52. Express the speed of the electron in the Bohr model interms of the fundamental constants (m, e, h, eg), thenuclear charge Z, and the quantum number n. Evaluatethe speed of an electron in the ground states of He* ionand U³1+. Compare these speeds with the speed of light c.As the speed of an object approaches the speed of light,relativistic effects become important. In which kinds ofatoms do you expect relativistic effects to be greatest?

Express the speed of the electron in the Bohr model in terms of the fundamental constants (m, e, h, eg), the nuclear charge Z, and the quantum number n. Evaluate the speed of an electron in the ground states of He* ion and U1+. Compare these speeds with the speed of light c. As the speed of an object approaches the speed of light, relativistic effects become important. In which kinds of atoms do you expect relativistic effects to be greatest?

Express the speed of the electron in the Bohr model in terms of the fundamental constants (m, e, h, eg), the nuclear charge Z, and the quantum number n. Evaluate the speed of an electron in the ground states of He* ion and U1+. Compare these speeds with the speed of light c. As the speed of an object approaches the speed of light, relativistic effects become important. In which kinds of atoms do you expect relativistic effects to be greatest?

Chemistry: An Atoms First Approach

2nd Edition

ISBN:9781305079243

Author:Steven S. Zumdahl, Susan A. Zumdahl

Publisher:Steven S. Zumdahl, Susan A. Zumdahl

Chapter2: Atomic Structure And Periodicity

Section: Chapter Questions

Problem 66E: Consider an electron for a hydrogen atom in an excited state. The maximum wavelength of...

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The ionization of the hydrogen atom can be calculated from Bohr's equation for the electron energy. E = (NARhc)(Z2/n2) where NARhc =1312 kJ/mol and Z is the atomic number. Let us use this approach to calculate a possible ionization energy for helium. First, assume the electrons of the He experience the full 2+ nuclear charge. This gives us the upper limit for the ionization energy. Next, assume one electron of He completely screens the nuclear charge from the other electrons, so Z = 1. This gives us a lower limit to the ionization energy. Compare these calculated values for the upper and lower limits to the experimental value of 2372.3 kJ/mol. What does this tell us about the ability of one electron to screen the nuclear charge?
Estimate the probability of finding an electron which is excited into the 2s orbital of the H atom, looking in a cubical box of volume 0.751036m3 centered at the nucleus. Then estimate the probability of finding the electron if you move the volume searched to a distance of 105.8 pm from the nucleus in the positive z direction. (Note that since these volumes are small, it does not matter whether the volume searched is cubical or spherical.)
The figure below represents part of the emission spectrum for a one-electron ion in lhe gas phase. All the lines result from electronic transitions from excited states to the n = 3 state. (See Exercise 160.) a. What electronic transitions correspond to lines A and B? b. If the wavelength of line B is 142.5 nm, calculate the wavelength of line A.
Imagine a world in which the rule for the l quantum number is that values start with 1 and go up to n. The rules for the n and mi quantum numbers are unchanged from those of our world. Write the quantum numbers for the first two shells (i.e., n = 1 and n = 2).
Consider the orbitals shown here in outline. (a) What is the maximum number of electrons contained in an orbital of type (x)? Of type (y)? Of type (z)? (b) How many orbitals of type (x) are found in a shell with n=2? How many of type (y)? How many of type (z)? (c) Write a set of quantum numbers for an electron in an orbital of type (x) in a shell with n=4, of an orbital of type (y) in a shell with n=2. Of an orbital of type (z) in a shell with n=3. (d) What is the smallest possible n value for an orbital of type (x)? Of type (y)? Of type (z)? (e) What are the possible I and ml values for an orbital of type (x)? Of type (y)? Of type (z)?
How does probability fit into the description of the atom?
The energies of macroscopic objects, as well as those of microscopic objects, are quantized, but the effects of the quantization are not seen because the difference in energy between adjacent states is so small. Apply Bohr’s quantization of angular momentum to the revolution of Earth (mass6.01024kg) , which moves with a speed of 3.0104ms1 in a circular orbit (radius1.51011m) about the sun. The sun can be treated as fixed. Calculate the value of the quantum number n for the present state of the Earthsun system. What would be the effect of an increase in n by 1?
For hydrogen atoms, the wave function for the state n = 3, l = 0, ml = 0 is 300=1813(1a0)3/2(2718+22)e where = r/a0 and a0 is the Bohr radius (5.29 1011 m). Calculate the position of the nodes for this wave function.
Use the mathematical expression for the 2pz wave function of a one-electron atom (see Table 5.2) to show that the probability of finding an electron in that orbital anywhere in the x-y plane is 0. What are the nodal planes for a dxz orbital and for a dx2y2 orbital?
Why is the electron in a Bohr hydrogen atom bound less tightly when it has a quantum number of 3 than when it has a quantum number of 1?
When metallic sodium is dissolved in liquid sodium chloride, electrons are released into the liquid. These dissolved electrons absorb light with a wavelength near 800 nm. Suppose we treat the positive ions surrounding an electron crudely as defining a three-dimensional cubic box of edge L, and we assume that the absorbed light excites the electron from its ground state to the first excited state. Calculate the edge length L in this simple model.
The wave function of an electron in the lowest (that is, ground) state of the hydrogen atom is (r)=( 1 a 0 3 )1/2exp(r a 0 )ao=0.5291010m (a) What is the probability of finding the electron inside a sphere of volume 1.0pm2 , centered at the nucleus (1pm=1012m) ? (b) What is the probability of finding the electron in a volume of 1.0pm2 at a distance of 52.9 pm from the nucleus, in a fixed but arbitrary direction? (c) What is the probability of finding the electron in a spherical shell of 1.0 pm in thickness, at a distance of 52.9 pm from the nucleus?