Transcript Lecture 15 (Slides) September 28

Wave Mechanics – Classical Systems
Standing waves.
Nodes do not undergo
displacement.
λ = 2L, n = 1, 2, 3…
n
FIGURE 8-18
•Standing waves in a string
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General Chemistry: Chapter 8
Slide 1 of 50
Dihydrogen Oxide Waves
Quantized Energies and Wave
Functions
• Line spectra tell us that atomic (and molecular)
energies are quantized. The switch from a
classical description of objects to a quantum
mechanical description becomes necessary as
we move to physically smaller and lighter
objects. In many cases these smaller particles
are confined to move in confined regions of
space (e.g. an electron in a H atom!).
Particle in a Box Model
• We live in a three dimensional world and
would expect the wave functions used to
describe atomic and molecular systems to have
three spatial coordinates and time. We could
write Ψ(x,y,z,t) and Ψ(r,ϴ,φ,t) for wave
functions. In some cases the properties and
energies of an atom do not change with time
and we can write suitable (time independent)
wave functions as Ψ(x,y,z) or Ψ(r,ϴ,φ).
Particle in a Box (One Dimension)
• Energy units (1 joule = 1 kg∙m2∙s-2) suggest
that second order differential equations are
needed to treat atomic and molecular energies.
Advanced mathematics allows us to solve 2nd
order differential equations and determine (a)
the functional form of wave functions for
particular objects/systems and (b) values for
properties such as atomic energies.
Wave and Quantum Mechanics
• We leave the detailed mathematics to higher
level courses. To begin, we consider an
approximate wave function that is a model for
some problems in chemistry and physics – the
so-called particle in a box (PIAB) model. This
crude model does provide insight into the
atomic/molecular world. One dimensional, two
dimensional and three dimensional PIAB
models are available. We will consider the one
dimensional PIAB model first.
PIAB – One Dimension
• The wave functions for that we require are
found by solving the so-called Schrodinger
equation, H ψn(x) = Eψn(x). This 2nd order
differential equation can be tackled fairly
simply (2nd year). It enables us to calculate
allowed energy values, positions of nodes
(zero probability of finding a particle) and
where a particle is most likely to be found.
PIAB – One Dimension
• The wave function(s) on the next slide are
written as Ψn(x) rather to emphasize that we
have different wave functions for various
values of the single quantum number, n. The
single quantum number is a direct
consequence of the use of a one dimensional
PIAB model. For energies we have the result
• En = n2h2 where h is Planck’s constant.
•
8mL2
Particle in Box: Standing Waves,
Quantum Particles, and Wave Functions
ψ, psi, the wave function.
Should correspond to a standing
wave within the boundary of
the system being described.
Particle in a box. Number of
nodes increases as n and
energy increase. Compare to
Slide 1?
FIGURE 8-20
•The standing waves of a particle in a one-dimensional box
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General Chemistry: Chapter 8
Slide 9 of 50
PIAB - Notes on the Ψn’s
• The PIAB wave functions exhibit nodes. As
we move to higher energy (higher n) states the
number of nodes increases. As well, as one
moves to higher n values the characteristic
wavelength decreases. (This is reminiscent of
light where, again, the energy of a photon
increases as the wavelength of the light
decreases).The wave functions can have both
positive and negative amplitude.
Amplitudes and Probabilities
• In quantum mechanics the probability of
finding a particle at a particular point in space
is proportional to the square of the amplitude
of the wave function at that particular point.
Probabilities must always be positive and the
squares of the amplitudes for the first few one
dimensional PIAB wave functions are shown
on the next slide.
FIGURE 8-21
The probabilities of a particle in a one-dimensional box
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General Chemistry: Chapter 8
Slide 12 of 50
Probabilities of Finding an e- ?
• The graphs on the previous slide describe the
probability of finding a particle (electrons are
especially of interest in chemistry) in different
parts of the “box”. In all cases there is a 50%
probability of finding a particle in the left half
of the box. Calculus (or a graphing calculator)
can tell us the probability of finding the
particle in any arbitrarily chosen part of the
box.
Probabilities – Class Examples
• We will use Slide 12 to determine when
calculus must be used to calculate the
probability of finding a particle in a given part
of the box and when simpler “symmetry
arguments” can be used. We’ll consider a
number of cases where symmetry arguments
can be used to specify exactly the probability
of finding a particle in a certain part of the
box.
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8 - 15
PIAB Energies – Mass & Particle Size
• The PIAB model allows us to calculate
energies for a particle in a particular quantum
state as a function of quantum number (n),
particle mass (m) and box length (L). As
values of n increase energies increase. As
particle mass increase and/or box size
increases energy level spacings decrease
rapidly. For the limiting case as m → ∞ and
L → ∞ the energies vary continuously
(familiar ground?).
PIAB – 3 Dimensions
• The Schrodinger Eqtn. for the 3-dimensional
PIAB can also be solved.
h2
n12
n22
n 32
• E=
+ 2 + 2
2
8m
a
c
b
• Here a, b and c are the lengths of the 3 sides of
the box and n1, n2 and n3 are a set of three
quantum numbers needed for this higher
dimensionality case. We will need three
quantum numbers when we discuss atoms!
{
}
Spherical Polar Coordinates
• When we move from the one dimensional
PIAB model to the wave functions for an
electron in a H atom it’s simplest to use
spherical polar coordinates when we
construct the H atom wave functions. The
correspondence between Cartesian and
spherical polar coordinates is shown on the
next slide.
Wave Functions of the Hydrogen Atom
Schrödinger, 1927
Eψ = H ψ
H (x,y,z) or H (r,θ,φ)
ψ(r,θ,φ) = R(r) Y(θ,φ)
R(r) is the radial wave function.
Y(θ,φ) is the angular wave function.
FIGURE 8-22
•The relationship between spherical polar coordinates and Cartesian coordinates
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General Chemistry: Chapter 8
Slide 19 of 50
H Atom Wave Functions
• The previous slide states that the H atom wave
functions, determined again by solving the
Schrodinger equation, can be factored into an
angular and a radial part if we employ
spherical polar coordinates. The use of these
coordinates makes it especially easy to locate
nodes (regions of zero “electron density”) and
to represent 3 dimensional probabilities (i.e.
represent in 3 dimensions the probability of
finding an electron in space/in an atom).
Sample Test 1 Examples
• A1. A 6.60 L tank containing oxygen gas at a
temperature of 46ºC is cooled at constant
volume to a final temperature of –10ºC and a
pressure of 88.8 kPa. What was the initial
pressure of the oxygen gas?
•
Sample Test 1 Examples
• A3. Calculate the value of ΔU in joules
associated with the compression of a gas from
17.8 L to 12.9 L by a constant external
pressure of 4.50 bar if, at the same time, the
gas releases 1,225 joules of heat to the
surroundings.
Sample Test 1 Examples
• A4. Write the balanced thermochemical
equation corresponding to the molar enthalpy
(heat) of formation of glucose, C6H12O6(s).
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Sample Test 1 Examples
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B1.
Calculate the standard enthalpy change ∆Hº for the reaction below:
N2O(g) + NO2(g) → 3 NO(g)
Data:
N2(g) + O2(g) → 2 NO(g)
∆Hº = +180.7 kJ
2 NO(g) + O2(g) → 2 NO2(g)
∆Hº = –113.1 kJ
2 N2O(g) → 2 N2(g) + O2(g)
∆Hº = –163.2 kJ