Ch.5 - Periodicity & Electronic Structure of Atoms

Problem 134

Chapter 5, Problem 134

In the Bohr model of atomic structure, electrons are constrained to orbit a nucleus at specific distances, given by the equation
where r is the radius of the orbit, Z is the charge on the nucleus, a0 is the Bohr radius and has a value of 5.292 * 10-11 m, and n is a positive integer (n = 1, 2, 3...) like a principal quantum number. Furthermore, Bohr concluded that the energy level E of an electron in a given orbit is
where e is the charge on an electron. Derive an equation that will let you calculate the difference ∆E between any two energy levels. What relation does your equation have to the Balmer–Rydberg equation?

Verified step by step guidance

Identify the given equations for the Bohr model: the radius of the orbit and the energy level of an electron.

Write down the equation for the energy level E of an electron in a given orbit: E = - (Z^2 * e^4) / (8 * ε0^2 * h^2 * n^2).

To find the difference in energy levels (ΔE) between two orbits, subtract the energy of the initial orbit (E_initial) from the energy of the final orbit (E_final): ΔE = E_final - E_initial.

Substitute the energy level equation into the ΔE equation: ΔE = - (Z^2 * e^4) / (8 * ε0^2 * h^2 * n_final^2) - [ - (Z^2 * e^4) / (8 * ε0^2 * h^2 * n_initial^2) ].

Simplify the equation to get the final form of ΔE: ΔE = (Z^2 * e^4) / (8 * ε0^2 * h^2) * (1/n_initial^2 - 1/n_final^2).

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Key Concepts

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

Bohr Model of the Atom

The Bohr model describes the atom as having electrons orbiting a central nucleus at fixed distances, or energy levels. Each orbit corresponds to a specific energy state, determined by the principal quantum number (n). This model was pivotal in explaining the quantized nature of electron energies and laid the groundwork for modern quantum mechanics.

Energy Level Differences (∆E)

The difference in energy (∆E) between two electron orbits can be calculated using the formula ∆E = E_n - E_m, where E_n and E_m are the energies of the respective orbits. This concept is crucial for understanding electron transitions, as energy is absorbed or emitted when an electron moves between these levels, resulting in spectral lines.

Balmer-Rydberg Equation

The Balmer-Rydberg equation describes the wavelengths of spectral lines emitted by hydrogen as electrons transition between energy levels. It relates the energy difference between levels to the wavelength of emitted light, demonstrating the quantized nature of electron transitions. The derived equation for ∆E in the Bohr model is directly related to this equation, as both describe the same underlying physical phenomena.

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Microwave ovens work by irradiating food with microwave radiation, which is absorbed and converted into heat. Assum-ing that radiation with l = 15.0 cm is used, that all the energy is converted to heat, and that 4.184 J is needed to raise the temperature of 1.00 g of water by 1.00 °C, how many photons are necessary to raise the temperature of a 350 mL cup of water from 20 °C to 95 °C?

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Textbook Question

The amount of energy necessary to remove an electron from an atom is a quantity called the ionization energy, Ei. This energy can be measured by a technique called photoelectron spectroscopy, in which light of wavelength l is directed at an atom, causing an electron to be ejected. The kinetic energy of the ejected electron (Ek) is measured by determining its veloc-ity, v (Ek = mv2/2), and Ei is then calculated using the conservation of energy principle. That is, the energy of the incident light equals Ei plus Ek. What is the ionization energy of selenium atoms in kilojoules per mole if light with l = 48.2 nm produces electrons with a velocity of 2.371 * 106 m/s? The mass, m, of an electron is 9.109 * 10-31 kg.

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Textbook Question

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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Textbook Question

Assume that the rules for quantum numbers are different and that the spin quantum number ms can have any of three values, ms = -1/2, 0, +1/2, while all other rules remain the same. (a) Draw an orbital-filling diagram for the element with Z = 25, showing the individual electrons in the outer-most subshell as up arrows, down arrows, or 0. How many partially filled orbitals does the element have?

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Textbook Question

A minimum energy of 7.21⨉10^-19 J is required to produce the photoelectric effect in chromium metal. (a) What is the minimum frequency of light needed to remove an electron from chromium?

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Textbook Question

A minimum energy of 7.21⨉10^-19 J is required to produce the photoelectric effect in chromium metal. (b) Light with a wavelength of 2.50⨉10^-7 m falls on a piece of chromium in an evacuated glass tube. What is the minimum de Broglie wavelength of the emitted electrons? (Note that the energy of the incident light must be conserved; that is, the photon's energy must equal the sum of the energy needed to eject the electron plus the kinetic energy of the electron.)

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