Atoms and Molecules By: Dr. Ashish Kumar LEARNING OBJECTIVES After completing this chapter, you will be able to understand the following: Concept of.

Atoms and Molecules

By: Dr. Ashish Kumar

LEARNING OBJECTIVES

After completing this chapter, you will be able to understand the following:• Concept of the structure of atom as a nucleus with orbital electrons.

• Progress towards quantum mechanical model of the atom.

• Quantum mechanical model of an atom and its important features.

• Application of Schrödinger equation to explain the structure of hydrogen atom.

• Radial and angular functions of wave equations.

• Statement of Pauli exclusion principle.• The features of quantum mechanical model.• Concept of orbitals and quantum numbers.• Electronic configuration of atoms.


Experiments to determine what an atom was

• J. J. Thomson- used Cathode ray tubesA cathode ray tube with an electric fieldperpendicular to the direction ofthe cathode rays and an externalmagnetic field. The symbols Nand S denote the north and southpoles of the magnet. The cathoderays will strike the end of thetube at A in the presence of amagnetic field, at C in thepresence of an electric field, andat B when there are no externalfields present or when the effectsof the electric field and magneticfield cancel each other.


Thomson’s Experiment

Voltage source

+-



Voltage source

+-



Voltage source

+-


Passing an electric current makes a beam appear to move from the negative to the positive end


Voltage source

+-




Voltage source

+-




Voltage source

+-




Voltage source

+-


Voltage source


• By adding an electric field


Voltage source


By adding an electric field

+

-


Voltage source



+

-


Voltage source



+

-


Voltage source



+

-


Voltage source



+

-


Voltage source


By adding an electric field he found that the moving pieces were negative

+

-


Thomsom’s Model

(a) A cathode ray produced in a discharge tube.The ray itself is invisible, but the fluorescence of a zinc sulfidecoating on the glass causes it to appear green.

(b) The cathode ray is bent in the presence of amagnet.


Thomsom’s Model• Found the electron• Couldn’t find positive (for a while) • Said the atom was like plum pudding• A bunch of positive stuff, with the electrons

able to be removed • Charge/mass = -1.76 X 108 C/g


Mullikan• Charge/mass = -1.76 X 108 C/g• Found by oil drop method• charge on electron= -1.6022 X 10-19 C


Rutherford’s Experiment

• Used uranium to produce alpha particles• Aimed alpha particles at gold foil by

drilling hole in lead block• Since the mass is evenly distributed in

gold atoms alpha particles should go straight through.

• Used gold foil because it could be made atoms thin


Lead block

Uranium

Gold Foil

Florescent Screen


What he expected


Because


Because, he thought the mass was evenly distributed in the atom


What he got


How he explained it

+

• Atom is mostly empty• Small dense,

positive piece at center

• Alpha particles are deflected by it if they get close enough


+


Modern View• The atom is mostly

empty space• Two regions• Nucleus- protons and

neutrons• Electron cloud- region

where you have a chance of finding an electron


Modern View


Properties of Waves

Wavelength (l) is the distance between identical points on successive waves.

Amplitude is the vertical distance from the midline of a wave to the peak or trough.

7.1By: Dr. Ashish Kumar

Properties of Waves

Frequency (n) is the number of waves that pass through a particular point in 1 second (Hz = 1 cycle/s).

The speed (u) of the wave = l x n7.1


Maxwell (1873), proposed that visible light consists of electromagnetic waves.

Electromagnetic radiation is the emission and transmission of energy in the form of electromagnetic waves.

Speed of light (c) in vacuum = 3.00 x 108 m/s

All electromagnetic radiationl x = n c



l x n = cl = c/nl = 3.00 x 108 m/s / 6.0 x 104 Hz l = 5.0 x 103 m

Radio wave

A photon has a frequency of 6.0 x 104 Hz. Convertthis frequency into wavelength (nm). Does this frequencyfall in the visible region?

l = 5.0 x 1012 nm

l

n


Mystery #1, “Black Body Problem”Solved by Planck in 1900

Energy (light) is emitted or absorbed in discrete units (quantum).

E = h x nPlanck’s constant (h)h = 6.63 x 10-34 J•s


Light has both:1. wave nature2. particle nature

hn = KE + BE

Mystery #2, “Photoelectric Effect”Solved by Einstein in 1905

Photon is a “particle” of light

KE = hn - BE

hn

KE e-


De Broglie (1924) reasoned that e- is both particle and wave.

Why is e- energy quantized?

7.4

u = velocity of e-

m = mass of e-

2pr = nl l = hmu


Let’s consider a base ball mass=0.142 kg and velocity = 42 m/s l = h /mv

l = 6.63 x 10-34 (kg m2 s-2)s / 0.142 kg x 42 m s-1

l = 1.1 x 10 -31 m --------Undetectably small

Why don’t we observe the wave nature of matter in day to day activities ?




l = h/mu

l = 6.63 x 10-34 / (2.5 x 10-3 x 15.6)

l = 1.7 x 10-32 m = 1.7 x 10-23 nm

What is the de Broglie wavelength (in nm) associated with a 2.5 g Ping-Pong ball traveling at 15.6 m/s?

m in kgh in J•s u in (m/s)


E = h x n

E = 6.63 x 10-34 (J•s) x 3.00 x 10 8 (m/s) / 0.154 x 10-9 (m)

E = 1.29 x 10 -15 J

E = h x c / l

7.2

When copper is bombarded with high-energy electrons, X rays are emitted. Calculate the energy (in joules) associated with the photons if the wavelength of the X rays is 0.154 nm.


Heisenberg’s Uncertainty Principle

According to Heisenberg’s uncertainty principle, it is difficult to measure both quantities accurately at the same time. Heisenberg stated that the more precisely we can define the position of an electron, the less certainly we are able to define its velocity, and vice versa.

If x is the uncertainty in defining the position and Δpx (or Δvx) the uncertainty in the momentum (or velocity), the uncertainty principle may be expressed mathematically as:

Δ Δ 4xhx p

Δ Δ( ) Δ Δ4 4x xh hx mv x v

m


Significance of Uncertainty Principle

The uncertainty principle formally limits the precision to which two complementary observables – position and momentum – of a subatomic particle can be measured.

It also leads to the conclusion that the properties of a subatomic particle observed are not inde pendent of the observer.

It also leads to the concept that the values of the position and momentum of a particle can take a range of values rather than a single, exact value.

Using Bohr’s atomic theory, we could calculate the radius of a one-electron system (such as H, He+ and Li2+) and the velocity of the electron in orbit.

For objects of large mass, the uncertainty in the position (∆x) and the uncertainty in the velocity (∆v) are insignificant because the value of h/4 is very small and is equal to 5.273 10−35 J s and its mass is very large compared to the value of h/4.By: Dr. Ashish Kumar

Heisenberg’s principle becomes significant in case of microscopic objects like an electron. The mass of an electron is 9.1 10−31 kg, then according to Heisenberg’s principle

Δ Δ 4xhx vm

345 2 1 2 –2

31

6.626 10 J sΔ Δ 5.79 10 m s (J = kg m s )

4 3.14 9.11 10 kgxx v

4 2 14 1

9

0.579 10 m sΔ Δ 5.79 10 ms

10 mxx v


QUANTUM MECHANICAL MODEL OF ATOM

Quantum mechanics is a theory of atomic structure based on the wave–particle duality of matter.

For a system, such as an atom, whose energy does not change with time, the Schrödinger equation is given by

is a mathematical operator called Hamiltonian which is constructed based on the total energy of the system.

Solutions to the wave equation are called wave functions and given by the symbol .

H E H


Hydrogen Atom and the Schrödinger Wave Equation

The application of Schrödinger’s wave equation to multi-electron systems is somewhat more complicated and its solution is not easily obtained. However, computer-aided calculations have shown that the orbitals in multi-electron atoms are similar to that in hydrogen, except that increased nuclear charge leads to their contraction.

The probability of finding an electron at a point x, y, z over all space

2 2 22

2

4 m vh

2 2 22

2

4 0m vh

2 212

2( )mv E V v E V

m

22

2

8( ) 0

mE V

h

2 d d d 1x y z


Schrodinger Wave EquationIn 1926 Schrodinger wrote an equation that described both the particle and wave nature of the e-

Wave function (Y) describes:

1 . energy of e- with a given Y

2 . probability of finding e- in a volume of space

Schrodinger’s equation can only be solved exactly for the hydrogen atom. Must approximate its solution for multi-electron systems.


Important Features of Quantum Mechanical Model

The energy of electrons in an atom is quantized, that is, they can have only specific values when bound to nucleus in an atom.

All the information about a particle (electron) in a given energy level is contained in the wave function, which corresponds to the allowed (physically possible) solutions of the Schrödinger wave equation.

The probability of finding an electron in an atom is proportional to | |2, that is, the square of wave function and is known as probability density. It always has a positive value and the value at different points in atom corresponds to region around the nucleus where the electron is most likely to be found.

The wave function and its square | |2 have values for all locations about the nucleus. For hydrogen atom, | |2 is large near the nucleus and decreases with increase in distance away from the nucleus.

The wave function obtained by solving Schrödinger equation for an electron in an atom corresponds to the atomic orbital. Since the equation can have many solutions, there are various possible wave functions for an electron and hence many possible atomic orbitals.


Difference between orbit and orbitals

Orbits Orbitals

They represent planar motion of electron orbits.

They represent three-dimensional motion of electron around the nucleus.

All orbits are circular or have disk-like shape.

Different orbitals have different shapes.

They do not have directional characteristics.

They have directional characteristics (apart from s orbital).

They can accommodate a maximum of 2n2 electrons.

They can accommodate a maximum of two electrons.

They are in accordance with classical theory of atom.

They are in accordance with wave theory (quantum mechanical model) of atom.


Schrodinger Wave Equation

= Y fn(n, l, ml, ms)

principal quantum number n

n = 1, 2, 3, 4, ….

n=1 n=2 n=3

7.6

distance of e- from the nucleus


Where 90% of thee- density is foundfor the 1s orbital

swave.gif


http://homepages.ius.edu/kforinas/physlets/quantum/hydrogen.html



Y = fn(n, l, ml, ms)

angular momentum quantum number l

for a given value of n, l = 0, 1, 2, 3, … n-1

n = 1, l = 0n = 2, l = 0 or 1

n = 3, l = 0, 1, or 2

Shape of the “volume” of space that the e- occupies

l = 0 s orbitall = 1 p orbitall = 2 d orbitall = 3 f orbital



Orbitals and Quantum Numbers

The principal quantum number (n) has positive integer values such as 1, 2, 3, etc.

Orbitals that have the same value of the principal quantum number form a shell. Orbitals within a shell are divided into subshells that are labeled with the same value of the angular quantum number.

The shells are sometimes denoted by the letters K, L, M, N, …, counting outwards from the nucleus.

The azimuthal or subsidiary quantum number (also known as orbital angular momentum, l) can have values 0, 1, 2, … (n 1).

The magnetic quantum number (ml) is used to describe the orientation in space of a particular orbital.

Value for l 0 1 2 3 4 5

Notation for subshell

s (spherical)

p (dumbbell)

d (double dumbbell)

f g h


l = 0 (s orbitals)

l = 1 (p orbitals)


l = 2 (d orbitals)



magnetic quantum number ml

for a given value of lml = -l, …., 0, …. +l

orientation of the orbital in space

if l = 1 (p orbital), ml = -1, 0, or 1if l = 2 (d orbital), ml = -2, -1, 0, 1, or 2



ml = -1 ml = 0 ml = 1

ml = -2 ml = -1 ml = 0 ml = 1 ml = 27.6


Atomic Orbitals

Principal quantum

number (n)

Subsidiary quantum

number (l)

Magnetic quantum numbers (ml)

Symbol

1 0 0 1s (one orbital)

22

01

01, 0, +1

2s (one orbital)2p (three orbitals)

333

012

01, 0, +12, 1, 0, +1, +2

3s (one orbital)3p (three orbitals)3d (five orbitals)

4444

0123

01, 0, +12, 1, 0, +1, +23, 2, 1, 0, +1, +2, +3

4s (one orbital)4p (three orbitals)4d (five orbitals)4f (seven orbitals)


Selection Rules Governing Allowed Combinations of Quantum Numbers

The three quantum numbers (n, l and ml) are all integers.

The principal quantum number (n) cannot be zero. The allowed values of n are therefore 1, 2, 3, 4, .

The angular quantum number (l) can be any integer between 0 and n 1. If n = 3, for example, l can be either 0, 1 or 2.

The magnetic quantum number (ml) can be any integer between l and +l. If l = 2, ml can be 2, 1, 0, 1 or 2.



spin quantum number ms

ms = +½ or -½


ms = -½ms = +½


Principal Quantum Number (n)

Azimuthal Quantum Number (l)

Magnetic Quantum Number (ml)

Spin Quantum Number (ms)

The main shell in which the electron resides.

The number of subshells present in the nth shell.

The number of orbitals present in each subshell or the orientation of the subshells.

The energy of the orbital.

The energy of the orbital in a multi-electron system.

Approximate distance of the electron from the nucleus.

Shape of the orbitals.

The maximum number of electrons present in the shell.

For a given value of n, l can have values from 0 to n 1. In each subshell, there are (2l + 1) types of orbitals.

For a given value of l, ml = l to +l.

For a particular value of ml, we have ms = +1/2 or 1/2.

Represented by integers 1, 2, 3, or K, L, M, N, etc.

l = 0, s subshelll = 1, p subshelll = 2, d subshelll = 3, f subshell

For s subshell: sFor p subshell: px, py, pz

For d subshell: dxy, dyz, dxz, ,

Represented by two arrows pointing in opposite directions, and .

Accounts for the main lines in the atomic spectrum.

Accounts for the fine lines in the atomic spectra.

Accounts for the splitting of lines in the atomic spectrum.

Accounts for magnetic properties of substances.

Information obtained from the four quantum numbers


Existence (and energy) of electron in atom is described by its unique wave function Y.

Pauli exclusion principle - no two electrons in an atomcan have the same four quantum numbers.







Shell – electrons with the same value of n

Subshell – electrons with the same values of n and l

Orbital – electrons with the same values of n, l, and ml

How many electrons can an orbital hold?

If n, l, and ml are fixed, then ms = ½ or - ½

Y = (n, l, ml, ½)or Y = (n, l, ml, -½)

An orbital can hold 2 electrons 7.6By: Dr. Ashish Kumar

How many 2p orbitals are there in an atom?

2p

n=2

l = 1

If l = 1, then ml = -1, 0, or +1

3 orbitals

How many electrons can be placed in the 3d subshell?

3d

n=3

l = 2

If l = 2, then ml = -2, -1, 0, +1, or +2

5 orbitals which can hold a total of 10 e-


Energy of orbitals in a single electron atom

Energy only depends on principal quantum number n

En = -RH ( )1n2

n=1

n=2

n=3


Energy of orbitals in a multi-electron atom

Energy depends on n and l

n=1 l = 0

n=2 l = 0n=2 l = 1

n=3 l = 0n=3 l = 1

n=3 l = 2


ARRANGEMENT OF THE ELEMENTS IN GROUPS IN THE PERIODIC TABLE

If the elements are arranged in groups which have the same outer electronic arrangement, then elements within a group should show similarities in chemical and physical properties.

Elements with one s electron in their outer shell are called Group 1 (the alkali metals) and elements with two s electrons in their outer shell are called Group 2 (the alkaline earth metals). These two groups are known as the s-block elements, because their properties result from the presence of s electrons.

Elements with three electrons in their outer shell (two s electrons and one p electron) are called Group 13, and similarly Group 14 elements have four outer electrons, Group 15 elements have five outer electrons. Group 16 elements have six outer electrons and Group 17 elements have seven outer electrons. Group 18 elements have a full outer shell of electrons. Groups 13–18 all have p orbitals filled and because their properties are dependent on the presence of p electrons, they are called jointly the p-block elements.


In a similar way, elements where d orbitals are being filled are called the d-block, or transition, elements. In these, d electrons are being added to the penultimate shell. For example, the element scandium Sc is the first transition element, and follows immediately after the element calcium Ca, which is in Group 2.

Finally, elements where f orbitals are filling are called the f-block, and here the f electrons are entering the antepenultimate (or second from the outside) shell.


Table 6 Periodic table


Electronic Configuration

Pauli Exclusion Principle

The Pauli’s exclusion principle states that no two electrons in one atom can have all four quantum numbers the same. By permutating the quantum numbers, the maximum number of electrons which can be contained in each main energy level can be calculated

Quantum numbers, the permissible number of electrons and the shape of the periodic table.


“Fill up” electrons in lowest energy orbitals (Aufbau principle)

H 1 electron

H 1s1

He 2 electrons

He 1s2

Li 3 electrons

Li 1s22s1

Be 4 electrons

Be 1s22s2

B 5 electrons

B 1s22s22p1

C 6 electrons

? ?


C 6 electrons

The most stable arrangement of electrons in subshells is the one with the greatest number of parallel spins (Hund’s rule).

C 1s22s22p2

N 7 electrons

N 1s22s22p3

O 8 electrons

O 1s22s22p4

F 9 electrons

F 1s22s22p5

Ne 10 electrons

Ne 1s22s22p6


Order of orbitals (filling) in multi-electron atom

1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s < 4d < 5p < 6s7.7


Electron configuration is how the electrons are distributed among the various atomic orbitals in an atom.

1s1

principal quantumnumber n

angular momentumquantum number l

number of electronsin the orbital or subshell

Orbital diagram

H

1s1


What is the electron configuration of Mg?

Mg 12 electrons

1s < 2s < 2p < 3s < 3p < 4s

1s22s22p63s2 2 + 2 + 6 + 2 = 12 electrons

7.8

Abbreviated as [Ne]3s2 [Ne] 1s22s22p6

What are the possible quantum numbers for the last (outermost) electron in Cl?

Cl 17 electrons 1s < 2s < 2p < 3s < 3p < 4s

1s22s22p63s23p5 2 + 2 + 6 + 2 + 5 = 17 electrons

Last electron added to 3p orbital

n = 3 l = 1 ml = -1, 0, or +1 ms = ½ or -½By: Dr. Ashish Kumar

Outermost subshell being filled with electrons



Paramagnetic

unpaired electrons

2p

Diamagnetic

all electrons paired

2p7.8


Atoms and Molecules By: Dr. Ashish Kumar LEARNING OBJECTIVES After completing this chapter, you will be able to understand the following: Concept of.

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