General chemistry 2

friction, drift and stokes law

particles in fluids lose energy due to friction, under a constant force, f a particle reaches a constant drift velocity Vd when friction balances the force

where e is the friction drag coefficient, constant for a given particle in a given medium and can be measured experimentally

for spherical particles stokes law applies

where n is the fluid visocity (10^-3 n/m²s for water) and r is the particle radius, larger particles have greater drag

flux and drift visocity

flux measures transported material per unit area per unit time, it is defined by counting particles crossing area A, in 1D flux J is above

sedimentation

particles heavier than the surrounding liquid settle under gravity

applications occur in geology, water treatment

the netforce on a particle is gravity minus the fluid’s bouyant force

m= mass particle

g= 9.81

pf=density surrounding fluid

v= volume particle

change in p= difference in density between particle and fluid

sedimentation velocity

for spheres V= 4/3 pi r³

heavy particles move down, lighter particles rise, small particles sediment slowly so ultracentrifuges use centrifugal acceleration up to 10^5

change in p >0 then it moves downwards

change in p < 0 then it moves upwards

spontaneous spreading

diffusion

even without external forces, random thermal motion causes particles to spread from high to low concentration until uniform

Ficks first law

negative sign ensures flux goes from high to low concentration (if concentration decreases in +x direction, gradient is negative giving positive flux)

D is diffusion coefficient around 10^-9 m²/s for small molecules in water, larger particles diffuse more slowly

J- diffusive flux (moles per unit area per unit time)

ac/ax- concentration gradient → conc. change/length

ion transport through a channel

membrane proteins forming narrow pores, a channel is approximated as a cyndrical tube

diameter d= 0.5nm

length L = 5nm (membrane thickness)

cells maintain concentration differences via active tranport

inside concentration 100mm greater than outside so potassium diffuses outwards

einstein relation

in most situations both directed motion (drift) and random motion (diffusion) occur simultaneously

drift arises from an external field, diffusion arises from conc gradients

Gas molecules in earth’s gravitational field

if molecules start uniformaly distributed, gravity pulls them downward (downward drift flux)

they accumulate near earth’s surface, creating a concentration gradient that produces an upward diffusive flux

total flux is sum of drift and diffusion

for gravity vd=mg/e

at equilibrium

nothing changes over time → Jtot=0

Einstein relation

stokes einstein relation

this gives us diffusion coefficients for spherical ions, proteins, dirt particles, bacteria

brownian motion

caused by collisions with thermally agitated water molecules

random molecular motion also causes diffusion

diffusion can be viewed as the longterm outcome of a random walk

has uncorrelated steps, taken in random directions, but the particle is unlikely to be found far from where it started due to frequent reversals

t=x/v

t is time taken, v is velocity and x is distance to travel

this connects average distance travelled during diffusion eith the diffusion coefficient

electric field setup

two parallel electrodes are placed distance l apart in a sample solution, applying a voltage Ecell creates an electric field

positive ions drift towards the negative electrode and negative towards the positive one

force on an ion

an ion with charge z e (valency z, elementary charge e=1.6 × 10^-19) experiences force-

drift velocity

positive ions move one way z>0 and negative ions move the other

mobility definition

ionic flux under an electric field

mobility in terms of stokes law

mobility depends on ionic charge, ionic radius and solvent visocity

why Li+ has lower mobility despite smaller radius

stokes radius is the hydrodynamic radius which includes hydration

small ions have strong electric fields → attracts more water → thicker hydration shell → effectively larger radius → lower mobility

why H+ and OH- have very high mobility

protons move via the Grotthuss mechanism; bond rearrangments transfer charge without moving mass

OH- moves similairly

diffusion coefficient relation

since mobility relates to friction and friction to diffusion diffusion is linked to mobility

thus one mobility is known, D is easily computed

electrophoresis

macro ions migrate in an electric field

electrophoresis uses this to seperate biomolecules

migration depends on charge and size, protein charge depends on pH (acidic, alkaline groups protonated or deprotonated)

each protein has an isoelectric point where net charge is zero

when crossing this pH drift slows → stops → reverses

this allows seperation of protein mixtures

the nernst equation

Nernst equation at chemical equilibrium

electrical conductance of a solution

When ions reach electrodes, they may not accumulate

often they accept or donate electrons at electrodes → continous current flows

migration rate depends on ionic mobility

current from ion flow, each ion type i contributes

Ji is flux mol/m²/s

Zi is charge per mole with F being faradys constant

conductance

c= 1/R

conductivity removes cell geometry dependence

k=c x l/A

electrochemical reactions always involve

the movement of electrons from one chemical species to another

the species that loses electrons is the reducing agent and is oxidized

the species that gains electrons is the oxidizing agent and is reduced

redox reaction is a sum of two half reactions as a singel half reaction does not occur by itself

oxidation state

the oxidation state of an atom is the charge that the atom would have if all shared electrons were assigned to the atom which attracts them most strongly

rules to assign oxidation states

the oxidation state of an atom in an element is zero

the oxidation state of a monoatomic ion is equal to its charge

the sum of all the oxidation states of all atoms in a nuetral molecule is equal to zero and in an ion it is equal to the charge of the ion

the following atoms in compounds have the following oxidation states

Balancing redox reactions

write the two unbalanced half reactions

balance the elements other than hydrogen and oxygen for each half reaction

balance O by adding H2O or OH-

Balance h by adding H+

if the solution is acidic get rid of any OH- by adding H+ to both sides

if the solution is alkaline get rid of H+ by adding OH- to both sides, reduce any redundant water molecules

balance the charge by adding electrons

make the number of electron in each half reaction equal by multiplying one or both half reaction by a small integer number

add or subtract the two half reactions in such a way that the electrons cancel

get rid of any redundant terms

voltaic cell

a cell that generates electricity through a spontaneous redox reaction

electrolytic cell

consumes electrical current generated by an external source to drive a non spontaneous chemical reaction

each of the two half cells in an electrochemical cell consists of an electrode that is immersed in an electrolyte solution

the electrode where reduction takes place

is called the cathode, electrons enter the cell at the cathode and are taken up by the substance being reduced

the electrode where oxidation takes place

called the anode, electrons are given up by the substance being oxidized and leave the cell at the anode

to enable the cell to operate

the accumulation of charge in the solution must be nuetralised

this is achieved by joining the two half cells by a salt bridge which acts as a pathway that allows ions in the solution to flow from one half cell to the other

negative ions move from the cathodic to the anodic half cell nuetralising the accumulation of positive charge at the minus pole while positive ions in the salt bridge move in the opposite direction (nuetralising the accumulation of negative charge at the plus pole)

for some redox reactions

reactants and products of half reaction are present in the same phase, in that case a conductive surface is needed for electron transfer to take place

usually an invert electrode is used as the anode or cathode, this inert electrode is not oxidised or reduced, it merely accepts or donates electrons

shorthand notation for electrochemical cell

the vertical lines in this diagram represent the boundaries between the solid electrodes and the solution; the double vertical line that seperates the two half reactions denotes the salt bridge

anode on the left, cathode on the right with the charge change shown

electrical potential difference

between the two electrodes of the cell E cell= Er - El

the half cell that donates the electrons where oxidation takes place is drawn as the left cell

the way an overall reaction is written from the cell diagram is done by writing the two half reactions as reduction reactions and then doing right minus left equation

the drive to advance the reaction is

this reaction thus generates an electrical potential difference that pushes the electrons through the external electrical circuit

if we supply and Ecell of exactly the equation, the electron flow stops and in this situation the cell is in equilibrium

the reaction is not but the cell as a whole is

for this reason this balancing E cell is called the equilibrium cell potential

E cell eq is just another way of expressing how desperate the reaction is to proceed

when Ecell is zero the reaction can go without any pushing back from the external circuit, the entropy production is then

to correct for non short circuited cells we just need to account for this pushing back effect

Ecell is the external potential difference, the voltage over the electrodes

Ecell eq is the potential that connects to the chemistry of the cell

the cell is used as a battery when E cell < E cell eq

when E cell > E cell eq the cell is charged

at equilbrium there is no entropy production so E cell = E cell eq

This equilibrium cell potential is also known as nernst potential at equilibrium there is no difference between Ecell and Ecell eq and therefore when it is indeed clear we are dealting with equilibrium we use Ecell = -gibbs energy/nF

the equilibrium constant K relates to the concentration in the same way as does the reaction quotient with the only exception that all values pertain to chemical equilbrium

the same line of reasoning can be done for the nernst equation

SHE

standard hydrogen electrode

H+/H2 half cell

the potential of any other half cell relative to the standard hydrogen electrode can then be obtained by constructing a voltaic cell in which the half cell of interest is measured against SHE

SHE is always the left cell in the cell diagram

since we are dealing with the standard hydrogen electrode you can eliminate PH2 and {H+} from the equation because the value is 1, the corresponding nernst equation for the equilibrium electrode potential is then given by

a species with a highly positive E standard eq has a high tendancy to attract electrons and is therefore a strong oxidising agent

a species with a highly negative E standard has a strong tedancy to repel electrons and is a strong reducing agent

the SHE serves as a common level to which all, what you can do to calculate equilbrium cell potentials from them

equilibrium electrode potentials are relative

start at the left cell decent to SHE level and climb to right cell level or do right minus left, standard cell potentials are just equilibrium cell potentials under standard conditions

therefore the standard cell potential of an electrochemical cell is found by subtracting the two standard electrode potentials

calculate equilibrium cell potential using the overall reaction equation or based on the equilibrium electrode potentials

To compute equilibrium cell potentials at arbitary concentrations use nernst equation ex

positive E cell eq indicates

The reaction is spontaneous to the right, the cell will be discharged

if set to a higher voltage (charging)

cell will be charged and electrons are forced to go from the RHS to LHS electrodes, reaction goes to the left

non spontaneous reaction

If Stot is negative

the reaction goes to the left and is not spontaneous, but if process naturally goes to the left like charging then Stot is actually positive and reaction will be spontaneous

If a reaction has a very large K

We can assume that the reaction goes to an end and stops working when the concentration of the reduced component Cu2+ equals zero

electrochemical cells can be used to measure concentrations

the potential of the indicator electrode is measured against a referance electrode of known composition

no electrical charge is allowed to flow so the referance electrode has a fixed equilibrium electrode potential

measure the potential of the indicator against the referance electrode so the indicator electrode is the half cell on the right in the cell diagram and the equilibrium electrode potential of the indicator electrode potential is found by adding the equilibrium electrode potential to the measured equilibrium cell potential

most used electrode

glass electrode used for measuring H+ concentrations

this electrode relies on membrane potential

when the glass electrode is placed in a solution of unknown pH, the measured electrical potential difference can be used to calculate the pH using the Nernst eqiation

In a potentiometric titration a redox reaction occurs between

the analyte and titrant

the equilbrium electrode potentials of the two half reactions that make up the redox reaction are mutually equal

the equilibrium electrode potential of the analyte solution is measured against a reference electrode

to measure the electrode potential we need a full electrochemical cell with two half cells

the second cell is some reference cell of constant composition

the two cells are connected by the salt bridge, since there flows no electrical current just to measure Ecell eq it is enough to know the potential of the reference electrode relative to SHE

as all components are in equilibrium throughout the titration both electrode potentials are equal to each other, they are the potential of the indicator electrode

in an electrolytic cell an external power source is used to drive

a non spontaneous redox reaction

by applying an external potential greater than the equilibrium potential, the reaction is forced to go in the other direction

in biological cells

glucose is oxidised in a number of steps, oxygen is the final electron acceptor

the energy that is released in this oxidation is used to produce ATP

If Ecell < Ecell eq

the term is positive → reaction proceeds forward (to the right)

If Ecell > E cell eq

reaction proceeds backwards (to the left) this is how recharging a battery forces reactions to reverse

atoms are made up of

a nucleus, consisting of positively charged protons and nuetral nuetrons, surrounded by a cloud of electrons

the nucleus occupies only a tiny part of the volume of the atom

atoms emission spectrum

discrete- only specific wavelengths are emitted

in bohrs model electrons can only occupy certain discrete orbits at specific distances from the nucleus

electrons can only gain and lose energy by jumping from one allowed orbit to another

while doing so they absorb or emit a quantum of electromagnetic radiation (a photon) with a wavelength that corresponds exactly to the energy difference between two orbits

regular pattern in the series of lines in the hydrogen spectrum

RH is a constant

n is which orbital it falls to and starts from

n=2 is visible light

light is a type of

electromagnetic radiation, form of energy associated with oscillating electric and magnetic fields

travels through space at a speed c= 3 × 10^8 m/s

electromagnetic radiation has both wave and particle nature; the wave particle duality

wave nature is characterised by a

wavelength and frequency which are related troughly in above equation

wave nature manifests itself in diffraction and interference

the particle nature of light is characterised by a specific amount of energy carried in small light particles called photons

Energy of a photon

h is planck’s constant

small particles also have wave length behaviour

their wavelength is given by de Broglies formula

h= plancks constant

m=mass

v=velocity

explains why electrons only have certain possible frequencies and energies

the wavelength of an object is inversely proportional to its mass, so heavy objects have wavelengths that are many magnitudes smaller than the object itself

A model for the structure of atoms in which the electrons were described as waves

made by combining Broglie’s wavelength with a standard wave equation

H stands for the Hamiltinoian operator

E stands for the actual energy level of the electron

Ψ is the wave function that represents the standing wave form of the electron

when the equation is analysed, a whole set of different solutions is found each solution consisting of a certain wave function Ψ that is called an orbital

Each orbital represents a

possible state of the electron and to each orbital corresponds a particular value of E

the wave function Ψ corresponding to a certain orbital is a function of the position in space, as specified by three spatial coordinates x,y,z

we can only know the probability of an electron to be somewhere

the square of the wave function Ψ² expresses the probabilty density, the probability per unit volume for an electron to be at a certain location

solutions of the schrodiner equation for the hydrogen atom

characterised as three quantum numbers n,l, and m

n is an integer with values 1,2…. which determines the size of the orbitals

l runs from 0 to n-1 and determines the shape of the orbitals

m runs from l to -l and determines its orientation (for a given l value there are 2l+1 given m values)

letter notations are used to refer to orbitals with certain l numbers;

Energy values for hydrogen that come out of Schrodingers equation given by

RH is the rydberg constant

h is plancks constant

At infinite distance the energy is zero, this is the ionized state (the electron is removed)

the level with n=1 has the lowest energy, ground state

this is the most favourable state for the electron so the electron is most likely to be found here

when an atom absorbs energy

an electron in a lower energy orbital can be excited or promoted to an orbital with a higher energy

in this excited state the atom is unstable, and the electron quickly falls back to a lower energy orbital

as it does so it releases energy in the form of electromagnetic radiation

the amount of energy that is released is equal to the energy difference between the two orbitals

periodic table

elements exhibit a periodic recurrence of similair properties

the quantum mechanical model gives a natural explanation for this ordering scheme

orbitals for many electron atoms are similair to those of hydrogen atom

therefore we can use the s,p,d, and f orbitals also for multi electron atoms

however in order to see how multiple electrons occupy these orbitals we need two additional features 1) electron spin , a fundamental property of electrons that determines the number of electrons that can occupy an orbital

and 2) a more complex set of orbital energy levels

The three quantum number n, l, and mi describe the size, shape, and orientation of an atomic orbital

an electron can have two different spin states producing two oppositely directed magnetic moments

the new quantum number used to describe this is called the spin quantum number ms, can only have values +1/2 or -1/2 as a result each electron in an atom is described completely by four quantum numbers

No two electrons can have the same

four quantum numbers; the pauli exclusion principle

since electrons in the same orbital have the same values of n, l , and mi, this postulate says that they must have different values ms, since there are only two allowed values of ms each orbital can only hold two electrons and they must have opposite spins

For multi electron atoms the pattern of energy levels

is much more complex due to additional repulsion between electrons, to account for this reduced binding of the electrons we imagine that the electrons are held by an effective nuclear charge Zeff which is smaller than the real charge of the nucleus Z

we can write the energy of an orbital as

the effective charge experienced by an electron depends on

the effect of nuclear charge- for one electron atoms the effective charge equals the real nuclear charfe (Zeff=z)

so when Zeff is twice as big, the orbital energy is two times lower

shielding

the effect of electron repulsion, for multielectron atoms, each electron not only feels the attractive positive charge of the nucleus but it also feels the repulsive negative charges of all the other electrons, these repulsive interactions between electrons increase the energy of the orbital

shielding of the nuclear charge by the inner electrons reduces the effective charge experienced by an electron in the outer shell

penetration

the effect of orbital shape; as the outer electron undergoes penetration into the region occupied by the inner electrons, it experuences a stronger attraction and therefore a lower energy

because electrons in s orbitals have a higher probability to be closer to the nucleus than p,d, or f orbitals (they penetrate more → higher Zeff) they have a lower energy

the aufbau principle can be built up as

we know that in the ground state electrons occupy the lowest energy orbitals available and that only two electrons with opposing spin are allowed per orbital, we can systematically build up the electron configuration for the elements

when two electrons occupy the same orbital

their spins must be opposite; they are paired

hunds rule

when orbitals of equal energy are available the electron configuration of lowest energy is the one with the maximum number of unpaired electrons with parallel spins

Electrons in different p orbitals are farther away from each other and therefore experience less repulsion → higher Zeff

Elements in the same group and same period

have the same electron configuration in their outer shell

valence electrons

have the same amount of shells (inner shielding)

ionization energy trend (amount of energy to remove one mole of electrons from one mole of gaseous atoms )

increases when going left to right across a period due to same shielding and increasing nuclear charge by the added protons so Zeff will increase so the orbital energy becomes lower

going down a group the ionisation decreases as the outer electrons are farther away from the nucleus as n increases

electron affinity trends

the energy released when an additional electron is added to a gaseous atom in its ground state

for most elements the electron affinity is positive, because the extra electron is attracted by positive nuclear charge

but if you have a full shell it wont be positive and will actually require energy to occur

This increases across a period as same inner shielding and more willingness to gain electrons and decreases down a group as the electrons are farther away from the nucleus

The octet rule

Atoms will tend to share, gain or lose electrons to acquire a noble gas configuration

three types of bonds

1- ionic bonds; these are formed between atoms with large differences in their tendancies to lose or gain electrons, typically a metal (low ionisation energy) and a non metal (high electron affinity) electrons trasnfer from metal to non metal giving both a noble gas configuration

2- covalent bonds; electrons are shared by the two atoms, they have the same high ionization energy, the shared electrons interact with the nuclei of both atoms, and thereby lower the potential energy

3- metallic bonds; metals have a low ionisation energy, so they lose electrons easily, the outer electrons are bound so weakly that they do not stay with a single metal atom but move freely throughout the entire metal

bond polarity

when a covalent bond is formed between two different kinds of atoms, the electrons are usually not shared equally

one atom attracts them more strongly, the electron is partially transferred, such a bond is said to be polar (having a positive and negative pole)

by contrast bonds in which the electrons are equally shared are non polar

electronegativity

the ability of an atom to attract electrons in chemical bonds

the bond polarity depends on the difference in electronegativity between the atoms involved in the bond

dipole moment occur when

there is a seperation of positive and negative charge

with what difference in electronegativity are certain bonds formed

Percent ionic character

u actual is the actual dipole moment of a bond

u complete is if the electron transfers completely from one atom to the other (q=e)

for molecules with multiple bonds

the net dipole moment follows from adding the dipole vectors of each of the bonds, can cancel out if all moving in opposite directions and then the molecule is non polar with polar bonds

Lewis structures

write down the atom symbols according to presumed molecular layout, put the atom with the lowest ionisation energy in central position

determine the total number of valence electrons in the molecule, add or subtract extra electrons according to the charge of the molecule, this number is N

Determine the total number of valence electrons in the molecule if each atom would have a noble gas configuration, this number is M

A good guess for the number of bonding molecules in M-N, use these electrons to draw bonds between the atoms, each line represents two electrons

distribute the remaining electrons over the molecule as to satisfy the octet rule