Radial Solutions to the Schrodinger Equation (7.2.2) Flashcards
• The solutions to Schrödinger’s equation provide descriptions of the energy and probable location of the electron in spherical coordinates.
• The solutions to Schrödinger’s equation provide descriptions of the energy and probable location of the electron in spherical coordinates.
• The square of the wave function predicts the probability of finding an electron at a given distance from the nucleus and describes distances where that probability is
zero.
• The square of the wave function predicts the probability of finding an electron at a given distance from the nucleus and describes distances where that probability is
zero.
The solutions to Schrödinger’s equation provide
descriptions of the energy and probable location of
the electron in spherical coordinates.
Spherical coordinates have three variables, the
radius (distance from the origin) and two angular
displacements. The solutions to the equation can
be decomposed into separate radial and angular
parts.
There are radial solutions to the equation for each
energy level of the electron.
The solutions to Schrödinger’s equation provide
descriptions of the energy and probable location of
the electron in spherical coordinates.
Spherical coordinates have three variables, the
radius (distance from the origin) and two angular
displacements. The solutions to the equation can
be decomposed into separate radial and angular
parts.
There are radial solutions to the equation for each
energy level of the electron.
The square of the wave function describes the
probability of finding an electron at a given distance
from the nucleus and describes distances where
that probability is zero (a node).
Multiplying the function by the surface area of a
sphere (4πr^2) gives the probability of finding the
electron in any direction (at a given radius).
For the first (1s) orbital, the probability of locating
the electron rises from near zero at the nucleus to a
peak at a distance of about 0.5 angstrom (Å) and
then declines exponentially.
For the second (2s) orbital there is a point where the
probability of finding an electron is equal to zero (a
node), while there are peaks of probability both
inside and outside of this radius.
The 3s orbital has two nodes, and as the energy
level increases the number of nodes in the s
sublevels increases.
The square of the wave function describes the
probability of finding an electron at a given distance
from the nucleus and describes distances where
that probability is zero (a node).
Multiplying the function by the surface area of a
sphere (4πr^2) gives the probability of finding the
electron in any direction (at a given radius).
For the first (1s) orbital, the probability of locating
the electron rises from near zero at the nucleus to a
peak at a distance of about 0.5 angstrom (Å) and
then declines exponentially.
For the second (2s) orbital there is a point where the
probability of finding an electron is equal to zero (a
node), while there are peaks of probability both
inside and outside of this radius.
The 3s orbital has two nodes, and as the energy
level increases the number of nodes in the s
sublevels increases.
Which of these expressions best represents the radial part of the wave function for a 1s orbital?
exp(-r) (B)
This expression represents exponential decay. In other words, this expression correctly indicates that the probability of finding the electron in a particular direction decreases as the distance from the nucleus increases.
How is an electron orbital different from a planetary orbit?
Electrons do not follow particular paths, but planets do. (D)
The electrons in orbitals do not follow specified paths. Therefore, you cannot predict exactly where an electron will be at any instant of time; you can only determine where it is likely to be. By contrast, planets do follow particular paths around the sun. These paths are predictable; therefore, the positions of the planets can be predicted at any particular instant of time.
The x-, y- and z-coordinates of a point are (3, 4, 0). What is the value of r for this point, using spherical coordinates?
5 (C)
The r-coordinate is equal to the distance of the point from the origin. Calculate the r-value by using the following formula and solving for r.
r^ 2 = x ^2 + y ^2 + z ^2 r^ 2 = (3)^2 + (4)^2 + (0)^2 r ^2 = 25 r = 5
The r-coordinate is 5, which means that the point is a distance of 5 units from the origin.
Which of the following graphs best represents the probability of finding an electron in any direction of a 2s orbital?
(C)
In a 2s orbital, the probability of finding an electron is relatively high close to the nucleus. This region of high probability is followed by a radial node, where the probability of finding the electron is zero. The node is then followed by a second region of fairly high probability. This graph correctly shows two regions of high probability with a radial node in between them.
Which of the following is the best explanation of why the probability of finding an electron in a 1s orbital is highest about 0.5 angstrom from the nucleus rather than closer to the nucleus?
The volume of space any closer to the nucleus is so small that the chance of finding the electron begins to decrease. (D)
The distance from the nucleus can be thought of as the radius of a spherical volume of space. As the distance from the nucleus decreases, the volume of this sphere also decreases. Eventually, the volume of space becomes so small that the electron is more likely to be found outside it than inside it. This distance corresponding to this volume happens to be about 0.5 angstrom for a 1s orbital.
Which of the following would be the most appropriate label for the horizontal axis of the following graph? (Ψ^2 1s represents the probability of finding the electron in a particular direction of a 1s orbital.)
r (A)
If the horizontal axis represents r, then the graph indicates that the probability of finding the electron in a particular direction decreases as r increases. Because r represents the distance of the electron from the nucleus, the graph correctly indicates that the electron is more likely to be found close to the nucleus than far from the nucleus. None of the other answer choices correctly match the behavior of an electron in a 1s orbital.
Which of the following types of information cannot be determined from examining the solutions to the Schrödinger equation?
the shape of the path an electron takes around the nucleus (C)
The Schrödinger equation does reveal the approximate shape of an electron orbital. However, an orbital represents a volume of space where an electron is likely to be found. It does not reveal the exact path the electron follows as it moves within the orbital. Therefore, the shape of the path taken by an electron cannot be determined from the Schrödinger equation.
How do the radial and angular parts of a wave function differ?
The radial part tells the size of the orbital, and the angular part tells the shape. (A)
The radial part of a wave function tells the probability of finding an electron a certain distance from the nucleus. Therefore, it can be used to determine the size of the orbital. The angular part of a wave function, by contrast, tells the probability of finding an electron in a certain direction from the nucleus. Therefore, it can be used to determine the shape of the orbital. (In an s orbital, for example, the wave function does not depend on the angular part. As a result, the probability of finding an electron is the same in every direction, and the orbital has the shape of a sphere. In this case, the angular part determines the shape of the orbital merely because it has no effect on the wave function.)
Suppose that electrons occupied fixed circular orbits around the nucleus. Which of these graphs shows the probability of finding an electron in a particular direction of such an orbit?
(D)
If an electron is traveling in a fixed orbit, its distance from the nucleus does not change. Therefore, there is only one possible distance at which the electron will be found. As a result, the probability of finding the electron at any other distance will be zero, and the probability of finding the electron at the distance of its orbit will be 100%. The graph of this situation will consist of a single point. This point corresponds to a 100% probability of finding the electron at the distance of its fixed orbit.
Which of the following is the best description of a radial node?
a spherical shell (C)
A radial node represents a distance from the nucleus at which the probability of finding the electron will be zero. Therefore, a radial node has the shape of a sphere that is centered on the nucleus and whose radius is equal to the distance at which at which the probability of finding the electron is zero. The node does not include any distances closer to the nucleus. Therefore, it has the shape of a hollow spherical shell.
How would a 2s orbital most likely be affected if the angular part of its wave function were to change?
The orbital would probably no longer be spherical. (C)
The angular part of a wave function determines the probability that an electron will be found in a particular direction from the nucleus. In other words, the angular part determines the shape of the orbital. Therefore, if the angular part of the wave function of a 2s orbital were different, the shape of the orbital would likely be different as well. The normal shape of a 2s orbital is spherical; therefore, it would probably not be spherical if the angular part of the wave function changed.
