Hydrogen atom electron probability

Here one sees the cause of the general repulsion, first, of all ions with closed electron configurations but, in addition, of all atoms which are not bound to each other by an electron pair. Indeed with two hydrogen atoms the probability is a priori 3 for the repulsive state against i for the attractive symmetrical state with antiparallel spins (p. 145). 

The idea of electrons existing in definite energy states was fine, but another way had to be devised to describe the location of the electron about the nucleus. The solution to this problem produced the modern model of the atom, often called the quantum mechanical model. In this new model of the hydrogen atom, electrons do not travel in circular orbits but exist in orbitals with three-dimensional shapes that are inconsistent with circular paths. The modern model of the atom treats the electron not as a particle with a definite mass and velocity, but as a wave with the properties of waves. The mathematics of the quantum mechanical model are much more complex, but the results are a great improvement over the Bohr model and are in better agreement with what we know about nature. In the quantum mechanical model of the atom, the location of an electron about the nucleus is described in terms of probability, not paths, and these volumes where the probability of finding the electron is high are called orbitals. 

Figure 3.7 Computer plot of a Is electron in a hydrogen atom. The probability of being at any point in a plane is represented by a height above the plane. The peak is above the nucleus.

Today Bohr s concept of an electronic orbit is no longer tenable, but the modern quantum theory substitutes for the orbit a probability distribution, in which, in the case of the hydrogen atom, the probability is concentrated in the region where the Bohr orbit was. For a free electron (i.e., one not bound to the hydrogen nucleus), the probability distribution looks like that of a wave—a confined region of oscillations, called a wave packet. These wavelike properties are extremely difficult to observe under normal conditions because typical wavelengths of electrons are extremely short—around 10 or m. Wavelike... 

By an obvious extension of our definition of the wave function for the hydrogen atom, the probability of simultaneously finding electron 1 in dz and electron 2 in dz2 must be given by... 

The reason a single equation = ( can describe all real or hypothetical mechanical systems is that the Hamiltonian operator H takes a different form for each new system. There is a limitation that accompanies the generality of the Hamiltonian and the Schroedinger equation We cannot find the exact location of any election, even in simple systems like the hydrogen atom. We must be satisfied with a probability distribution for the electron s whereabouts, governed by a function (1/ called the wave function. 

The characteristic feature of valence bond theory is that it pictures a covalent bond between two atoms in terms of an m phase overlap of a half filled orbital of one atom with a half filled orbital of the other illustrated for the case of H2 m Figure 2 3 Two hydrogen atoms each containing an electron m a Is orbital combine so that their orbitals overlap to give a new orbital associated with both of them In phase orbital overlap (con structive interference) increases the probability of finding an electron m the region between the two nuclei where it feels the attractive force of both of them... 

Hydrogen Abstra.ction. These important reactions have been carried out using a variety of substrates. In general, the reactions involve the removal of hydrogen either direcdy as a hydrogen atom or indirectly by electron transfer followed by proton transfer. The products are derived from ground-state reactions. For example, chlorarul probably reacts with cycloheptatrienyl radicals to produce ether (50) (39). This chemistry contrasts with the ground-state reaction in which DDQ produces tropyhum quinolate in 91% yield (40). 

Another, more common way of showing the electron distribution in the ground state of the hydrogen atom is to draw the orbital (Figure 6.4b) within which there is a 90% chance of finding the electron. Notice that the orbital is spherical, which means that the probability is independent of direction the electron is equally likely to be found north, south, east, or west of the nucleus. 

Quantum mechanics provides a mathematical framework that leads to expression (4). In addition, for the hydrogen atom it tells us a great deal about how the electron moves about the nucleus. It does not, however, tell us an exact path along which the electron moves. All that can be done is to predict the probability of finding an electron at a given point in space. This probability, considered over a period of time, gives an averaged picture of how an electron behaves. This description of the electron motion is what we have called an orbital. 

A.33 Calculate the energy released when an electron is brought from infinity to a distance of 53 pm from a proton. (That distance is the most probable distance of an electron from the nucleus in a hydrogen atom.) The actual energy released when an electron and a proton form a hydrogen atom is 13.6 electronvolts (eV 1 eV = 1.602 X 10 19 J). Account for the difference. 

All s-orbitals are independent of the angles 0 and c[>, so we say that they are spherically symmetrical (Fig. 1.31). The probability density of an electron at the point (r,0,ct>) when it is in a ls-orbital is found from the wavefunction for the ground state of the hydrogen atom. When we square the wavefunction (which was given earlier, but can also be constructed as RY from the entries for R and V in Tables 1.2a and 1.2b) we find that... 

Suppose the electron is in a Is orbital of a hydrogen atom. What is the probability of finding the electron in a small region a distance a0 from the nucleus relative to the probability of finding it in the same small region located right at the nucleus ... 

FIGURE 1.34 The radial wavefunctions of the first three s-orbitals of a hydrogen atom. Note that the number of radial nodes increases (as n 1), as does the average distance of the electron from the nucleus (compare with Fig. 1.32). Because the probability density is given by ip3, all s-orbitals correspond to a nonzero probability density at the nucleus. 

FIGURE 1.42 The radial distribution functions for s-, p-, and cf-orbitals in the first three shells of a hydrogen atom. Note that the probability maxima for orbitals of the same shell are close to each other however, note that an electron in an ns-orbital has a higher probability of being found close to the nucleus than does an electron in an np-orbital or an nd-orbital. 

The quantity p2 as a function of the coordinates is interpreted as the probability of the corresponding microscopic state of the system in this case the probability that the electron occupies a certain position relative to the nucleus. It is seen from equation 6 that in the normal state the hydrogen atom is spherically symmetrical, for p1M is a function of r alone. The atom is furthermore not bounded, but extends to infinity the major portion is, however, within a radius of about 2a0 or lA. In figure 3 are represented the eigenfunction pm, the average electron density p = p]m and the radial electron distribution D = 4ir r p for the normal state of the hydrogen atom. 

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