Ch.5 - Periodicity & Electronic Structure of Atoms

Problem 133

Chapter 5, Problem 133

X rays with a wavelength of 1.54 * 10-10 m are produced when a copper metal target is bombarded with high-energy electrons that have been accelerated by a voltage difference of 30,000 V. The kinetic energy of the electrons equals the product of the voltage difference and the electronic charge in coulombs, where 1 volt-coulomb = 1 J. (a) What is the kinetic energy in joules and the de Broglie wavelength in meters of an electron that has been accel-erated by a voltage difference of 30,000 V?

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Calculate the kinetic energy (KE) of the electron using the formula KE = e * V, where 'e' is the elementary charge (approximately 1.602 x 10^-19 coulombs) and 'V' is the voltage difference (30,000 V).

Convert the kinetic energy from electron volts (eV) to joules (J) if necessary, using the conversion factor 1 eV = 1.602 x 10^-19 J.

Calculate the momentum (p) of the electron using the formula p = \sqrt{2mKE}, where 'm' is the mass of the electron (approximately 9.109 x 10^-31 kg).

Determine the de Broglie wavelength (\lambda) of the electron using the formula \lambda = \frac{h}{p}, where 'h' is the Planck constant (approximately 6.626 x 10^-34 Js).

Substitute the values of 'h' and 'p' into the de Broglie wavelength formula to find the wavelength in meters.

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Key Concepts

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

Kinetic Energy of Electrons

The kinetic energy (KE) of an electron accelerated through a voltage difference (V) can be calculated using the formula KE = eV, where e is the elementary charge (approximately 1.602 x 10^-19 coulombs). This relationship shows that the energy gained by the electron is directly proportional to the voltage it is accelerated through, allowing us to convert voltage into energy measured in joules.

de Broglie Wavelength

The de Broglie wavelength (λ) of a particle, such as an electron, is given by the formula λ = h/p, where h is Planck's constant (6.626 x 10^-34 Js) and p is the momentum of the particle. For an electron, momentum can be expressed as p = mv, where m is the mass of the electron and v is its velocity. This concept illustrates the wave-particle duality of matter, indicating that particles can exhibit wave-like properties.

Relationship Between Voltage and Electron Velocity

When an electron is accelerated through a voltage, it gains kinetic energy that can be related to its velocity. The relationship can be derived from the kinetic energy formula, KE = 0.5mv^2, allowing us to express the velocity in terms of the voltage. This understanding is crucial for calculating the de Broglie wavelength, as it requires knowledge of the electron's velocity after being accelerated by the given voltage.

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