# Chemistry 361 Quantum Chemistry And Chemical Kinetics

## Transcript Of Chemistry 361 Quantum Chemistry And Chemical Kinetics

CHEM 366 FALL 2006

CHEMISTRY 361 QUANTUM CHEMISTRY

AND CHEMICAL KINETICS

Enrique Peacock-L´opez1, ∗ 1Department of Chemistry Williams College Williamstown, MA 01267

(Dated: June 17, 2006)

Abstract

The early XX century saw the birth of the theory of relativity. This theory was developed after the 1887 Michelson and Morley’58 experiment which was designed to reveal the earth’s motion relative to the ether. In an eﬀort to explain the experimental facts, Poincare and Einstein altered the ideas and concept of space, time and astronomical distances. Most of the development of the the theory of relativity was carried out by the latter scientists. In contrast Quantum chemistry took several decades and many contributors to be developed. This theory can be seen as an extension of classical mechanics to the subatomic, atomic and molecular sizes and distances.

In this course we will study the fundamental theory of electrons, atoms and molecules known as Quantum Chemistry. In our approach we will follow the historical development of Quantum Chemistry. In this approach we will see how Quantum Chemistry gradually evolves from confusion and dilemma into a formal theory.

∗Author to whom correspondence should be addressed. Electronic address: [email protected]

1

Lecture 1

As we have seen in previous chapters, the world we experience with our senses is a world described by Thermodynamics, Classical Mechanics and Classical Electromagnetism. It is a macroscopic world based in macroscopic laws. But scientist have long recognized experimentally the existence of small particles which are the components of matter. Therefore it is natural to look ﬁrst for microscopic laws that explain the particle’s behavior, and second we have to look for a theoretical approach that will link the microscopic laws with the macroscopic laws.

Towards the end of the XIX century, many physicist felt that all the principles of physics had been discovered and little remainded to be understood except for few minor problems. At that time our world was understood using Classical Mechanics, Thermodynamics and Classical Electromagnetism. For example in Classical Mechanics or Newtonian Mechanics we needed to ﬁnd the dynamical variables of the system under study. Once done this, we needed to construct the equation of motion which predicted the system’s evolution in time. The predicted behavior was ﬁnally compared with the experimental observations.

At the time the Universe was divided into matter and radiation. Matter was ruled by Newtonian mechanics and thermodynamics, and radiation obeyed Maxwell’s laws of Electromagnetism. A controversy whether light was wave-like or corpuscular-like existed since Newton’s days, who proposed the corpuscular theory of light. The wave-like theory was developed by Huggens based in the constructive and destructive interference of light; a property which is characteristic of waves.

The interaction between matter and radiation was not well understood. For example, Earnshaw’s theorem states that a system of charged particles can not remain at rest in stable equilibrium under the inﬂuence of purely electrostatic forces. Moreover, according to Electromagnetism an accelerated charged particles radiate energy in the form. of electromagnetic waves. Thus how is that molecules are stable? From this minor problems the theory of Quantum chemistry was developed.

2

Lecture 2

During the 17th century Galileo Galilei and Issac Newton postulated a corpuscular theory of light. After 200 years of work, in the 19th century we have a logical theory with deterministic equation that explain planetary motion as well as classical mechanics, thermodynamics, optics, electricity and magnetism.

In the mid 1800s, Thomas Young reported a pattern consisting of dark and bright fringes, which were produces after passing light through two narrow closely spaced holes or slits. This interference pattern is a natural wave behavior. By 1860, James Clark Maxwell developed a classical electromagnetic theory, which combines electricity and magnetism in a single theory. Also, Maxwell’s theory predicts that the properties of light can be explained if we consider light as electromagnetic radiation. The experimental conﬁrmation of maxwell equation was reported by Michael Faraday, and 1886, although assuming a luminiferous (ether) medium, Hertz conﬁrmed experimentally that light is electromagnetic radiation giving the wave theory of light a solid foundation.

In 1887, Albert Michelson and edward Morley’1860, conﬁrmed experimentally the electromagnetic waves traveled in vacuum by ﬁnding no diﬀerence in the speed of light relative to the motion of Earth. In other word, there is no need of the luminiferous or any other medium. Without a medium, light is subject to a relativistic description that was developed by Poincar`e and Einstein.

From the classical Maxwell equations, we can prove Earnshaw’s Theorem, which states that a system of charged particles can not remain at rest in stable equilibrium under the inﬂuence of purely electrostatic forces. In other words, accelerated charged particles radiate energy in the form of electromagnetic waves. So how atoms and molecules are stable? or how atoms and molecules absorb and emit radiation only at certain frequencies?

Electromagnetic waves

Electromagnetic waves are made up of an oscillating electric (E), and a perpendicular magnetic (B) and are produced by accelerated charges, where the electric ﬁeld E displaces charged particles along the direction of the ﬁeld, and the magnetic ﬁeld B rotates charged particles around the direction of the ﬁeld.

3

The electric ﬁeld in units of volt m−1 and pointing in the Z-direction is described

Ez(y, t) = Eo Sin(k y − w t)

(1)

where the angular frequency is deﬁned as

w = 2πν‘,

(2)

the wave vector as

2π

k= ,

(3)

λ

and Eo is the amplitude of the electric ﬁeld. The magnetic ﬁeld pointing in the X-direction

is described Bx(y, t) = Eo Sin(k y − w t)‘, (4) c

where

w

λν ‘= c =

(5)

k

A single wavelength, λ, implies monochromatic light, where the range of wave lengths varies

from:

λ = 3 10−24 m

Cosmic Rays ν = 1032 s−1

4x1017 eV

Long radio waves λ = 3 106 m ν = 102 s−1 4x10−13 eV

When Electro Magnetic radiations (light) interacts with matter, it gets scattered. In the case of scattering from many centers, we say that light gets diﬀracted. In contrast, when light gets scattered by few centers, we say that light shows interference.

4

Lecture 3

Blackbody radiation

Experimentally we observe that when a metallic object is heated it would change its color. As the temperature is increased, ﬁrst the object changes its color into a dull red and, progressively, becomes more and more red. Eventually it changes from red to blue. As it is well established experimentally, the radiated energy is associated with a color which is associated to a frequency or wavelength. To understand this phenomenon, we will consider a so-called black-body

An ideal object that absorbs and emits all frequencies is deﬁned as a black-body. It is a theoretical model invented by theorist to study the emision and absorbtion of radiation. Its experimental counterparts consist of an insulated box which can be heated. On one of its faces a small pinhole allows radiation to enter or leave. This radiation is equivalent to that of a perfect black-body.

One of the “minor” problems mention earlier relates to the density of radiated energy per unit volume per unit frequency interval dν at temperature T, ρ(ν, T ). The search for an expression for ρ(ν, T ) backs to 1860 when Gustav Kirchhoﬀ recognized the need of a theoretical approach to blackbody radiation. Also the relation between ρ(ν, T ) with certain empirical equations was not well understood. In 1899 Wien noticed that the product of temperature ,T, and maximum wavelength, λmax was always a constant,

T λmax = constant.

(6)

Equation 6 is called the displacement law. Also the total energy radiated per unit area per unit time, R, from a blackbody followed Stefan-Boltzman law,

R = σ T4 ,

(7)

where σ is a constant. These two empirical law needed to be explained from ﬁrst principles.

In 1896 Wilhem Wien derived the following expression for ρ:

ρ(ν, T ) = C ν3 exp

ν −a

,

(8)

T

where C and a are constants. Wien’s result was experimentally conﬁrmed for high frequencies

by Freiderich Paschen in 1897. But in 1900, Otto Lummer and Ernest Pringsheim found

5

that Wien’s expression failed in the low frequency regime. The ﬁrst attempts to explain

1.5 6000 K

Classical 3000 K

1.0

Energy density per wavelength (10 J /m )

5000 K 0.5

0.0 0.0

4000 K

3000 K

500.0

1000.0 Wavelength / nm

1500.0

2000.0

FIG. 1: Plank’s radiation density for a blackbody.

the previous empirical laws from ﬁrst principles used the classical knowledge at the time.

For example in 1900 Rayleigh assumed that the radiation trapped in the box interacted

with the walls. On the walls small oscillators which in turm vibrate emiting radiation.

Finally the equilibrium between the “oscillators” and the radiation trapped in the box is

responsable of the properties of black-body radiation. Under these assuptions, the density

of radiated energy per unit volume per unit frequency at temperature T is given by the

following equation:

ρ(ν, T ) dν = Nν dν ¯(ν, T ) ,

(9)

where Nν dν represents the number of oscillators between ν and ν + dν, and ¯(ν, T ) is the average energy radiated by the “electronic oscillator” at frequency ν and temperature T.

The number density was widely accepted to be

8π ν2

Nν = c3

(10)

6

where c is the speed of light. In the calculation of ¯(ν, T ) Reyleigh assumed that the“oscillators” could achieve any

possible energy and the equipatition theorem. In other words Rayleigh assumed a continuous energy spectrum for the oscillators and

¯(ν, T ) = kBT ,

(11)

where kB is Boltzman constant. With these assumptions, Rayleigh obtained the following expression for the amount of radiative energy between ν and ν + dν:

8π ν2

ρ(ν, T ) dν = c3 kBT dν .

(12)

If we express Eq. 12 as radiated energy per unit volume per unit wavelength we get

8π

ρ (λ, T ) dλ = λ4 kBT dλ ,

(13)

where we have used the relation c = λν. This latter result agreed with the low frequency observation but did not ﬁt the experimental observationsm at high frequencies.

From the low and high frequency expressions,

kB T

for low ν

¯(ν, T ) ≈

,

(14)

ν exp − a Tν for high ν

Planck derived an expression consistent with both limiting behaviors,

hν

¯(ν, T ) =

.

(15)

exp khBνT − 1

This expression required only one constant, h, which Plank determined by ﬁtting the expression to the experiemntal data. Equation 15 is called the Plank distribution law. Plank obtained an excellent agreement with experiments using

h = 6.626 × 10−34 J s .

(16)

Although Eq. 15 ﬁtted the experiemntal data extraordinary well, Plank wanted to understand why Eq. 15 worked so well. First Plank calculated the entropy of the system from 15. Second, he calculated the entropy using Boltzman mechanistic approach. When these to independent expression for the entropy were compared, Plank concluded the the oscillators

7

spectrum had to be discrete. In other words the “oscillators” could only achive the following

energy values:

n = nhν,

(17)

where n is a positive integer and h is Plank’s constant. Thus both of Plank’s entropy expression were consisten if energy was quantized.

From Max Planck’s underlying assuption that the energies of the “electronic oscillators” could have only a discrete set of values we can derived the semiempirical Plank’s distribution using statistical methods. In this approach we assumed that only jumps of ∆n = ±1 can occur, so the change of energy is given by:

∆ = hν.

(18)

This means that energy is absorbed or emitted only in packets. In order to obtain an expression for the density of radiative energy between ν and ν + dν

we have to calculate the average energy, ¯(ν), at frequency ν. First we consider the Boltzman probability for the energy level of each oscillator at frequency ν, i.e.,

n hν Pn(ν) ≈ exp kBT . (19)

Using Eq. 19 we get for the average oscillator’s energy

¯(ν) =

∞ n=1 nhν exp nkBhTν ∞ n=1 exp nkBhTν

∞ nhν Xn

=

n=1 ∞

n=1 X n

= hν (1−XX)2

1

1−X

hν X =

1−X

hν =

exp khBνT

. −1

(20)

Finally using Eqs. 9 and 10 the Plank’s expression for the radiative energy density is equal

to

8π ν2

hν

ρ(ν, T ) dν =

dν .

(21)

c3 exp h ν − 1

kB T

8

Also as a function of wavelength we get

8π hc

1

ρ (λ, T ) dλ =

dλ .

(22)

λ5 exp h c − 1

kB T λ

From Eq. 22 we can obtain the Wien displacement law and the Stefan-Boltzman law.

9

Lecture 4

Photoelectric eﬀect

Late in the 19th century a series of experiments revealed that electrons are emitted from a metal surface when light of suﬃcient high frequency falls upon it. This is known as the photoelectric eﬀect. Classical theory predicted that the energy of the ejected electron is proportional to the intensity of the light; electrons are ejected for any frequency radiated. Both prediction are not support by the experients.

The experiemnts show the existance of a threshold frequency, ν◦. Einstein in 1905 assumed that radiation consists of little packets of energy = hν. If this is true the kinetic energy of the ejected electron is given by the following equation:

KE = 1 mv2 = hν − φ

(23)

2

where φ is the work function which is usually expressed in electron volts, eV. Note that the threshhold frequency, ν◦, implies no kinetic energy. Thus

hν◦ = φ,

(24)

and the photoelectric eﬀect is observed only if hν ≥ φ. Equation 23 represents a straight

line with slope h. Actually in an experiment one measures the stopping potential, VS, such

that KE = − e VS = 21 mv2 . (25)

10

CHEMISTRY 361 QUANTUM CHEMISTRY

AND CHEMICAL KINETICS

Enrique Peacock-L´opez1, ∗ 1Department of Chemistry Williams College Williamstown, MA 01267

(Dated: June 17, 2006)

Abstract

The early XX century saw the birth of the theory of relativity. This theory was developed after the 1887 Michelson and Morley’58 experiment which was designed to reveal the earth’s motion relative to the ether. In an eﬀort to explain the experimental facts, Poincare and Einstein altered the ideas and concept of space, time and astronomical distances. Most of the development of the the theory of relativity was carried out by the latter scientists. In contrast Quantum chemistry took several decades and many contributors to be developed. This theory can be seen as an extension of classical mechanics to the subatomic, atomic and molecular sizes and distances.

In this course we will study the fundamental theory of electrons, atoms and molecules known as Quantum Chemistry. In our approach we will follow the historical development of Quantum Chemistry. In this approach we will see how Quantum Chemistry gradually evolves from confusion and dilemma into a formal theory.

∗Author to whom correspondence should be addressed. Electronic address: [email protected]

1

Lecture 1

As we have seen in previous chapters, the world we experience with our senses is a world described by Thermodynamics, Classical Mechanics and Classical Electromagnetism. It is a macroscopic world based in macroscopic laws. But scientist have long recognized experimentally the existence of small particles which are the components of matter. Therefore it is natural to look ﬁrst for microscopic laws that explain the particle’s behavior, and second we have to look for a theoretical approach that will link the microscopic laws with the macroscopic laws.

Towards the end of the XIX century, many physicist felt that all the principles of physics had been discovered and little remainded to be understood except for few minor problems. At that time our world was understood using Classical Mechanics, Thermodynamics and Classical Electromagnetism. For example in Classical Mechanics or Newtonian Mechanics we needed to ﬁnd the dynamical variables of the system under study. Once done this, we needed to construct the equation of motion which predicted the system’s evolution in time. The predicted behavior was ﬁnally compared with the experimental observations.

At the time the Universe was divided into matter and radiation. Matter was ruled by Newtonian mechanics and thermodynamics, and radiation obeyed Maxwell’s laws of Electromagnetism. A controversy whether light was wave-like or corpuscular-like existed since Newton’s days, who proposed the corpuscular theory of light. The wave-like theory was developed by Huggens based in the constructive and destructive interference of light; a property which is characteristic of waves.

The interaction between matter and radiation was not well understood. For example, Earnshaw’s theorem states that a system of charged particles can not remain at rest in stable equilibrium under the inﬂuence of purely electrostatic forces. Moreover, according to Electromagnetism an accelerated charged particles radiate energy in the form. of electromagnetic waves. Thus how is that molecules are stable? From this minor problems the theory of Quantum chemistry was developed.

2

Lecture 2

During the 17th century Galileo Galilei and Issac Newton postulated a corpuscular theory of light. After 200 years of work, in the 19th century we have a logical theory with deterministic equation that explain planetary motion as well as classical mechanics, thermodynamics, optics, electricity and magnetism.

In the mid 1800s, Thomas Young reported a pattern consisting of dark and bright fringes, which were produces after passing light through two narrow closely spaced holes or slits. This interference pattern is a natural wave behavior. By 1860, James Clark Maxwell developed a classical electromagnetic theory, which combines electricity and magnetism in a single theory. Also, Maxwell’s theory predicts that the properties of light can be explained if we consider light as electromagnetic radiation. The experimental conﬁrmation of maxwell equation was reported by Michael Faraday, and 1886, although assuming a luminiferous (ether) medium, Hertz conﬁrmed experimentally that light is electromagnetic radiation giving the wave theory of light a solid foundation.

In 1887, Albert Michelson and edward Morley’1860, conﬁrmed experimentally the electromagnetic waves traveled in vacuum by ﬁnding no diﬀerence in the speed of light relative to the motion of Earth. In other word, there is no need of the luminiferous or any other medium. Without a medium, light is subject to a relativistic description that was developed by Poincar`e and Einstein.

From the classical Maxwell equations, we can prove Earnshaw’s Theorem, which states that a system of charged particles can not remain at rest in stable equilibrium under the inﬂuence of purely electrostatic forces. In other words, accelerated charged particles radiate energy in the form of electromagnetic waves. So how atoms and molecules are stable? or how atoms and molecules absorb and emit radiation only at certain frequencies?

Electromagnetic waves

Electromagnetic waves are made up of an oscillating electric (E), and a perpendicular magnetic (B) and are produced by accelerated charges, where the electric ﬁeld E displaces charged particles along the direction of the ﬁeld, and the magnetic ﬁeld B rotates charged particles around the direction of the ﬁeld.

3

The electric ﬁeld in units of volt m−1 and pointing in the Z-direction is described

Ez(y, t) = Eo Sin(k y − w t)

(1)

where the angular frequency is deﬁned as

w = 2πν‘,

(2)

the wave vector as

2π

k= ,

(3)

λ

and Eo is the amplitude of the electric ﬁeld. The magnetic ﬁeld pointing in the X-direction

is described Bx(y, t) = Eo Sin(k y − w t)‘, (4) c

where

w

λν ‘= c =

(5)

k

A single wavelength, λ, implies monochromatic light, where the range of wave lengths varies

from:

λ = 3 10−24 m

Cosmic Rays ν = 1032 s−1

4x1017 eV

Long radio waves λ = 3 106 m ν = 102 s−1 4x10−13 eV

When Electro Magnetic radiations (light) interacts with matter, it gets scattered. In the case of scattering from many centers, we say that light gets diﬀracted. In contrast, when light gets scattered by few centers, we say that light shows interference.

4

Lecture 3

Blackbody radiation

Experimentally we observe that when a metallic object is heated it would change its color. As the temperature is increased, ﬁrst the object changes its color into a dull red and, progressively, becomes more and more red. Eventually it changes from red to blue. As it is well established experimentally, the radiated energy is associated with a color which is associated to a frequency or wavelength. To understand this phenomenon, we will consider a so-called black-body

An ideal object that absorbs and emits all frequencies is deﬁned as a black-body. It is a theoretical model invented by theorist to study the emision and absorbtion of radiation. Its experimental counterparts consist of an insulated box which can be heated. On one of its faces a small pinhole allows radiation to enter or leave. This radiation is equivalent to that of a perfect black-body.

One of the “minor” problems mention earlier relates to the density of radiated energy per unit volume per unit frequency interval dν at temperature T, ρ(ν, T ). The search for an expression for ρ(ν, T ) backs to 1860 when Gustav Kirchhoﬀ recognized the need of a theoretical approach to blackbody radiation. Also the relation between ρ(ν, T ) with certain empirical equations was not well understood. In 1899 Wien noticed that the product of temperature ,T, and maximum wavelength, λmax was always a constant,

T λmax = constant.

(6)

Equation 6 is called the displacement law. Also the total energy radiated per unit area per unit time, R, from a blackbody followed Stefan-Boltzman law,

R = σ T4 ,

(7)

where σ is a constant. These two empirical law needed to be explained from ﬁrst principles.

In 1896 Wilhem Wien derived the following expression for ρ:

ρ(ν, T ) = C ν3 exp

ν −a

,

(8)

T

where C and a are constants. Wien’s result was experimentally conﬁrmed for high frequencies

by Freiderich Paschen in 1897. But in 1900, Otto Lummer and Ernest Pringsheim found

5

that Wien’s expression failed in the low frequency regime. The ﬁrst attempts to explain

1.5 6000 K

Classical 3000 K

1.0

Energy density per wavelength (10 J /m )

5000 K 0.5

0.0 0.0

4000 K

3000 K

500.0

1000.0 Wavelength / nm

1500.0

2000.0

FIG. 1: Plank’s radiation density for a blackbody.

the previous empirical laws from ﬁrst principles used the classical knowledge at the time.

For example in 1900 Rayleigh assumed that the radiation trapped in the box interacted

with the walls. On the walls small oscillators which in turm vibrate emiting radiation.

Finally the equilibrium between the “oscillators” and the radiation trapped in the box is

responsable of the properties of black-body radiation. Under these assuptions, the density

of radiated energy per unit volume per unit frequency at temperature T is given by the

following equation:

ρ(ν, T ) dν = Nν dν ¯(ν, T ) ,

(9)

where Nν dν represents the number of oscillators between ν and ν + dν, and ¯(ν, T ) is the average energy radiated by the “electronic oscillator” at frequency ν and temperature T.

The number density was widely accepted to be

8π ν2

Nν = c3

(10)

6

where c is the speed of light. In the calculation of ¯(ν, T ) Reyleigh assumed that the“oscillators” could achieve any

possible energy and the equipatition theorem. In other words Rayleigh assumed a continuous energy spectrum for the oscillators and

¯(ν, T ) = kBT ,

(11)

where kB is Boltzman constant. With these assumptions, Rayleigh obtained the following expression for the amount of radiative energy between ν and ν + dν:

8π ν2

ρ(ν, T ) dν = c3 kBT dν .

(12)

If we express Eq. 12 as radiated energy per unit volume per unit wavelength we get

8π

ρ (λ, T ) dλ = λ4 kBT dλ ,

(13)

where we have used the relation c = λν. This latter result agreed with the low frequency observation but did not ﬁt the experimental observationsm at high frequencies.

From the low and high frequency expressions,

kB T

for low ν

¯(ν, T ) ≈

,

(14)

ν exp − a Tν for high ν

Planck derived an expression consistent with both limiting behaviors,

hν

¯(ν, T ) =

.

(15)

exp khBνT − 1

This expression required only one constant, h, which Plank determined by ﬁtting the expression to the experiemntal data. Equation 15 is called the Plank distribution law. Plank obtained an excellent agreement with experiments using

h = 6.626 × 10−34 J s .

(16)

Although Eq. 15 ﬁtted the experiemntal data extraordinary well, Plank wanted to understand why Eq. 15 worked so well. First Plank calculated the entropy of the system from 15. Second, he calculated the entropy using Boltzman mechanistic approach. When these to independent expression for the entropy were compared, Plank concluded the the oscillators

7

spectrum had to be discrete. In other words the “oscillators” could only achive the following

energy values:

n = nhν,

(17)

where n is a positive integer and h is Plank’s constant. Thus both of Plank’s entropy expression were consisten if energy was quantized.

From Max Planck’s underlying assuption that the energies of the “electronic oscillators” could have only a discrete set of values we can derived the semiempirical Plank’s distribution using statistical methods. In this approach we assumed that only jumps of ∆n = ±1 can occur, so the change of energy is given by:

∆ = hν.

(18)

This means that energy is absorbed or emitted only in packets. In order to obtain an expression for the density of radiative energy between ν and ν + dν

we have to calculate the average energy, ¯(ν), at frequency ν. First we consider the Boltzman probability for the energy level of each oscillator at frequency ν, i.e.,

n hν Pn(ν) ≈ exp kBT . (19)

Using Eq. 19 we get for the average oscillator’s energy

¯(ν) =

∞ n=1 nhν exp nkBhTν ∞ n=1 exp nkBhTν

∞ nhν Xn

=

n=1 ∞

n=1 X n

= hν (1−XX)2

1

1−X

hν X =

1−X

hν =

exp khBνT

. −1

(20)

Finally using Eqs. 9 and 10 the Plank’s expression for the radiative energy density is equal

to

8π ν2

hν

ρ(ν, T ) dν =

dν .

(21)

c3 exp h ν − 1

kB T

8

Also as a function of wavelength we get

8π hc

1

ρ (λ, T ) dλ =

dλ .

(22)

λ5 exp h c − 1

kB T λ

From Eq. 22 we can obtain the Wien displacement law and the Stefan-Boltzman law.

9

Lecture 4

Photoelectric eﬀect

Late in the 19th century a series of experiments revealed that electrons are emitted from a metal surface when light of suﬃcient high frequency falls upon it. This is known as the photoelectric eﬀect. Classical theory predicted that the energy of the ejected electron is proportional to the intensity of the light; electrons are ejected for any frequency radiated. Both prediction are not support by the experients.

The experiemnts show the existance of a threshold frequency, ν◦. Einstein in 1905 assumed that radiation consists of little packets of energy = hν. If this is true the kinetic energy of the ejected electron is given by the following equation:

KE = 1 mv2 = hν − φ

(23)

2

where φ is the work function which is usually expressed in electron volts, eV. Note that the threshhold frequency, ν◦, implies no kinetic energy. Thus

hν◦ = φ,

(24)

and the photoelectric eﬀect is observed only if hν ≥ φ. Equation 23 represents a straight

line with slope h. Actually in an experiment one measures the stopping potential, VS, such

that KE = − e VS = 21 mv2 . (25)

10