In a technique used for surface analysis called Auger electron spectroscopy (AES), electrons are accelerated toward a metal surface. These electrons cause the emissions of secondary electrons—called Auger electrons—from the metal surface. The kinetic energy of the Auger electrons depends on the composition of the surface. The presence of oxygen atoms on the surface results in Auger electrons with a kinetic energy of approximately 506 eV. What is the de Broglie wavelength of one of these electrons? [KE = 1/2mv^2; 1 electron volt (eV) = 1.602 * 10^(-19) J]

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Convert the kinetic energy from electron volts (eV) to joules (J) using the conversion factor: 1 eV = 1.602 \times 10^{-19} J.

Use the kinetic energy formula KE = \frac{1}{2}mv^2 to solve for the velocity (v) of the electron. Rearrange the formula to v = \sqrt{\frac{2 \times KE}{m}}, where m is the mass of an electron (9.109 \times 10^{-31} kg).

Substitute the kinetic energy in joules and the mass of the electron into the velocity equation to find the velocity.

Use the de Broglie wavelength formula \lambda = \frac{h}{mv}, where h is Planck's constant (6.626 \times 10^{-34} m^2 kg/s).

Substitute the values for Planck's constant, the mass of the electron, and the velocity into the de Broglie wavelength formula to find the wavelength.

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Kinetic Energy and Electron Mass

Kinetic energy (KE) is the energy an object possesses due to its motion, calculated using the formula KE = 1/2 mv^2, where m is mass and v is velocity. For electrons, which have a very small mass (approximately 9.11 x 10^-31 kg), their kinetic energy can be converted from electron volts (eV) to joules (J) for calculations. Understanding this relationship is crucial for determining the velocity of the electrons based on their kinetic energy.

de Broglie Wavelength

The de Broglie wavelength is a concept that describes the wave-like behavior of particles, including electrons. It is given by the formula λ = h/p, where λ is the wavelength, h is Planck's constant (6.626 x 10^-34 Js), and p is the momentum of the particle. For electrons, momentum can be expressed as p = mv, linking their mass and velocity to their wave properties, which is essential for understanding their behavior in quantum mechanics.

Conversion of Energy Units

In the context of this problem, converting energy from electron volts (eV) to joules (J) is necessary for calculations involving kinetic energy and de Broglie wavelength. The conversion factor is 1 eV = 1.602 x 10^-19 J. This understanding is vital for accurately applying the kinetic energy in the de Broglie wavelength formula, ensuring that all units are consistent and calculations yield correct results.

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