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**Physical Chemistry 2nd Edition**

Chapter 20 The Hydrogen Atom Physical Chemistry 2nd Edition Thomas Engel, Philip Reid

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Objectives Solve the Schrödinger equation for the motion of an electron in a spherically symmetric Coulomb potential. Emphasize the similarities and differences between quantum mechanical and classical models. Comparison is made between the quantum mechanical picture of the hydrogen atom.

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**Outline Formulating the Schrödinger Equation**

Solving the SchrödingerEquation for the Hydrogen Atom Eigenvalues and Eigenfunctions for the Total Energy The Hydrogen Atom Orbitals The Radial Probability Distribution Function The Validity of the Shell Model of an Atom

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**20.1 Formulating the Schrödinger Equation**

Hydrogen atom as made up of an electron moving about a proton located at the origin of the coordinate system. The two particles attract one another and the interaction potential is given by a simple Coulomb potential: where e = electron charge me = electron mass ε0 = permittivity of free space

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**20.1 Formulating the Schrödinger Equation**

As the potential is spherically symmetrical, we choose spherical polar coordinates to formulate the Schrödinger equation.

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**20.2 Solving the Schrödinger Equation for the Hydrogen Atom**

Separation of variables Thus the differential equation for R(r) is obtained. The second term can be viewed as an effective potential.

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**20.2 Solving the Schrödinger Equation for the Hydrogen Atom**

Each of the terms that contribute to Veff (r) and their sums can be graphed as a function of distance.

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**20.3 Eigenvalues and Eigenfunctions for the Total Energy**

Note that the energy, E, only appears in the radial equation and not in the angular equation. R(r) can be well behaved at large values of r [R(r) → 0 as r → ∞]. where (Bohr radius) The definition leads to

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**20.3 Eigenvalues and Eigenfunctions for the Total Energy**

The other two quantum numbers are l and ml, which arise from the angular coordinates. Their relationship is given by

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**20.3 Eigenvalues and Eigenfunctions for the Total Energy**

The quantum numbers associated with the wave functions are

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**20.3 Eigenvalues and Eigenfunctions for the Total Energy**

The quantum numbers associated with the wave functions are

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**Example 20.1 Normalize the functions in three-dimensional**

spherical coordinates.

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Solution In general, a wave function is normalized by multiplying it by a constant N defined by . In three-dimensional spherical coordinates, it is The normalization integral For the first function,

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**Solution We use the standard integral**

Integrating over the angles , we obtain Evaluating the integral over r,

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**Solution For the second function, This simplifies to**

Integrating over the angles using the result , we obtain

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Solution Using the same standard integral as in the first part of the problem,

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**20.3 Eigenvalues and Eigenfunctions for the Total Energy**

The angular part of each hydrogen atom total energy eigenfunctions is a spherical harmonic function.

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Example 20.2 a. Consider an excited state of the H atom with the electron in the 2s orbital.Is the wave function that describes this state,an eigenfunction of the kinetic energy? Of the potential energy? b. Calculate the average values of the kinetic and potential energies for an atom described by this wave function.

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**Solution a. We know that this function is an eigenfunction of the**

total energy operator because it is a solution of the Schrödinger equation. You can convince yourself that the total energy operator does not commute with either the kinetic energy operator or the potential energy operator by extending the discussion of Example Problem Therefore, this wave function cannot be an eigenfunction of either of these operators.

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Solution b. The average value of the kinetic energy is given by

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Solution We use the standard integral, Using the relationship

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Solution The average potential energy is given by

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Solution We see that The relationship of the kinetic and potential energies is a specific example of the virial theorem and holds for any system in which the potential is Coulombic.

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**20.3 Eigenvalues and Eigenfunctions for the Total Energy**

The radial distribution function is used to extract information from the H atom orbitals. We first look at the ground-state (lowest energy state) wave function for the hydrogen atom, We need a four-dimensional space to plot as a function of all its variables.

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**20.4 The Hydrogen Atom Orbitals**

Since such a space is not readily available, the number of variables is reduced. It is reduced by evaluating in one of the x–y, x–z, or y–z planes by setting the third coordinate equal to zero. r are spherical nodal surfaces rather than nodal points (one-dimensional) potentials.

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**Example 20.3 Locate the nodal surfaces in Solution:**

The radial part of the equations is zero for finite values of for This occurs at

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**20.5 The Radial Probability Distribution Function**

3D perspective plots of the square of the wave functions for the orbitals is indicated.

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**Example 20.4 a. At what point does the probability density for the**

electron in a 2s orbital have its maximum value? b. Assume that the nuclear diameter for H is 2 × m. Using this assumption, calculate the total probability of finding the electron in the nucleus if it occupies the 2s orbital.

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Solution a. The point at which and , therefore, has its greatest value is found from the wave function: which has its maximum value at r=0, or at the nucleus

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Solution b. The result obtained in part (a) seems unphysical, but is a consequence of wave-particle duality in describing electrons. It is really only a problem if the total probability of finding the electron within the nucleus is significant. This probability is given by

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Solution Because , we can evaluate the integrand by assuming that is constant over the interval

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Solution Because this probability is vanishingly small, even though the wave function has its maximum amplitude at the nucleus, the probability of finding the electron in the nucleus is essentially zero.

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**20.5 The Radial Probability Distribution Function**

It is most meaningful for the s orbitals whose amplitudes are independent of the angular coordinates. The radial distribution P(r) is the probability function of choice to determine the most likely radius to find the electron for a given orbital

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Example 20.6 Calculate the maxima in the radial probability distribution for the 2s orbital. What is the most probable distance from the nucleus for an electron in this orbital? Are there subsidiary maxima?

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**Solution The radial distribution function is**

To find the maxima, we plot P(r) and versus and look for the nodes in this function.

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Solution These functions are plotted as a function of in the following figure:

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Solution The resulting radial distribution function only depends on r, and not on Therefore, we can display P(r)dr versus r in a graph as shown

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**20.6 The Validity of the Shell Model of an Atom**

The idea of wave-particle duality is that waves are not sharply localized.

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