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]

Name: Chemistry: A Molecular Approach
Author: Tro
ISBN: 9780134874371

검증된 단계별 안내

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.

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.