Crude Approximation of Electron Position in a Hydrogen Atom
To what extent will quantum mechanics permit us to pinpoint the position of an electron when it is bound to an atom? We can obtain an order of magnitude answer to this question by applying the uncertainty principle
[Delta x Delta p ge dfrac{hbar}{2}]
Electron affinity of Hydrogen is 72.8 kJ/mol. In chemistry and atomic physics, the electron affinity of an atom or molecule is defined as: the change in energy (in kJ/mole) of a neutral atom or molecule (in the gaseous phase) when an electron is added to the atom to form a negative ion. THE HYDROGEN ATOM; ATOMIC ORBITALS Atomic Spectra When gaseous hydrogen in a glass tube is excited by a 5000-volt electrical discharge, four lines are observed in the visible part of the emission spec-trum: red at 656.3 nm, blue-green at 486.1 nm, blue violet at 434.1 nm and violet at 410.2 nm: Figure 1. Visible spectrum of atomic hydrogen. A hydrogen atom is an atom of the chemical element hydrogen.The electrically neutral atom contains a single positively charged proton and a single negatively charged electron bound to the nucleus by the Coulomb force. The ground state energy level of the electron in a hydrogen atom is −13.6 eV, which is equivalent to an ultraviolet photon of roughly 91 nm wavelength. The energy levels of hydrogen can be calculated fairly accurately using the Bohr model of the atom, which conceptualizes the electron as 'orbiting' the proton in analogy to the Earth's orbit of the Sun.
[ approx h]
Hydrogen Atomic Emission Spectrum
to estimate (Delta x), which represents the minimum uncertainty in our knowledge of the position of the electron. The momentum of an electron in an atom is of the order of magnitude of (9 times 10^{-19}; g, cm/sec). The uncertainty in the momentum (Delta p) must necessarily be of the same order of magnitude. Thus
[ Delta x =dfrac{ 7 times 10^{-27}}{9 times 10^{-19}} approx 10^{-8} ,cm]
The uncertainty in the position of the electron is of the same order of magnitude as the diameter of the atom itself. As long as the electron is bound to the atom, we will not be able to say much more about its position than that it is in the atom. Certainly all models of the atom which describe the electron as a particle following a definite trajectory or orbit must be discarded.
We can obtain an energy and one or more wavefunctions for every value of (n), the principal quantum number, by solving Schrödinger's equation for the hydrogen atom. A knowledge of the wavefunctions, or probability amplitudes (psi_n), allows us to calculate the probability distributions for the electron in any given quantum level. When n = 1, the wave function and the derived probability function are independent of direction and depend only on the distance r between the electron and the nucleus. In Figure (PageIndex{1}), we plot both (psi_1) and (P_1) versus (r), showing the variation in these functions as the electron is moved further and further from the nucleus in any one direction. (These and all succeeding graphs are plotted in terms of the atomic unit of length, a0 = 0.529 ´ 10-8 cm.)
Figure (PageIndex{1}): The wave function and probability distribution as functions of r for the n = 1 level of the H atom. The functions and the radius r are in atomic units in this and succeeding figures.
Two interpretations can again be given to the (P_1) curve. An experiment designed to detect the position of the electron with an uncertainty much less than the diameter of the atom itself (using light of short wavelength) will, if repeated a large number of times, result in Figure (PageIndex{1}) for (P_1). That is, the electron will be detected close to the nucleus most frequently and the probability of observing it at some distance from the nucleus will decrease rapidly with increasing (r). The atom will be ionized in making each of these observations because the energy of the photons with a wavelength much less than 10-8 cm will be greater than (K), the amount of energy required to ionize the hydrogen atom. If light with a wavelength comparable to the diameter of the atom is employed in the experiment, then the electron will not be excited but our knowledge of its position will be correspondingly less precise. In these experiments, in which the electron's energy is not changed, the electron will appear to be 'smeared out' and we may interpret (P_1) as giving the fraction of the total electronic charge to be found in every small volume element of space. (Recall that the addition of the value of Pn for every small volume element over all space adds up to unity, i.e., one electron and one electronic charge.)
When the electron is in a definite energy level we shall refer to the (P_n) distributions as electron density distributions, since they describe the manner in which the total electronic charge is distributed in space. The electron density is expressed in terms of the number of electronic charges per unit volume of space, e-/V. The volume V is usually expressed in atomic units of length cubed, and one atomic unit of electron density is then e-/a03. To give an idea of the order of magnitude of an atomic density unit, 1 au of charge density e-/a03 = 6.7 electronic charges per cubic Ångstrom. That is, a cube with a length of (0.52917 times 10^{-8}; cm), if uniformly filled with an electronic charge density of 1 au, would contain 6.7 electronic charges.
(P_1) may be represented in another manner. Rather than considering the amount of electronic charge in one particular small element of space, we may determine the total amount of charge lying within a thin spherical shell of space. Since the distribution is independent of direction, consider adding up all the charge density which lies within a volume of space bounded by an inner sphere of radius r and an outer concentric sphere with a radius only infinitesimally greater, say (r + Dr). The area of the inner sphere is 4pr2 and the thickness of the shell is Dr. Thus the volume of the shell is (4pi r^2 Delta r) and the product of this volume and the charge density P1(r), which is the charge or number of electrons per unit volume, is therefore the total amount of electronic charge lying between the spheres of radius (r) and (r + Dr). The product (4pr^2P_n) is given a special name, the radial distribution function, which we shall label (Q_n(r)).
Volume Element for a Shell in Spherical Coordinates
The reader may wonder why the volume of the shell is not taken as:
Hydrogen Atomic Number
[ dfrac{4}{3} pi left[ (r + Delta r)^3 -r^3 right]]
the difference in volume between two concentric spheres. When this expression for the volume is expanded, we obtain
[dfrac{4}{3} pi left(3r^2 Delta r + 3r Delta r^2 + Delta r^3right)]
and for very small values of (Delta r) the (3r Delta r^2) and (Delta r^3) terms are negligible in comparison with (3r^2Delta r). Thus for small values of (Delta r), the two expressions for the volume of the shell approach one another in value and when (Delta r) represents an infinitesimal small increment in (r) they are identical.
The radial distribution function is plotted in Figure (PageIndex{2}) for the ground state of the hydrogen atom.
Figure (PageIndex{2}): The radial distribution function (Q_1(r)) for an H atom. The value of this function at some value of r when multiplied by (delta r) gives the number of electronic charges within the thin shell of space lying between spheres of radius (r) and (r + delta r).
The curve passes through zero at r = 0 since the surface area of a sphere of zero radius is zero. As the radius of the sphere is increased, the volume of space defined by 4pr2Dr increases. However, as shown in Figure (PageIndex{1}):, the absolute value of the electron density at a given point decreases with r and the resulting curve must pass through a maximum. This maximum occurs at rmax = a0. Thus more of the electronic charge is present at a distance a0, out from the nucleus than at any other value of r. Since the curve is unsymmetrical, the average value of r, denoted by , is not equal to rmax. The average value of r is indicated on the figure by a dashed line. A 'picture' of the electron density distribution for the electron in the (n = 1) level of the hydrogen atom would be a spherical ball of charge, dense around the nucleus and becoming increasingly diffuse as the value of r is increased.
We could also represent the distribution of negative charge in the hydrogen atom in the manner used previously for the electron confined to move on a plane (Figure (PageIndex{1})), by displaying the charge density in a plane by means of a contour map. Imagine a plane through the atom including the nucleus. The density is calculated at every point in this plane. All points having the same value for the electron density in this plane are joined by a contour line (Figure (PageIndex{3})). Since the electron density depends only on r, the distance from the nucleus, and not on the direction in space, the contours will be circular. A contour map is useful as it indicates the 'shape' of the density distribution.
Figure (PageIndex{3}): (a) A contour map of the electron density distribution in a plane containing the nucleus for the (n = 1) level of the H atom. The distance between adjacent contours is 1 au. The numbers on the left-hand side on each contour give the electron density in au. The numbers on the right-hand side give the fraction of the total electronic charge which lies within a sphere of that radius. Thus 99% of the single electronic charge of the H atom lies within a sphere of radius 4 au (or diameter = (4.2 times 10^{-8}; cm)). (b) This is a profile of the contour map along a line through the nucleus. It is, of course, the same as that given previously in Figure (PageIndex{1}) for (P_1), but now plotted from the nucleus in both directions.
This completes the description of the most stable state of the hydrogen atom, the state for which (n = 1). Before proceeding with a discussion of the excited states of the hydrogen atom we must introduce a new term. When the energy of the electron is increased to another of the allowed values, corresponding to a new value for (n), (y_n) and (P_n) change as well. The wavefunctions (y_n) for the hydrogen atom are given a special name, atomicorbitals, because they play such an important role in all of our future discussions of the electronic structure of atoms. In general the word orbital is the name given to a wavefunction which determines the motion of a single electron. If the one-electron wave function is for an atomic system, it is called an atomic orbital.
Do not confuse the word orbital with the classical word and notion of an orbit. First, an orbit implies the knowledge of a definite trajectory or path for a particle through space which in itself is not possible for an electron. Secondly, an orbital, like the wave function, has no physical reality but is a mathematical function which when squared gives the physically measurable electron density distribution.
For every value of the energy En, for the hydrogen atom, there is a degeneracy equal to (n^2). Therefore, for n = 1, there is but one atomic orbital and one electron density distribution. However, for n = 2, there are four different atomic orbitals and four different electron density distributions, all of which possess the same value for the energy, E2. Thus for all values of the principal quantum number n there are n2 different ways in which the electronic charge may be distributed in three-dimensional space and still possess the same value for the energy. For every value of the principal quantum number, one of the possible atomic orbitals is independent of direction and gives a spherical electron density distribution which can be represented by circular contours as has been exemplified above for the case of n = 1. The other atomic orbitals for a given value of n exhibit a directional dependence and predict density distributions which are not spherical but are concentrated in planes or along certain axes. The angular dependence of the atomic orbitals for the hydrogen atom and the shapes of the contours of the corresponding electron density distributions are intimately connected with the angular momentum possessed by the electron.