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Chapter 20 The Hydrogen Atom. Physical Chemistry 2 nd Edition. Thomas Engel, Philip Reid. Objectives. Solve the Schrödinger equation for the motion of an electron in a spherically symmetric Coulomb potential.

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physical chemistry 2 nd edition

Chapter 20

The Hydrogen Atom

Physical Chemistry 2nd Edition

Thomas Engel, Philip Reid

objectives
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.
outline
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
20 1 formulating the schr dinger equation
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 massε0 = permittivity of free space

20 1 formulating the schr dinger equation1
20.1 Formulating the Schrödinger Equation
  • As the potential is spherically symmetrical, we choose spherical polar coordinates to formulate the Schrödinger equation.
20 2 solving the schr dinger equation for the hydrogen atom
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.
20 2 solving the schr dinger equation for the hydrogen atom1
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.
20 3 eigenvalues and eigenfunctions for the total energy
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
20 3 eigenvalues and eigenfunctions for the total energy1
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
20 3 eigenvalues and eigenfunctions for the total energy2
20.3 Eigenvalues and Eigenfunctions for the Total Energy
  • The quantum numbers associated with the wave functions are
20 3 eigenvalues and eigenfunctions for the total energy3
20.3 Eigenvalues and Eigenfunctions for the Total Energy
  • The quantum numbers associated with the wave functions are
example 20 1
Example 20.1

Normalize the functions in three-dimensional

spherical coordinates.

solution
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,

solution1
Solution

We use the standard integral

Integrating over the angles , we obtain

Evaluating the integral over r,

solution2
Solution

For the second function,

This simplifies to

Integrating over the angles using the result , we obtain

solution3
Solution

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

20 3 eigenvalues and eigenfunctions for the total energy4
20.3 Eigenvalues and Eigenfunctions for the Total Energy
  • The angular part of each hydrogen atom total energy eigenfunctions is a spherical harmonic function.
example 20 2
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.

solution4
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 20.1. Therefore, this wave function cannot

be an eigenfunction of either of these operators.

solution5
Solution

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

solution6
Solution

We use the standard integral,

Using the relationship

solution7
Solution

The average potential energy is given by

solution8
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.

20 3 eigenvalues and eigenfunctions for the total energy5
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.
20 4 the hydrogen atom orbitals
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.
example 20 3
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 .

20 5 the radial probability distribution function
20.5 The Radial Probability Distribution Function
  • 3D perspective plots of the square of the wave functions for the orbitals is indicated.
example 20 4
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 × 10-15 m. Using this assumption, calculate the total

probability of finding the electron in the nucleus if it

occupies the 2s orbital.

solution9
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

solution10
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

solution11
Solution

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

solution12
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.

20 5 the radial probability distribution function1
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
example 20 6
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?

solution13
Solution

The radial distribution function is

To find the maxima, we plot P(r) and

versus and look for the nodes in this function.

solution14
Solution

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

solution15
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

20 6 the validity of the shell model of an atom
20.6 The Validity of the Shell Model of an Atom
  • The idea of wave-particle duality is that waves are not sharply localized.