One way to give someone directions is to tell them to go three blocks East and five blocks South. Another way to give directions is to point and say “Go a half mile in that direction.” Polar graphing is like the second method of giving directions. Each point is determined by a distance and an angle. Initial ray A polar coordinate pair determines the location of a point. Polar Coordinates

To define the Polar Coordinates of a plane we need first to fix a point which will be called the Pole (or the origin) and a half-line starting from the pole. This half-line is called the Polar Axis. θ r P(r, θ) Polar Axis Polar Angles The Polar Angle θ of a point P, P ≠ pole, is the angle between the Polar Axis and the line connecting the point P to the pole. Positive values of the angle indicate angles measured in the counterclockwise direction from the Polar Axis. Polar Coordinates The Polar Coordinates (r,θ) of the point P, P ≠ pole, consist of the distance r of the point P from the Pole and of the Polar Angle θ of the point P. Every (0, θ) represents the pole. A positive angle.

More than one coordinate pair can refer to the same point. All of the polar coordinates of this point are:

The connection between Polar and Cartesian coordinates θ r (x,y)(x,y) y x From the right angle triangle in the picture one immediately gets the following correspondence between the Cartesian Coordinates (x,y) and the Polar Coordinates (r,θ) assuming the Pole of the Polar Coordinates is the Origin of the Cartesian Coordinates and the Polar Axis is the positive x-axis. x = r cos(θ) y = r sin(θ) r 2 = x 2 + y 2 tan(θ) = y/x Using these equations one can easily switch between the Cartesian and the Polar Coordinates.

3) The hydrogen atom orbitals Radial components of the hydrogen atom wavefunctions look as follows: R(r) = for n = 1, l = 0 1s orbital R(r) = for n = 2, l = 02s orbital R(r) = for n = 2, l = 1, m l = 02p z orbital R(r) = n = 3, l = 03s orbital R(r) = n = 3, l = 1, m l = 03p z orbital Here a 0 =, the first orbit radius (0.529Å). Z is the nuclear charge (+1)

THE ELECTRONIC STRUCTURE OF THE HYDROGEN ATOM 1) Theoretical description of the hydrogen atom: Schrödinger equation: H  = E  H H is the Hamiltonian operator (kinetic + potential energy of an electron) E – the energy of an electron whose distribution in space is described by  - wave function  * is proportional to the probability of finding the electron in a given point of space If  is normalized, the following holds: The part of space where the probability to find electron is non-zero is called orbital When we plot orbitals, we usually show the smallest volume of space where this probability is 90%. : 1) Theoretical description of the hydrogen atom: Schrödinger equation: H  = E  H H is the Hamiltonian operator (kinetic + potential energy of an electron) E – the energy of an electron whose distribution in space is described by  - wave function  * is proportional to the probability of finding the electron in a given point of space If  is normalized, the following holds: The part of space where the probability to find electron is non-zero is called orbital When we plot orbitals, we usually show the smallest volume of space where this probability is 90%. :

2) Solution of the Schrödinger equation for the hydrogen atom. Separation of variables r,  For the case of the hydrogen atom the Schrödinger equation can be solved exactly when variables r,  are separated like  = R(r)  Three integer parameters appear in the solution, n (principal quantum number), while solving R- equation l (orbital quantum number), while solving  - equation m l (magnetic quantum number),while solving  - equation n = 1, 2, … ∞; defines energy of an electron

4) Radial components of the hydrogen atom orbitals The exponential decay of R(r) is slower for greater n At some distances r where R(r) is equal to 0, we have radial nodes, n-l- 1 in total The exponential decay of R(r) is slower for greater n At some distances r where R(r) is equal to 0, we have radial nodes, n-l- 1 in total 1s 2s 2p node

5) Radial probability function Radial probability function is defined as [r R(r)] 2 – probability of finding an electron in a spherical layer at a distance r from nucleus Radial probability plots for hydrogen atom: Radial probability function is defined as [r R(r)] 2 – probability of finding an electron in a spherical layer at a distance r from nucleus Radial probability plots for hydrogen atom: 1s 2s 2p

6) Angular components Angular components of the hydrogen atom wave functions,  for l = 0, m l = 0;  = s orbitals for l = 1, m l = 0;  = p z orbitals Angular components of the hydrogen atom wave functions,  for l = 0, m l = 0;  = s orbitals for l = 1, m l = 0;  = p z orbitals

4) Radial components of the hydrogen atom orbitals The exponential decay of R(r) is slower for greater n At some distances r where R(r) is equal to 0, we have radial nodes, n-l- 1 in total The exponential decay of R(r) is slower for greater n At some distances r where R(r) is equal to 0, we have radial nodes, n-l- 1 in total 1s 2s 2p node

5) Radial probability function Radial probability function is defined as [r R(r)] 2 – probability of finding an electron in a spherical layer at a distance r from nucleus Radial probability plots for hydrogen atom: Radial probability function is defined as [r R(r)] 2 – probability of finding an electron in a spherical layer at a distance r from nucleus Radial probability plots for hydrogen atom: 1s 2s 2p

Summary Exact energy and spatial distribution of an electron in the hydrogen atom can be found by solving Shrödinger equation; Three quantum numbers n, l, m l appear as integer parameters while solving the equation; Each hydrogen orbital can be characterized by energy (n), shape (l), spatial orientation (m l ), number of nodal surfaces (n-1) Exact energy and spatial distribution of an electron in the hydrogen atom can be found by solving Shrödinger equation; Three quantum numbers n, l, m l appear as integer parameters while solving the equation; Each hydrogen orbital can be characterized by energy (n), shape (l), spatial orientation (m l ), number of nodal surfaces (n-1)

Electrons are part of what makes an atom an atom atom But where exactly are the electrons inside an atom?

Orbitals are areas within atoms where there is a high probablility of finding electrons.

Knowing how electrons are arranged in an atom is important because that governs how atoms interact with each other

Knowing how electrons are arranged in an atom is important because that governs how atoms interact with each other

Knowing how electrons are arranged in an atom is important because that governs how atoms interact with each other

Knowing how electrons are arranged in an atom is important because that governs how atoms interact with each other

Knowing how electrons are arranged in an atom is important because that governs how atoms interact with each other

Knowing how electrons are arranged in an atom is important because that governs how atoms interact with each other

Knowing how electrons are arranged in an atom is important because that governs how atoms interact with each other

Let’s say you have a room with marbles in it

The marbles are not just anywhere in the room. They are inside boxes in the room.

You know where the boxes are, and you know the marbles are inside the boxes, but…

you don’t know exactly where the marbles are inside the boxes

The room is an atom The marbles are electrons The boxes are orbitals

The room is an atom The marbles are electrons The boxes are orbitals

The room is an atom The marbles are electrons The boxes are orbitals Science has determined where the orbitals are inside an atom, but it is never known precisely where the electrons are inside the orbitals

So what are the sizes and shapes of orbitals?

The area where an electron can be found, the orbital, is defined mathematically, but we can see it as a specific shape in 3-dimensional space…

x y z

The 3 axes represent 3-dimensional space x y z

For this presentation, the nucleus of the atom is at the center of the three axes. x y z

The “1s” orbital is a sphere, centered around the nucleus

The 2s orbital is also a sphere.

The 2s electrons have a higher energy than the 1s electrons. Therefore, the 2s electrons are generally more distant from the nucleus, making the 2s orbital larger than the 1s orbital.

1s orbital

2s orbital

Don’t forget: an orbital is the shape of the space where there is a high probability of finding electrons

The s orbitals are spheres Don’t forget: an orbital is the shape of the space where there is a high probability of finding electrons

There are three 2p orbitals

The three 2p orbitals are oriented perpendicular to each other

This is one 2p orbital (2p y ) x y z

another 2p orbital (2p x ) x y z

the third 2p orbital (2p z ) x y z

Don’t forget: an orbital is the shape of the space where there is a high probability of finding electrons

This is the shape of p orbitals Don’t forget: an orbital is the shape of the space where there is a high probability of finding electrons

x y z

2p x x y z

2p x and 2p z x y z

The three 2p orbitals, 2p x, 2p y, 2p z x y z

once the 1s orbital is filled,

the 2s orbital begins to fill

once the 2s orbital is filled,

the 2p orbitals begin to fill

each 2p orbital intersects the 2s orbital and the 1s orbital

each 2p orbital gets one electron before pairing begins

once each 2p orbital is filled with a pair of electrons, then

the 3s orbital gets the next two electrons

the 3s electrons have a higher energy than 1s, 2s, or 2p electrons,

so 3s electrons are generally found further from the nucleus than 1s, 2s, or 2p electrons

The order of filling orbitals Electrons fill low energy orbitals (closer to the nucleus) before they fill higher energy ones. Where there is a choice between orbitals of equal energy, they fill the orbitals singly as far as possible. This filling of orbitals singly where possible is known as Hund's rule. It only applies where the orbitals have exactly the same energies (as with p orbitals, for example), and helps to minimise the repulsions between electrons and so makes the atom more stable.

Observing the Effect of Electron Spin

Table 8.2 Summary of Quantum Numbers of Electrons in Atoms NameSymbolPermitted ValuesProperty principalnpositive integers(1,2,3,…)orbital energy (size) angular momentum l integers from 0 to n-1orbital shape (The l values 0, 1, 2, and 3 correspond to s, p, d, and f orbitals, respectively.) magnetic mlml integers from - l to 0 to + l orbital orientation spin msms +1/2 or -1/2direction of e - spin

Pauli Exclusion Principle no two e - in an atom can have the same four quantum numbers. Result – an orbital can hold a maximum of 2 electrons no two e - in an atom can have the same four quantum numbers. Result – an orbital can hold a maximum of 2 electrons

Illustrating Orbital Occupancies The electron configuration n l # of electrons in the sublevel as s,p,d,f The orbital diagram (box or circle) Figure 8.7 Order for filling energy sublevels with electrons order of filling

Electron Configurations order of filling 7s 7p s 6p 6d 6f.... 5s 5p 5d 5f.. 4s 4p 4d 4f 3s 3p 3d 2s 2p 1s order of filling 7s 7p s 6p 6d 6f.... 5s 5p 5d 5f.. 4s 4p 4d 4f 3s 3p 3d 2s 2p 1s

dark - filled, spin-paired light - half-filled no color-empty A vertical orbital diagram for the Li ground state

Figure 8.9 Orbital occupancy for the first 10 elements, H through Ne.

Figure 8.13 The relation between orbital filling and the periodic table

Categories of electrons Core (inner) electrons – all those shared by the previous noble gas Outer electrons – those in the highest occupied energy level Similar chemical properties of elements in groups is a result of similar outer electron configurations In the main group, the group number equals the number of outer electrons Valence electrons – those involved in bonding In the main group, the outer electrons are valence In the transition metals can include some d electrons

Atomic radii of the main- group and transition elements.

Figure 8.16 Periodicity of atomic radius

Ionization energy energy required to remove an electron from 1 mole of the element (to make it positive or more positive) always positive value i.e. endothermic First ionization energy IE 1 – to remove 1st e- – Elements with small IE 1 tend to form cations, large IE 1 anions energy required to remove an electron from 1 mole of the element (to make it positive or more positive) always positive value i.e. endothermic First ionization energy IE 1 – to remove 1st e- – Elements with small IE 1 tend to form cations, large IE 1 anions

Ionization energy Trends: – Generally as size decreases, IE 1 increases. – 1. IE 1 generally increases from left to right, some exceptions – 2. IE 1 generally decreases going down in a group – 3. transition and f-block elements have much smaller variances in IE 1 But Why? – Across - increasing Z eff and smaller size(e- closer to nucleus) – exceptions - Be to B because s shields p and lowers Z eff – N to O because repulsions in first paired electron – Down - Z eff constant and larger so easier to ionize Trends: – Generally as size decreases, IE 1 increases. – 1. IE 1 generally increases from left to right, some exceptions – 2. IE 1 generally decreases going down in a group – 3. transition and f-block elements have much smaller variances in IE 1 But Why? – Across - increasing Z eff and smaller size(e- closer to nucleus) – exceptions - Be to B because s shields p and lowers Z eff – N to O because repulsions in first paired electron – Down - Z eff constant and larger so easier to ionize

Figure 8.17 Periodicity of first ionization energy (IE 1 )

Figure 8.18 First ionization energies of the main-group elements

Figure 8.21 Trends in three atomic properties

Trends in metallic behavior

Main-group ions and the noble gas configurations

Magnetic properties Spectral and magnetic properties of atoms can be used to confirm electron configurations. Paramagnetism – attraction by an external magnetic field – elements with unpaired electrons – The more unpaired electrons there are, the stronger the attraction Diamagnetism – Not attracted by an external magnetic field – elements with only paired electrons Spectral and magnetic properties of atoms can be used to confirm electron configurations. Paramagnetism – attraction by an external magnetic field – elements with unpaired electrons – The more unpaired electrons there are, the stronger the attraction Diamagnetism – Not attracted by an external magnetic field – elements with only paired electrons

Apparatus for measuring the magnetic behavior of a sample

SAMPLE PROBLEM 8.7Writing Electron Configurations and Predicting Magnetic Behavior of Transition Metal Ions PLAN: SOLUTION: PROBLEM:Use condensed electron configurations to write the reaction for the formation of each transition metal ion, and predict whether the ion is paramagnetic. (a) Mn 2+ (Z = 25)(b) Cr 3+ (Z = 24)(c) Hg 2+ (Z = 80) Write the electron configuration and remove electrons starting with ns to match the charge on the ion. If the remaining configuration has unpaired electrons, it is paramagnetic. paramagnetic(a) Mn 2+ (Z = 25) Mn([Ar]4s 2 3d 5 ) Mn 2+ ([Ar] 3d 5 ) + 2e - (b) Cr 3+ (Z = 24) Cr([Ar]) Cr 3+ ([Ar] ) + 3e - paramagnetic (c) Hg 2+ (Z = 80) Hg([Xe]6s 2 4f 14 5d 10 ) Hg 2+ ([Xe] 4f 14 5d 10 ) + 2e - not paramagnetic (is diamagnetic)

سه جزيره ايرانی ابوموسی، تنب بزرگ، و تنب کوچک جزء لاينفک سرزمين ايران است.