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The concept of electron momentum is central to understanding the behavior of electrons in quantum mechanics. The momentum of an electron is directly tied to its de Broglie wavelength through the formula \( \lambda = \frac{h}{p} \). Here, \( \lambda \) represents the wavelength, \( h \) stands for Planck's constant \((6.63 \times 10^{-34} \, \text{J s})\), and \( p \) is the momentum.

To solve for the momentum when given the de Broglie wavelength, rearrange the equation to \( p = \frac{h}{\lambda} \). Plug in the given values:

• Planck's constant: \(6.63 \times 10^{-34} \text{J s}\)
• Wavelength: \(2.80 \times 10^{-10} \text{m}\)

Calculating the momentum gives \( p \approx 2.37 \times 10^{-24} \, \text{kg m/s} \).

This value provides insight into the moving state of the electron, underpinning both classical concepts and confirming aspects of wave-particle duality.

Kinetic energy is a measure of an electron’s motion and is derived using the electron's momentum. The kinetic energy \( KE \) can be calculated by the formula \( KE = \frac{p^2}{2m} \), where \( m \) is the mass of an electron, approximately \(9.11 \times 10^{-31} \, \text{kg}\).

First, substitute the known momentum value \( p = 2.37 \times 10^{-24} \, \text{kg m/s} \) into the equation to find

• \( KE = \frac{(2.37 \times 10^{-24})^2}{2 \times 9.11 \times 10^{-31}} \approx 3.10 \times 10^{-19} \, \text{J}\)

The result in joules gives a sense of how much energy the moving electron carries due to its velocity.

This energy is fundamental in fields such as thermodynamics and electrodynamics, where electron behavior often dictates energy transfer processes.

Converting energy from joules to electron volts (eV) is a common requirement in physics, especially when dealing with subatomic particles like electrons. This unit conversion is crucial because it provides a more intuitive scale for quantum mechanics experts and researchers.

The conversion factor is:

• 1 eV = \( 1.60 \times 10^{-19} \, \text{J} \)

To convert the calculated kinetic energy from joules to electron volts, use the equation:

• \[ KE \approx \frac{3.10 \times 10^{-19}}{1.60 \times 10^{-19}} \approx 1.94 \, \text{eV} \]

Using eV makes the results more accessible and relevant to experimental contexts where energy levels are often discussed in electron volts.

This simple conversion process is critical for ensuring clarity and communication within the scientific community.