Dr Christian Clausen III Department of Chemistry
CALCULUS TOPIC: POLAR COORDINATES Section 11.3 (page 67 5 in your textbook)
In your previous math and science courses you have probably always used Cartesian coordinates to describe points in a plane. However, there are lots of other coordinate systems in use as well, particularly in specialized applications. One
specialized application in Chemistry that you will learn about in this section concerns the allowed energies and regions in space that an electron can occupy in a hydrogen-like atom. The coordinate system that is used in performing the calculations about the electron is known as Polar coordinates. This is the coordinate system that you studied in this section. Let’s review what you have learned before moving on to the Chemistry application.
In the Cartesian system for a position of a point in a plane we fix an origin, a point O and then take two lines which are perpendicular to each other passing through O. These two lines are the axes. The x-axis is horizontal, and the y-axis is vertical. The position of any point P in the plane is then described by noting its displacement from O in the direction of the x-axis, the x coordinate, and also its displacement from O in the direction of the y-axis, the y coordinate as shown in Figure 1.
Figure 1. Cartestian coordinate system
Notice that we talk about the displacement, and not the distance. This is because, in this system, we need to use the directions indicated by the arrows to help us to be able to specify a point uniquely. Thus moving up from the origin is a positive displacement, whereas moving down from the origin is a negative displacement. Similarly, moving to the right from the origin gives a positive displacement, and moving to the left gives a negative displacement.
In the Polar coordinate system we take an origin (or pole) O, and a fitted line OA. A point P is then described by specifying a distance O to P along the radius direction, and the angle θ where we have had to turn from the initial line to be looking along the radius direction as shown in Figure 2. The point then has polar coordinates (r, θ).
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Figure 2. Polar coordinate system
There are similar conventions about direction with polar coordinates as there are with Cartesian coordinates. We use the convention that an angle is positive if measured in the counter-clockwise direction from the polar axis and negative in the clockwise direction. We can also extend the meaning of polar coordinates (r, θ) to the case in which
r is negative by agreeing that, as in Figure 3, the points (-r, θ) and (r, θ) lie on the same line through O and at the same distance │r│ from O, but on opposite sides of O, the point (r, θ) lies in the same quadrant θ; if r < 0, it lies in the quadrant on the opposite side of the pole. Notice that (-r, θ) represents the same point as (r, θ + π).
Figure 3. (-r, θ) and (r, θ) Example: Problem 11.3-1(c).
Plot the point whose polar coordinates are given. Then find two other pairs of polar coordinates of this point, one with r > 0 and one with r < 0.
Solution:
Also (1, 3π/2) and (-1, 5π/2)
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Let’s consider the conversion of a point P with Cartesian coordinates (x, y) to polar coordinates (r, θ). We shall identify the pole with the origin of Cartesian
coordinates, and the initial line with the x-axis. From Figure 4, we see that there are two
Figure 4. Conversion between Polar and Cartesian coordinate systems equations we can use to help us to move from one system to another:
x = rcos(θ), y = rsin(θ)
to convert from polar coordinates to Cartesian coordinates and we can use the following equation to convert from Cartesian coordinates to polar coordinates.
r2 = x2 + y2, tan(θ) = y/x
Both of these sets of equations will be useful, but they must be used with care, as it is easy to obtain an incorrect value of θ when the point does not lie in the first quadrant. So it is always a good idea to plot the points first so that the point (r, θ) lies in the correct quadrant.
Example:
Assuming a common origin and the x-axis as the initial line, find the Cartesian coordinates of the point with the following polar coordinate: (2, - π/2)
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Curves in Polar Coordinates
The graph of a polar equation r = f(θ) consists of all points P that have at least one polar representation (r,θ) whose coordinates satisfy the equation. For example let’s plot the curve represented by the polar equation r = 4. The curve consists of all points (r, θ) with r = 4. Since r represents the distance from the point to the origin, the curve r = 4 represents the circle with center O and radius 4. In general, the equation r = a represents a circle with center O and radius │a│ as shown in Figure 5.
Figure 5. Polar curve of radius a
When the polar equations become more complicated than the example we just demonstrated then it sometimes helps to sketch out the equation in Cartesian coordinates first to see how r changes with increasing values of θ. This is illustrated in the next example:
Example:
Sketch the curve r = cos(2θ) first in Cartesian coordinates and then in polar coordinates. Solution:
First we sketch r = cos(2θ), 0 ≤θ≤ 2π, in Cartesian coordinate as shown in Figure 6. As
θ increase from 0 to π/4, Figure 6 shows that r decreases from 1 to 0 and so we draw the corresponding portion of the polar curve in Figure 7 (this is indicated by [1]). As θ
increases from π/4 to π/2, r goes from 0 to -1. This means that the distance from O increases from 0 to 1, but instead of being in the first quadrant this portion of the polar curve (indicated by [2]) lies on the opposite side of the origin in the third quadrant. The remainder of the curve is drawn in a smaller fashion with the final result being the four- leaf clover design shown in Figure 7.
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Figure 6. r = cos(2θ) in Cartesian coordinates
Figure 7. Four-leaf clover r = cos(2θ)
In this section of your book you have only covered the use of polar coordinates in two dimensions, later on in Calculus III you will add the third dimension and it then will be called spherical polar coordinates. This is covered in section 16.8 of your text book. In Chemistry we will be using polar coordinates in three dimensions to calculate the allowed energy levels and locations of electrons in an atom. Even though we can use a two dimensional model using r and θ a more complete picture of the atomic model can be attained by adding the 3rd dimension. Thus it is useful to discuss the addition of this 3rd dimension at this time. The addition of this new dimension to the polar coordinate system involves the distance along the z-axis. This distance is related to the angle φ as shown in Figure 8.
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Figure 8. Spherical polar coordinates
The relationship of the polar coordinates to the Cartesian coordinate system for x, y, and z
is:
x = rcos(θ)
y = rsin(θ)
z = rcos(φ)
To convert from spherical to rectangular coordinates we use the equations:
x = rsin(φ)cos(θ)
y = rsin(φ)sin(θ)
z = rcos(φ)
We have now reached the point where we can begin to discuss the application of Polar coordinates in Chemistry. But first we will look at a brief historical discussion about the evolution of the structure of the atom
Background/Motivation
Chemistry has a rich, colorful history, even some concepts and discoveries that led temporarily along confusing paths have contributed to the heritage of chemistry. This brief overview of early breakthroughs and false directions provides some insight into how modern chemistry arose and how science progresses.
Chemistry has its origin in a pre-scientific past that incorporated three
overlapping traditions, alchemy, medicine, and technology. Alchemy refers to the occult study of nature practiced in the 1st century AD by Greeks living in northern Egypt. Alchemists were influenced by the Greek idea that matter naturally strives toward perfection, and they searched for ways to change less valued substances into precious ones. Thus alchemy’s legacy to chemistry is mixed at best, but it did dominate the thinking about matter for some 1500 years.
Chemical investigation in the modern sense – inquiry into the causes of changes in matter – began in the early 18th century. A young French chemist Antoine Lavoister put chemistry on a scientific basis by performing experiments using quantitative, reproducible measurements to prove his theories. Thus laws about matter began to develop that still are true today, such as the law of mass conservation: the total mass of substances does not change during a chemical reaction.
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The mass laws led to John Dalton’s theory about the structure of matter in 1808 which he proposed in a series of postulates. One of the postulates being that all matter consists of atoms, tiny indivisible particles of an element that cannot be created or destroyed. This model of the atom lasted until 1897, when J. J. Thomspson discovered the presence of electrons in the atom and thus disproved Dalton’s theory that the atom could not be broken down into smaller particles. In 1910 Ernest Rutherford, through use of alpha particle scattering experiments, established that most of the mass of an atom was located in a tiny region of space within the atom, which he called the nucleus. Thus the model of the atom now became one of a tiny, very dense, positive nucleus surrounded by electrons.
One of several experimental observations about matter that the Rutherford model of the atom could not explain is that when elements are heated to a high temperature they emit electromagnetic radiation of certain characteristic frequencies. An example of this is the color of a neon sign. However, soon after Rutherford proposed his nuclear model, Niels Bohr, a young Danish physicist working in Rutherford’s laboratory, suggested a model for the H atom that predicted the existence of line spectra. In his model, Bohr used Planck’s and Einstein’s ideas about quantized energy and proposed three postulates: 1) the H atom has only certain allowable energy levels, 2) the atom does not radiate energy while in one of its stationary states, and 3) the atom changes to another stationary state (the electron moves to another orbit) only by absorbing or emitting a photon whose energy equals the difference in energy between the two states. In Bohr’s model, the quantum number n (1, 2, 3,…) is associated with the radius of an electron’s orbit, which is directly related to the electron’s energy: the lower the n value, the smaller the radius of the orbit, and the lower the energy level. Thus the Bohr model of the atom can be
envisioned as shown in Figure 9.
Figure 9. Bohr model of the atom
Despite its great success in accounting for the spectral lines of the H atom, the Bohr model failed to predict the spectrum of any other atom, even that of helium, the next simplest element. Thus, the conclusion can be made that the Bohr model of the atom is incorrect. So what is the correct model? We will see that it took several amazing
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discoveries in the early part of the 20th century before the correct model would be proposed.
Without going into any detail to describe the reasons, it has been established that as a result of quantum theory and relativity theory, we can no longer view matter and energy as distinct entities. This idea is embodied in Einstein’s famous equation E = mc2. Another amazing theory proposed by Louis deBroglie is that if energy is particle-like, perhaps matter is wave-like. The deBroglie wavelength is the idea that electrons (and all matter) have wave-like motion. Thus, allowed atomic energy levels are related to
allowed wavelengths of the electron’s motion. Electrons exhibit diffraction patterns, as do waves of energy, and photons can exhibit transfer of momentum, as do particles of mass. The wave-particle duality of matter and energy exists at all scales but is observable only on the atomic scale. Now if an electron has the properties of both a particle and a wave, what can we determine about its position in the atom. In 1927, the German physicist Werner Heisenburg postulated the uncertainty principle, which states that it is impossible to know the exact position and momentum of the electron simultaneously. Thus the electron cannot be in an orbit because we would know both its position and momentum (i.e. energy).
Acceptance of the dual nature of matter and energy and of the uncertainty
principle culminated in the field of quantum mechanics, which examines the wave nature of objects on the atomic scale. In 1926, Erwin Schrödinger derived an equation that is the basis for the quantum-mechanical model of the hydrogen atom. The model describes an atom that has certain allowed quantities of energy due to the allowed wave-like behavior of an electron whose exact location is impossible to know. In order to account for these requirements Schrödinger introduced the wave function Ψ (psi), a mathematical function of the position coordinates x, y, and z. The Schrödinger equation, of which the wavefunction is a solution, is:
In this equation Ψ is the amplitude of the wave function associated with the electron, E is the total energy of the system, V is the potential energy of the system (equal to –e2/r for hydrogen), m is the mass of the electron, h is Planck’s constant, and x, y, and z are the usual Cartesian coordinates.
Since the system described by the Schrödinger equation is spherically symmetrical, it is most convenient to use spherical polar coordinates. When these coordinates are substituted for the Cartesian coordinates the Schrödinger equation takes the form:
Although this equation at first appears more cumbersome than the Cartesian coordinated equation it leads to a much simpler solution because it may be solved by the well-known method of separation of variable, which leads to three ordinary differential equations that can be exactly solved. Each of the three equations has only one of the
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variables r, θ, and φ. To solve these we write the wave function Ψ, which is a function of
r, θ, and φ, as the product of three functions R(r), Θ(θ), and Φ(φ):
Ψ(r, θ, φ) = R(r)*Θ(θ)*Φ(φ)
Substitution of this expression into the above equation and division by RΘΦ gives:
The partial derivatives have been replaced by ordinary derivatives since each function now depends only on a single variable. Without going through the actual manipulations, the Schrödinger equation as it is written above can be split into three equations, one involving Φ and φ, one involving Θ and θ, and one involving R and r. These equations can be solved so as to eliminate the differentials.
In solving the differential equations, definite boundary conditions require that certain constants that enter into the solution of the wave equation take on only integral values. These constants are called quantum numbers and are designated by n, l, and ml. The letter n is called the principal quantum number and may take on the values 1, 2, 3, …; l may have the values of 0, 1, … up to n – 1 and ml can have values ranging from –l through 0 to +l. The wave functions, Ψ, which are solutions of the Schrödinger equation, are commonly called orbitals. Orbitals for which l = 0, 1, 2, 3, and 4 respectively, are called s, p, d, f, and g orbitals. Solutions of the Θ equation leads to the values for a
hydrogen-like atom shown in Table 1 for five different values of the quantum number ml. Table 1. Solutions of the Θ equation
Value of ml Function 0 Θ0 = π 2 1 1 Θ1 = 1 θ πcos -1 Θ-1 = 1 θ πsin 2 Θ2 = π θ) 2 cos( -1 Θ-2 = π θ) 2 sin(
Plots of these solutions in planar polar coordinates are shown in Figure 10. The reason why plots of both θ and θ2 are shown is that θ2 is proportional to the region in space
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Figure 10. Plots of θ and θ2
given in Table 2. These solutions are given for l values of 0, 1, and 2 and the corresponding permitted ml values. Because the sine and cosine functions can have positive and negative values, there are positive and negative regions of the wave
functions. A plot of φ and φ2 for the functions listed in Table 2 are shown in Figure 11. Table 2. Solutions of the Φ equation
l ml Function 0 0 Φ = 1π 1 0 Φ = 26cosϕ 1 +1, -1 Φ = 23sinϕ 2 0 Φ = 410
(
3cos2 ϕ−1)
2 +1, -1 Φ = 215sinϕcosϕ 2 +2, -2 Φ = 415sin2ϕApplication of Polar Coordinates in Chemistry 4-12
Figure 11. Plots of φ and φ2
The θ and φ wave functions are best considered together, since both are concerned with the angular dependence of the orbitals. Plots of [Θ(θ)Φ(φ)]2 for various l and ml values are shown in Figure 12. It is to be emphasized that these plots are in no way related to distance from the nucleus but only to the variation of the wave function with the angles θ and φ. For given values of θ and φ, the length of the line joining the origin to the surface of the solid figure is the relative probability that the electron is to be found in that direction. These Ψ(θ, φ) surfaces are often referred to as orbitals and indeed serve very satisfactorily for most arguments concerned with chemical bonding. It should be realized, however, that the variation of Ψ in space is the product of Ψ(R) and Ψ(θ, φ) at every point in space.
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Figure 12. Total angular dependence of the wave function Ψ(θ, φ) with all Cartesian coordinates fixed
It is now time to look at the solution to the radial part of the wave equation R(r). The radial part of the wave function depends only on the n and l values and has an exponential terme−r/ na0 where a0 is 0.529Å, and a pre-exponential term involving a
polynomial of the n -1 degree. The radial functions for n = 1, l = 0; n = 2, l = 0; and n =