eV} = 1.602 \times 10^{-19} \text{ J} \]To convert a given energy from eV to J, multiply by this factor. For example, for the energy of 13.65 eV given in the exercise, we have:\[ 13.65 \text{ eV} \times 1.602 \times 10^{-19} \text{ J/eV} = 2.18643 \times 10^{-18} \text{ J} \] Converting energy into joules ensures that our calculations remain consistent and aligns with the constants we use.

Planck's constant, denoted as h, is a fundamental constant in quantum mechanics. It relates the energy of a photon to the frequency of its associated electromagnetic wave via the equation:\[ E = h u \] Here:

• h is Planck's constant (\( 6.626 \times 10^{-34} \text{ J s}\)).
• E is the energy (in joules).
• ν (nu) is the frequency of the radiation (in Hz).

Planck's constant bridges the behavior of particles and waves, showing that higher frequency light has higher energy photons. To find the wavelength (\( \text{wavelength} \)), you rearrange the energy-wavelength equation to get\( \text{wavelength} = \frac{hc}{E} \). Utilizing Planck's constant is essential in accurately calculating the wavelength of emitted radiation.

Critical potential energy is the minimum energy required to ionize an electron from an atom. For hydrogen, this value is 13.65 eV. When an electron with this energy collides with a hydrogen atom, it excites the atom, causing it to emit radiation. This energy input leads to the emission of a photon with a specific wavelength. This principle is foundational in understanding emission spectra in quantum mechanics.

• Knowing the critical potential helps us calculate the exact wavelength of the resulting radiation.
• The energy is directly used in the energy-wavelength relationship to find the wavelength of emitted photons.

Thus, accurately determining critical potential energy is crucial for predicting and confirming the behavior of atomic emissions in various contexts.