Skip to main content
Ch.6 - Electronic Structure of Atoms
Brown - Chemistry: The Central Science 15th Edition
Brown15th EditionChemistry: The Central ScienceISBN: 9780137542970Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBrown 15th EditionCh.6 - Electronic Structure of AtomsProblema 95
Capitolo 6, Problema 95

An electron is accelerated through an electric potential to a kinetic energy of 1.6 * 10^-15 J. What is its characteristic wavelength? [Hint: Recall that the kinetic energy of a moving object is E = 1/2 mv^2, where m is the mass of the object and v is the speed of the object.]

Guida verificata passo dopo passo
1
Step 1: Identify the relevant equations. The de Broglie wavelength equation is \( \lambda = \frac{h}{p} \), where \( \lambda \) is the wavelength, \( h \) is Planck's constant, and \( p \) is the momentum of the electron. The momentum \( p \) can be expressed as \( p = mv \), where \( m \) is the mass and \( v \) is the velocity of the electron.
Step 2: Relate kinetic energy to velocity. The kinetic energy \( E \) of the electron is given by \( E = \frac{1}{2}mv^2 \). Rearrange this equation to solve for \( v \): \( v = \sqrt{\frac{2E}{m}} \).
Step 3: Calculate the momentum \( p \) using the velocity. Substitute the expression for \( v \) from Step 2 into the momentum equation \( p = mv \) to get \( p = m \sqrt{\frac{2E}{m}} \). Simplify this to \( p = \sqrt{2mE} \).
Step 4: Substitute the expression for momentum \( p \) into the de Broglie wavelength equation. This gives \( \lambda = \frac{h}{\sqrt{2mE}} \).
Step 5: Substitute known values into the equation. Use Planck's constant \( h = 6.626 \times 10^{-34} \text{ J s} \), the mass of an electron \( m = 9.109 \times 10^{-31} \text{ kg} \), and the given kinetic energy \( E = 1.6 \times 10^{-15} \text{ J} \) to calculate \( \lambda \).

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Kinetic Energy

Kinetic energy is the energy possessed by an object due to its motion, calculated using the formula E = 1/2 mv^2, where m is the mass and v is the velocity of the object. In this context, the kinetic energy of the electron is given as 1.6 * 10^-15 J, which will be used to determine its speed and subsequently its wavelength.
Video consigliato:
Percorso guidato
00:34
Kinetic & Potential Energy

De Broglie Wavelength

The De Broglie wavelength is a concept in quantum mechanics that relates the wavelength of a particle to its momentum, expressed as λ = h/p, where h is Planck's constant and p is the momentum. For an electron, this wavelength can be calculated after determining its velocity from the kinetic energy, illustrating the wave-particle duality of matter.
Video consigliato:
Percorso guidato
00:58
De Broglie Wavelength Formula

Planck's Constant

Planck's constant (h) is a fundamental constant in quantum mechanics, approximately equal to 6.626 x 10^-34 Js. It plays a crucial role in the relationship between energy and frequency of photons, as well as in calculating the De Broglie wavelength of particles, linking classical and quantum physics.
Video consigliato:
Percorso guidato
00:50
Photons and Planck's Constant
Pratica correlata
Domanda del libro di testo

As discussed in the A Closer Look box on 'Measurement and the Uncertainty Principle,' the essence of the uncertainty principle is that we can't make a measurement without disturbing the system that we are measuring. (a) Why can't we measure the position of a subatomic particle without disturbing it?

479
views
Domanda del libro di testo

The series of emission lines of the hydrogen atom for which nf = 3 is called the Paschen series. (b) Calculate the wavelengths of the first three lines in the Paschen series—those for which ni = 4, 5, and 6.

710
views
Domanda del libro di testo

The series of emission lines of the hydrogen atom for which nf = 3 is called the Paschen series. (a) Determine the region of the electromagnetic spectrum in which the lines of the Paschen series are observed.

759
views
Domanda del libro di testo

Determine whether each of the following sets of quantum numbers for the hydrogen atom is valid. If a set is not valid, then indicate which of the quantum numbers has a value that is not valid: e. n = 2, l = 2, ml = 1, ms = +1/2

2
views
Domanda del libro di testo

The Chemistry and Life box in Section 6.7 described the techniques called NMR and MRI. (c) When the 450-MHz photon is absorbed, does it change the spin of the electron or the proton on a hydrogen atom?

401
views