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

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© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 20: The Hydrogen Atom 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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© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 20: The Hydrogen Atom Outline 1.Formulating the Schrödinger Equation 2.Solving the SchrödingerEquation for the Hydrogen Atom 3.Eigenvalues and Eigenfunctions for the Total Energy 4.The Hydrogen Atom Orbitals 5.The Radial Probability Distribution Function 6.The Validity of the Shell Model of an Atom

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© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 20: The Hydrogen Atom 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 m e = electron mass ε 0 = permittivity of free space

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© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 20: The Hydrogen Atom 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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© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 20: 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.

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© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 20: The Hydrogen Atom 20.2 Solving the Schrödinger Equation for the Hydrogen Atom Each of the terms that contribute to V eff (r) and their sums can be graphed as a function of distance.

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© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 20: The Hydrogen Atom 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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© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 20: The Hydrogen Atom 20.3 Eigenvalues and Eigenfunctions for the Total Energy The other two quantum numbers are l and m l, which arise from the angular coordinates. Their relationship is given by

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© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 20: The Hydrogen Atom 20.3 Eigenvalues and Eigenfunctions for the Total Energy The quantum numbers associated with the wave functions are

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© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 20: The Hydrogen Atom 20.3 Eigenvalues and Eigenfunctions for the Total Energy The quantum numbers associated with the wave functions are

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© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 20: The Hydrogen Atom Example 20.1 Normalize the functions in three-dimensional spherical coordinates.

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© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 20: The Hydrogen Atom 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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© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 20: The Hydrogen Atom Solution We use the standard integral Integrating over the angles, we obtain Evaluating the integral over r,

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© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 20: The Hydrogen Atom Solution For the second function, This simplifies to Integrating over the angles using the result, we obtain

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© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 20: The Hydrogen Atom Solution Using the same standard integral as in the first part of the problem,

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© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 20: The Hydrogen Atom 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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© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 20: The Hydrogen Atom 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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© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 20: The Hydrogen Atom 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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© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 20: The Hydrogen Atom Solution b. The average value of the kinetic energy is given by

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© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 20: The Hydrogen Atom Solution We use the standard integral, Using the relationship

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© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 20: The Hydrogen Atom Solution The average potential energy is given by

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© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 20: The Hydrogen Atom 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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© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 20: The Hydrogen Atom 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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© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 20: The Hydrogen Atom 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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© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 20: The Hydrogen Atom 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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© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 20: The Hydrogen Atom 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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© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 20: The Hydrogen Atom 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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© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 20: The Hydrogen Atom 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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© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 20: The Hydrogen Atom 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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© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 20: The Hydrogen Atom Solution Because, we can evaluate the integrand by assuming that is constant over the interval

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© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 20: The Hydrogen Atom 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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© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 20: The Hydrogen Atom 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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© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 20: The Hydrogen Atom 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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© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 20: The Hydrogen Atom 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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© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 20: The Hydrogen Atom Solution These functions are plotted as a function of in the following figure:

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© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 20: The Hydrogen Atom 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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© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 20: The Hydrogen 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.

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