Science fundamentals with point by point knowledge record. Exhaustive list of learnt points. Condensed high school science; chemistry, physics and mathematics.

## Mathematics

21 Chapters

### Algebra 1

• For direct proof: assume P is true and then use P to show that Q must be true.
• Proof by exhaustion. Show cases are exhaustive and prove each cases.
• Proof by counter example. Give an example that disproves the statement.
• $x^{a} \times x^{b} = x^{a+b}$, $x^{a} \div x^{b} = x^{a-b}$, $(x^{a})^{b} = x^{ab}$
• $x^{0} = 1, x^{-n} = \frac{1}{x^{n}}, x^{\frac{1}{n}} = \sqrt[n]{x}, x^{\frac{p}{r}} = \sqrt[n]{x^{p}} = (\sqrt[n]{x})^{p}$
• A Rational Number can be made by dividing an integer by an integer.
• $\sqrt[]{A} \times \sqrt[]{B} = \sqrt[]{AB}$
• Rationalise $\frac{k}{\sqrt{a}}$ means multiply the top and bottom by $\sqrt{a}$.
• Rationalise $\frac{k}{a \pm \sqrt{b}}$ means multiply the top and bottom by $a \mp \sqrt{b}$.
• Rationalise $\frac{k}{\sqrt{a} \pm \sqrt{b}}$ means multiply the top and bottom by $\sqrt{a} \mp \sqrt{b}$.
• A quadratic function of the form: $ax^{2} + bx + c = 0$, where $a\neq 0$
• You can solve a quadratic equation $f(x) = 0$ using the following formula: $x = \frac{-b \pm \sqrt{b^{2} - 4ac} }{2a}$, by completing the square or graphically.
• If the discriminant $\Delta = b^{2} - 4ac > 0$, the quadratic has two different roots. If $\Delta = 0$ the quadratic has one repeated root and if $\Delta < 0$ the quadratic has not real roots.
• You can use gradients of two straight lines to decide if they are parallel, perpendicular, or neither.
• The equation of a circle, centre $(a,b)$ and radius r, is $(x-a)^2 + (y-b)^2 = r^2$
• If multiplying or dividing an inequality by a negative number you reverse the inequality sign.

### Polynomials the binomial theorem

• The highest power of a polynomial is called it’s degree.
• When adding or subtracting polynomials, expand brackets before collecting like terms.
• Identities use the $\equiv$ sign. Identities are true for all values of the variable(s).
• For $n = 0,1,2,3,...$, the binomial expansions are $(1+x)^n \equiv 1 + nx + \frac{n(n-1)}{2!}x^2+ \frac{n(n-1)(n-2)}{3!}x^3 +...+x^n$
• … and $(a+b)^n \equiv a^n + ^{n}C_{1}a^{(n-1)}b+$ $^{n}C_{2}a^{(n-2)}b^2$ $+...+ ^{n}C_{r}a^{(n-r)}b^r + ...+b^{n}$
• The coefficients of these expansions can be found from Pascal’s triangle or from $^{n}C_{r} \equiv \frac{n!}{(n-r)!r!}$
• One can divide algebraically using the same technique as for long division in arithmetic.
• The factor theorem: If $f(a)=0$, then $(x-a)$ is a factor of $f(x)$.
• To sketch a graph you need to consider the symmetry, $x-$ and $y-$ intercepts, asymptote, behaviour as x and/or y approach $\pm\infty$, and any other obvious critical points. It can also be useful to think in terms of transformations.

### Trigonometry

• Sine, cosine and tangent are periodic functions. Their graphs have line and rotational symmetry.
• The sine, cosine and tangent functions can be expressed in terms of an acute angle.
• The sign and size of the sine, cosine and tangent functions of any angle can also be found using a sketch of the graph of the function.
• $\sin^{2}\theta + \cos^{2}\theta \equiv 1$ and $\tan\theta \equiv \frac{\sin\theta}{\cos\theta}$
• To solve a trigonometric equation, use identities to simplify it then use a quadrant diagram or graph to find all possible angles.
• The sine rule: $\frac{a}{\sin{A}} =\frac{b}{\sin{B}}=\frac{c}{\sin{C}}$, used when two opposite side angle pairs are known.
• The cosine rule: $a^2 = b^2 +c^2 - 2bc\cos{A}$, used when 3 sides are known or two sides and the angle between them.
• 🛠️ Formula for the area of a triangle using two sides and the included angle: $A = \frac{1}{2}ab\sin{C}$.

### Differentiation and integration (Calculus)

• The gradient of the tangent of a curve known as the derivative, $f(x)$, can be calculated form first principles using the formula: $f^'(x)=\lim{h \to 0} \frac{f(x+h) - f(x)}{h}$
• If $y=f(x)$ then $\frac{dy}{dx}=f^'(x)$
• The derivative gives the instantaneous rate of change of $y$ with respect to $x$.
• The derivative of y = ax^{n}, where a is a constant, is $\frac{dy}{dx}=nax^{n-1}$
• $f(x)=ag(x)+bh(x) \Rightarrow f^'(x)=ag^'(x)+bh^'(x)$, where $a$ and $b$ are constants.
• $f^'(x)$ gives the rate of change of function $f$ with respect to $x$. If $f^'(x)$ is positive if the function is increasing, negative if the function is descreasing and 0 if the function is stationary. This can also be observed graphically.
• At a stationary or turning point $f^'(x)=0$.
• Maxima are peaks and minima are troughs.
• The gradient of the normal to the curve, $m_{normal} =-\frac{1}{f^'(a)}=-\frac{1}{m_{tangent}}$
• The second derivative is denoted by $f^{''}(x) = \frac{d^{2}y}{dx^{2}}$
• $\frac{d^{2}y}{dx^{2}}$ gives the rate of change of the gradient with respect to x. Assuming $\frac{dy}{dx} = 0$, if $\frac{d^{2}y}{dx^{2}} > 0$, the turning point is a maximum, if $\frac{d^{2}y}{dx^{2}} < 0$, then the turning point is a minimum.
• Integration is the reverse of differentiation. $F^'(x) = f(x) \Rightarrow F(x) = \int f(x)dx +c$
• $\int ax^n dx = \frac{ax^{n+1}}{n+1}$
• $\int(af(x) + bg(x))dx = a\int f(x) dx + b \int g(x) dx$
• $\int_a^b f(x) dx$ is called the definite integral with lower bound $a$ and upper bound $b$.
• If $F(x) +c = \int f(x)dx \Rightarrow \int_a^b f(x) dx = F(b) - F(a)$. AKA the first fundamental theorem of calculus.
• $A(x) = \int_a^b f(x) dx$, where the positive value of $A(x)$ is the area bounded by the curve in the interval $a\leq x\leq b$ assuming all points on the curve in this interval are on the same side of the $x$-axis.

### Exponentials and Logarithms

• $x=a^n$ and $n = \log_{a}x$ are equivalent statements.
• $\log_{a}x$ in English is equivalent to saying ‘$a$ to the power of what number equals $x$
• The inverse of $y = a^x$ is $y=\log_{a}x$.
• The Log laws:
• Law 1: $\log_{a}(xy) = \log_{a}x + \log_{a}y$ -substances Law 2: $\log_{a}(\frac{x}{y}) = \log_ax - \log_ay$
• Law 3: $\log_{a}(x^k) = k\log_ax$
• Logs can have different bases. 10 and $e=2.71828...$ are common.
• The general exponential function is $a^x$. The exponential function is $e^x$.
• The inverse of $y=e^x$ is $y=\ln x$.
• The gradient of $y=e^x$ is $e^x$ and the gradient of $y=e^{kx}$ is $ke^x$.
• Mathmematical models using exponential functions can be used to describe real-world event. One common model is $y=Rae^{kt}$.
• 🛠️ Two common non-linear relationships can be transformed into straight lines using logs.
• $y=ax^n$ becomes $Y=nX+c$, where $Y=\log y,X=\log x$ and $c=\log a$.
• $y=kb^x$ becomes $Y=mx+c$, where $Y=\log y,m=\log b$ and $c=\log k$.
• These straight lines are lines of best fit through the data points.

### Vectors 🛠️

• A vector quantity has both magnitude and direction.
• A scalar quantity has magnitude only.
• Equal vectors have the same magnitude and direction.
• $k$a is parallel to a and has magnitude $k$|a|.
• Two or more points are collinear if a single vector, or multiple parallel vectors, pass through the those points.
• The unit vector has a magnitude of 1 in the direction of a is $\hat{a}=\frac{a}{|a|}$.
• $AC = AB + BC$, $AC$ is the resultant of $AB$ and $BC$
• Making a vector negative reverses its direction: e.g. $BA = -AB$
• A vector is written in component form $xi + yj$ or $\begin{pmatrix}x\\y\end{pmatrix}$.
• For magnitude $r$ and direction $\theta$ (the positive rotation from the x-direction):
• $x=r\cos{\theta}$ and $y=r\sin{\theta}$
• For components $x$ and $y$, $r=\sqrt[]{x^2 + y^2}$ and $\tan{\theta}=\frac{y}{x}$.
• Treat components separately in calculations and equations.
• The position vector of a point $A$ relative to an origin $O$ is $OA$, often labelled a or $\bm{r}_a$
• If points $A$ and $B$ have position vectors a and b then vector $AB = \bm{b} - \bm{a}$ and distance $AB = |\bm{b} - \bm{a}|$

### Units and kinematics

• Initial SI base units: kilograms (mass), meters (length) and seconds (time).
• Kinematics quantities:
Vector Quantity Scalar Quantity SI Units
Displacement Distance $m$
Velocity Speed $ms^{-1}$
Acceleration $ms^{-2}$

• Formulae should be dimensionally consistent and you must use the same units throughout.
• Position is a vector - the distance and direction from the origin.
• Displacement is a vector - the change of position.
• Velocity is a vector - the rate of change of displacement.
• Speed is a scalar - the magnitude of velocity.
• Acceleration is a vector - the rate of change of velocity.
• $\text{Average velocity} = \frac{\text{resultant displacement}}{\text{total time}}$ and $\text{Average speed} = \frac{\text{total distance}}{\text{total time}}$.
• The gradient of a displacement-time ($s-t$) graph is velocity.
• The gradient of a velocity-time ($v-t$) graph is acceleration.
• The area between a $v-t$ graph and the t-axis is the displacement.
• In straight-line graphs, changes in motion are assume to be instantaneous. In reality, this is usually not possible.
• $s$ = displacement, $u$ = initial velocity, $v$ = final velocity, $a$ = acceleration, $t$ = time.
• For constant acceleration:
• $v=u+at$, $v^2=u^2 + 2as$, $s=ut+\frac{1}{2}at^2$, $s=vt-\frac{1}{2}at^2$, $s =\frac{1}{2}(u+v)t$.
• The equations of motion for constant acceleration assume objects to be particles with tiny or irrelevant size. In reality, the mass of an object may affect calculations.
• For variable acceleration, use calculus: $v=\frac{ds}{dt}$, $a=\frac{dv}{dt}=\frac{d^2s}{dt^2}$, $s=\int{v} \,dt$ and $v=\int{a} \,dt$.

### Forces and Newton’s Laws

• Newton’s first law of motion: An object will remain at rest or continue to move with constant velocity unless an external force is applied to it.
• Force is a vector; it has both magnitude and direction.
• An object is in equilibrium if the resultant force acting on the object is 0 and the object will be at rest or moving at a constant velocity.
• The resultant force is the single force equivalent to all the forces acting on the object.
• Newton’s second law of motion: $F=ma$.
• Friction always opposes motion. If a particle is moving to the right, friction will act on the left.
• Deceleration of $a$ in one direction is acceleration of $-a$ in the opposite direction.
• An object of mass $m$ $kg$ has weight $mg$, where $g=9.81ms^{-2}$ on the earth’s surface. $g$ decreases as it moves away from the surface.
• Newton’s third law of motion: For every action, there is an equal and opposite reaction. If A exerts a force on B, then B exerts an equal and opposite force on A.
• If two objects are connected, internal forces can be ignored when the two objects are considered as a whole.
• Assumptions often made in problems using Newton’s Laws:
• Objects are particles; there is no turning effect and mass acts at one point.
• Strings are light and inextensible.
• The acceleration is constant through the string.
• The tension is constant through a string.
• The tension in the string is the same on both sides of a pulley.
• Pulleys and smooth surfaces are perfectly smooth. There is no resistance force acting.

### Collecting, representing and interpreting data

• A population is the set of things you are interested in. A sample is a subset of the population. Different sampling methods include:
• If every member can be listed:
• ‘Simple random sampling’: Every member is equally likely to be chosen. Random numbers are used to choose a sample of a desired size.
• ‘Systematic sampling’: Find a sample of size $n$ from population of size $N$ by taking every $k^{th}$ member of the population where $k=\frac{N}{n}$.
• ‘Stratified sampling’: Sampling within distinct groups producing sub-samples proportional to their relative group size in the population.
• If listing every member is not possible:
• ‘Opportunity sampling’, taking samples from accessible population members.
• ‘Quota sampling’: Taking certain numbers of distinct groups from a population.
• ‘Cluster sampling’: Split the population into clusters that you expect to be similar to each other then take a sample from each of these clusters.
• A parameter is a number that describes the entire population. These can often be very difficult to find. Instead statistic’s are used to estimate parameters. For example, a sample mean may be used to estimate the population mean.
• Sampling methods can be biased in some situations.
• The mode of a set of observations it the most frequently occurring value.
• The mean is calculated by: $\mu=\frac{\sum_{k=1}^n x_i}{n}$
• The median of a set is the middle value of the ordered set.
• The range of a set is the largest observation minus the smallest.
• The interquartile range is the difference between the first and third quartiles.
• The variance of a set of data measures the degree of spread.
• Outliers are points that diverge from a pattern.
• Data is discrete or continuous.
• Continuous data should be summarised for representation:
• Using a five-number summary to form a box-and-whisker plot.
• Group data to draw a histogram or a cumulative frequency diagram.
• Variables whose values show a linear relationships are correlated.
• Correlation can be positive, negative or zero and can be characterised as weak, moderate or strong.
• Scatter diagrams can show strength and type of relationship between two variables.
• Correlation coefficient, $r$, takes values from -1 to +1 inclusive.
• Correlation does not mean causation.

### Probability

• $P(A\cup B) = P(A) + P(B)$ if $A$ and $B$ are mutually exclusive.
• The total probability of all mutually exclusive events is 1
• If A is an event associated with a random experiment, then $P(\neg A)=1-P(A)$
• For any sample space $N$ equally probable outcomes, if an event $A$ can occur in $n(A)$ of these outcomes, the probability of $A$ is given by $P(A)=\frac{n(A)}{N}$.
• If $A$ and $B$ are independent, $P(A\cap B)=P(A)\times P(B)$
• if $A$ and $B$ are mutually exclusive, $P(A\cap B)=0$
• A probability distribution for a random experiment shows how the total probability of 1 is distributed between all the possible outcomes.
• The conditions for a binomial probability distribution are:
• Two possible outcomes of each trial
• Fixed number of trials
• Independent trials
• Identical trials ($p$ is the same for each trial)
• If the above is true; the binomial probability distribution can be used to calculate the probabilities of events expressed in terms of the number of ‘successes’ in a set of trials.
• If $X~B(n,p)$ then $P(X=x)=^nC_rp^x(1-p)^{n-x}$
• $P(X\leq x-1)=P(X and $P(X>x)=1-P(X \leq x)$

## Physics

29 Chapters

### Particles and nuclides

• The Rutherford-Bohr atom: a small dense positively charged nucleus, comprised of neutrons and protons, surrounded by negatively charged electrons.
• The specific charge of a particle is the charge per unit mass: $= \frac{Q}{m}$.
• $^A_ZX$ notation for a nuclei or isotope. $A$ is the nucleon number, $Z$ is the proton number, $X$ is the chemical symbol.
• The strong nuclear force is one of the four fundamental forces. It acts between nucleons.
• It is attractive up to distances of 3fm (3E-15 m), and repulsive below 0.4fm.
• At 0.4fm the resultant force is 0. This is the distance of separation of atoms in a stable nucleus.
• Alpha decay: Involves the emission of an alpha particle ($\equiv$ to a helium nucleus).
• Beta decay: Involves the decay of a neutron into a proton, an electron and antineutrino which are all emitted from the nucleus. ($\beta^-$ decay). There is also ($\beta^+$ decay), where a positron is emitted.
• Gamma emission involves nucleons losing energy and the emission of a gamma ray photon.
• The total emission energy from a beta particle emission, $E_{tot}=E_{\beta}+E_{\bar{v}}$. $E_{tot}$ is constant for any one radioactive beta-emitting nuclide.
• Spark counters detect highly ionising alpha particles. Beta and gamma do not ionise enough of the air between the metal gauze and the thin wire underneath. When air particles are ionised by the alpha particles, charged particles cause a spark. The spark jumps the 5000V gap between the gauze and the wire.
• Geiger-Muller Counters: The low-pressure inert gas inside the detector. Ionising radiation passes through a mica window and ionises the gas producing a cascade of particles that are attracted to oppositely charged electrodes. The small pulse of current is detected by an electronic counter. Geiger counters can detect Gamma, Beta and Alpha radiation.
• (Wilson) Cloud chamber: Alpha, produces straight broad definite length tracks; Beta, produces thin straight or curved tracks (depending on energy). Gamma does not produce any effect.
• Photon model of em radiation: $E=hf$, where $f$ is the frequency of radiation absorbed or emitted and $h=6.63\times10^{-34} Js$, is the Planck constant.
• Mega electron-volts, the usual units of energy of nuclear particles: One electron is 1.6E-19 J, and defined as the amount of energy needed to accelerate an electron of charge ‘e’ through a potential difference of 1 volt. One mega electron volt is 1.6E-13.
• This unit is derived from the definition of the volt: A volt is the amount of energy per unit charge or $volts=\frac{energy}{charge}$ so $energy=volts\times charge$.
• Particle-antiparticle annihilation. When a particle meets its antiparticle; they will annihilate. The mass of the two particles is converted into energy in the form of two gamma ray photons.The production of the photons conserves momentum. The total energy of the photons is equal to the total rest energies of the particle-antiparticle pair.
• Pair production: A photon with enough energy can interact with a large nucleus and be converted directly into a particle-antiparticle pair.
• The wavelength of a photon can be related to energy with: $E=\frac{hc}{\lambda}$.

### Fundamental particles

• For every particle there is a corresponding antiparticle.
• Particles and antiparticles have: rest mass (in $MeV/c^2$); charge (in $C$) and rest-energy (in $MeV$).
• The positron, the antiproton, the the antineutron and the electron antineutrino are the antiparticles of the electron, proton, neutron and electron neutrino respectively.
• The four fundamental interactions are: gravity, electromagnetic, weak and strong.
• Exchange particles are used to explain the forces between elementary particles on the quantum scale.
• The virtual photon is the exchange particles of the electromagnetic force.
• Examples of the weak interaction are $\beta^{-}$ and $\beta^{+}$ decay, electron capture and electron-proton collisions.
• $W^+$ and $W^-$ are the exchange particles of the weak interaction.
• Feynman diagrams are used to represent reactions or interactions in terms of particles going in and out, and exchange particles.
• Hadrons are particles that are subject to the strong interaction.
• There are two classes of hadrons:
• baryons (proton, neutron) and their corresponding antibaryons.
• mesons (pion, kaon).
• Baryon number, $B$, is a quantum number that describes baryons. Baryons have $B=+1$; antibaryons have $B=-1$; and non-baryons have $B=0$.
• The baryon number is always conserved during particle interactions.
• The proton is the only stable baryon and all other baryons will eventually decay into protons.
• Free neutrons are unstable and decay via the weak interaction forming a proton, $\beta^-$ particle and an electron antineutrino.
• Pions and kaons are examples of mesons. The pion is the exchange particle of the strong nuclear force between baryons. The kaon is a particle that can decay into pions.
• Leptons are particles that are subject to the weak interaction.
• They include: the electron, muon, tau and neutrinos (their respective types) and their antiparticles.
• Lepton number $L$ is a quantum number that describes leptons.
• For lepton’s $L=+1$; for antileptons $L=-1$; for non-leptons $L=0$.
• $L$ is always conserved in particle interactions.
• Muons are particles that decay into electrons.
• Strange particles are particles that are produced through the strong interaction and decay through the weak interaction (e.g kaons).
• Strangeness (symbol $S$) is a quantum number to describe strange particles. Strange particles are always created in pairs by the strong interaction.
• Quarks have charge: ($+\frac{1}{3}, -\frac{1}{3}, +\frac{2}{3}, -\frac{2}{3}$), baryon number ($+\frac{1}{3}, -\frac{1}{3}$), and strangeness (+1, 0, or -1).
• Hadrons have the following quark structures:
• baryons: proton, $uud$; neutron, $udd$; antiproton $\bar{u}\bar{u}\bar{d}$; antineutron, $\bar{u}\bar{d}\bar{d}$.
• mesons:
• pions: Pion$^+(\pi^+)$, $u\bar{d}$; Pion$^-(\pi^-)$, $d\bar{u}$; Pion$^0(\pi^0)$, $d\bar{d}$ or $u\bar{u}$.
• kaons: Kaon$^+(\pi^+)$, $u\bar{s}$; Kaon$^-(\pi^-)$, $s\bar{u}$; Kaon$^0(\pi^0)$, $d\bar{s}$ or $s\bar{d}$.
• During $\beta^-$ decay a d quark changes into a u quark, and during $\beta^+$ decay a u quark changes into a d quark.
• Conservation laws for charge, $B$, $L$, and $S$ can be applied to particle interactions.

### Waves

• The Frequency, $f$, of a wave is the number of cycles per second. It is measured in hertz.
• The Period, $T$, of a wave is the length in seconds of one cycle.
• Amplitude is the maximum displacement of a wave.
• Wavelength is the distance between to successive equivalent points on a wave, such as two peaks.
• $c=\lambda \times f$
• $f=\frac{1}{T}$
• The vertical displacement of the particles in a wave is described by the equation: $y=A\sin{\frac{2\pi t}{T}}$
• Phase of a wave: the fraction of the cycle completed at a particular time.
• Phase difference is measured as a fraction of the wave cycle: $\phi = \frac{2\pi x}{\lambda}$ rad.
• If two waves are in phase, it means particles in parts of the wave are travelling at the same speed in the same direction.
• Path difference is $n\lambda$.
• If two waves are out of phase, it means that particles in parts of the wave are at different points in their cycle at a particular time.
• TODO If two waves are antiphase, it means they are travelling at the same speed in opposite directions.
• Path difference is: $\lambda(n+\frac{1}{2})$
• Path difference: The difference in distance travelled by two waves.
• TODO **Polarisation*
• The refractive index of air = 1.
• The law of refraction: $n_1 \sin{\theta_1}=n_1 \sin{\theta_1}$. Where:
• $n_1$ = the refractive index of material 1, and $n_2$ = the refractive index of material 2.
• This can be rearranged to: $_1n_2=\frac{\sin{\theta_1}}{\sin{\theta_2}}$, where $_1n_2$ is the refractive index between the two materials.
• $\theta_1$ is the angle of incidence in material 1.
• $\theta_2$ is the angle of refraction in material 2.
• Dispersion: As the refractive index varies with wavelength different colours of light slow down by different amounts when refracted.
• Material dispersion: Refractive index of an optical fibre varies with frequency. This causes pulse broadening, where the duration of each pulse increases. Pulse broadening limits the max frequency of pulses and the hence the maximum bandwidth available.
• Modal dispersion: This occurs when rays inside an optical fibre take slightly different paths. Variations in path length can also cause pulse broadening.
• Pulse broadening in Multi-mode fibres (with wider cores) is significant as they are broad enough to allow rays to take different paths. Vs mono-mode fibre.
• Absorption: some wavelengths are absorbed strongly in materials used to make optical fibres so the signal strength falls. Choice of material and signal amplification can mitigate this.

### Combining waves

• LASER - Light Amplification by Stimulated Emission of Radiation.
• Coherence - Two waves are coherent if they both have the same wavelength (and therefore the same frequency) and there is a constant phase relationship between the two sources.
• Superposition
• Superposition of waves is the phenomenon whereby two waves, of the same type, meet and overlap creating a resultant displacement that is the vector sum of the displacements of each wave.
• It can be constructive, if the two waves interact and a larger wave is created, and destructive, where the two waves cancel each other out.
• A fixed pattern of superposition, where interactions create positions of high and low intensity in fixed positions is called interference.
• Interference
• A stable pattern of superposition or interference, will only occur if two waves sources are coherent.
• An example of interference is the pattern created when two coherent light sources interact with each other creating minima and maxima of intensity.
• Another example is interference patterns from sound waves when two identical notes are played through two loudspeakers. The loudness of the sound oscillates depending on position relative to the two speakers.
• Young’s double slit experiment:
1. A coherent source of light waves is shone through two very narrow slits.
2. The light diffracts (spreads) through the slits creating a interference pattern of fringes on a screen.
• Fringe spacing formula: $w=\frac{\lambda D}{s}$, where $D$ is the distance from the slits to the screen and $s$ is the spacing between the slits.
• This is an example of a stable pattern of superposition.
• Stationary waves (AKA standing waves)
• Stationary waves occur when two progressive waves of the same frequency and amplitude but moving in opposite directions superpose.
• They often occur when reflections superpose.
• When a wave reflects off a rigid boundary it becomes 180 degrees out of phase with the original wave, but keeps the same amplitude and frequency.
• They have nodes and antinodes. A node is a point of zero amplitude. An antinode is a point of maximum amplitude.
• A harmonic is a mode of vibration that is a multiple of a first harmonic.
• For waves travelling along a string in tension, the speed $v$ is given by: $v=\sqrt[]{ \frac{T}{\mu} }$. Where $T$ is the tension in N and $\mu$ is the mass per unit length of the string in $kg$ $m^{-1}$.
• This equation can be used to calculate the frequency of a harmonic: $f=\frac{1}{2l}\sqrt[]{\frac{T}{\mu}}$ gives the first harmonic where the wavelength is twice the length of a string of length $l$.
• Stationary sound waves can occur in open-ended tubes. In a one end open tube: There is an antinode at the closed end and a node at the open end. In a tube open at both ends there are nodes at each open end and an antinode in the middle.
• Different notes are created from the different harmonics created from stationary waves in air columns.
• Diffraction occurs when waves spread around an obstacle or through a gap. Can easily be shown using a ripple tank.
• The following equations relate the slit width, slit spacing width and angle of diffraction:
• $\sin{\theta}=\frac{\lambda}{a}$, where $a$ is the slit width and $\theta$ is the angle of diffraction of the first minima.
• $\sin{\theta _{n}}=\frac{n \lambda}{d}$, where $\theta$ is the angle of diffraction of each maxima and $d$ is spacing between slits and $n$ is a whole number and the order of the maximum.

## Chemistry

30 Chapters

### Glossary

• Sublimation: is the change of state directly from a solid to a gas.
• A mechanism: is a detailed step by step sequence illustrating the how an overall chemical reaciton occurs.
• Free radical: an atom or group with an unpaired electron.
• Homolytic fission: is the breaking down of a covalent bond to create two free radicals.
• Heterolytic fission occurs when a covalent bond breaks and both electrons in the bond move to one of the atoms. Two oppositely charged ions are formed.
• Nucleophile: an electron pair donor.
• Anhydrous means without water.
• Stereoisomerism are molecules which have the same molecular formula and structural formula but a different three-dimensional arrangement of their atoms in space.
• E-Z isomerism is stereoisomerism where the difference is due to positions around the carbon-carbon double bond.
• Electrophiles are substances which can accept pairs of electrons.
• Carbocation: a species which contains a positive charge on a carbon atom.
• Hydration is where water is added to a substance.
• Hydrolysis is the decomposition of a chemical compound with water.

### Common ions

Here are some common ions that are useful to know.

Positive ions Formula Negative ions Formula
Hydrogen $H^+$ Chloride $Cl^{-}$
Sodium $Na^+$ Bromide $Br^-$
Potassium $K^+$ Fluoride $F^-$
Lithium $Li^+$ Iodide $I^-$
Ammonium $NH_4^-$ Hydroxide $OH^-$
Silver $Ag^+$ Nitrate $NO_3^-$
Barium $Ba^{2+}$ Oxide $O^{2-}$
Calcium $Ca^{2+}$ Sulfide $S^{2-}$
Copper (II) $Cu^{2+}$ Sulfate $SO_4^{2-}$
Magnesium $Mg^{2+}$ Carbonate $CO_3^{2-}$
Zinc $Zn^{2+}$ Hydrogen carbonate $HCO_3^{-}$
Lead $Pb^{2+}$ - -
Iron (II) $Fe^{2+}$ - -
Iron (III) $Fe^{3+}$ - -
Aluminium $Al^{3+}$ - -

### Basic atomic structure

• Rutherford-Bohr model.
• The carbon-12 standard. The masses of all atoms are measured relative to one atom of carbon 12, which is given a value of 12.0000.
• Relative isotopic mass: The mass of a single isotope of an element relative to 1/12 the mass of an atom of carbon-12. For now: this number is the same as the mass number of a particular isotope.
• Relative atomic mass ($A_r$): is the mass of an atom of an element relative to 1/12 the mass of an atom of carbon-12.
• $A_r=\frac{\sum{}{}(isotope_i\text{ mass number}\times\text{% abundance of }isotope_i)}{100}$
• TOF mass spectrometer:
1. Electrospray ionisation: A high voltage is applied to the tip of a capillary to produce highly charged droplets. The solvent evaporates to form gaseous charged ions. In this simple treatment we assume all ions are mono-nuclear and have a single, positive charge.
2. Acceleration: An electric field is applied to give all the ions with the same charge constant kinetic energy. As Ke=0.5massv2 heavier particles move more slowly than lighter particles.
3. Ion drift: The ions enter a region with no electric field called the flight tube. Here ions separate based on their velocities. Smaller ones faster, larger ones slower.
4. Ion detection: Positively charged ions arrive at the detector and cause a small electric current. The detector records the flight time of the ions.
5. Data analysis: The flight times are analysed and recorded as mass spectrum by the data analyser. Relative abundance is plotted against a m/z (mass to charge) ratio.
• Mass spectrum of a compound:
• The horizontal axis is usually the: mass to charge ratio: m/z.
• The molecular ion peak $M^+$ is usually the last major peak in the spectra. The m/z value of this peak is the relative molecular mass of the compound.
• The peak with the greatest abundance is usually not the molecular ion peak. This is called the base peak and corresponds to the most stable ion as the molecule breaks up in the mass spectrometer.
• Electron config: There is 1 s orbital at each energy level. There are 3 p orbitals at each energy level starting at n = 2. There are 5 d orbitals at each energy level, starting at n=3. An f sub-level has 7 orbitals. A diagram can be easily drawn to show the order in which electrons fill the sub-shells. Chromium and copper have slightly different configurations both with an even number of electrons in the 3d sub-shell leaving one electron in the 4s sub-shell, this structure gives a more stable and full 3d sub-shell. Take note!
• Chromium configuration: $1s^2 2s^2 2p^6 3s^2 3p^6 3d^5 4s^1$. Filling up the 3d sub-shell and leaving 1 electron in the 4s makes for a more stable configuration.
• Copper configuration: $1s^2 2s^2 2p^6 3s^2 3p^6 3d^10 4s^1$. Similar electron variation to chromium.
• Particles which have the same electron configuration are isoelectronic.
• Metal atoms tend to lose electrons to become positive ions.
• Non-metal atoms tend to gain electrons to become negative ions.
• Hydrogen can either lose and electron to become a hydrogen ion, H+ (lost electron), or it can lose an electron to become a hydride ion, H-.
• First ionisation energy: the energy required to remove one mole of electrons from one mole of gaseous atoms to form 1 mole of gaseous 1+ ions.
• Second ionisation energy
• Third ionisation energy… and so on…
• Values for ionisation energies are measured in $kJ mol^-1$.
• E.g: $Na (g) \Rightarrow Na^+ (g) + e^-$, $\Delta H = +496 \, kJ mol^-1$.
• Three general patterns for elements 1 to 36
• Ionisation energy decreases down a group:
• Ionisation energy shows a general increase across a period.
• Within short periods (periods 2 and 3), there is a zig-zag pattern.
• Explaining trends in ionisation energies:
• The atomic radius of beryllium is less than the atomic radius of magnesium.
• Atomic radius increases down a group and decreases across a period.
• The further an outer electron is from the attractive power of the nucleus the less energy is required to ionise it.
• Nuclear charge:
• Nuclear charge increases with the number of protons.
• A greater nuclear charge means the attractive forces to outer electrons are stronger and more energy is required to ionise it.
• Shielding by inner electrons:
• The attractive power of the nucleus can be shielded by inner electrons.
• The more inner electrons there are the less energy is required to ionise the outer electron.
• Group 3 and 6 shower lower than expected first ionisation energies. Repulsion between group 6 3p orbital electrons means they ionise more easily. Group 3 elements have a lone electron in the outer 3p orbital which due to distance from the nucleus and shielding makes ionisation easier.

• TODO

### Covalent Bonding and structures

• Covalent bonding, is a strong electrostatic attraction between a shared pair of electrons and the nuclei of the bonded atoms.
• Usually occurs between 2 non-metals.
• Pairs of electrons are shared and allow atoms to gain a full outer shell.
• Covalent bonds can occur singly or in multiples.
• [in] Dative covalent bonds both electrons come from one atom.
• Dot-cross diagrams can be used to show covalent bonds.
• Between covalent molecules there are weaker intramolecular forces of attraction.
• Simple molecular structures have low melting and boiling points.
• Only a small amount of energy is needed to disrupt these forces.
• Solubility: Simple covalent substances are soluble in non-polar solvents.
• Weak intermolecular forces of attraction between molecules of non-polar solvents are similar to that in simple molecular substances.
• These forces are able to act between the molecules of the simple molecular substance of the solvent.
• Electrical conductivity. Simple molecular substances are non-conductors.
• Giant covalent substances have high melting and boiling points.
• Lots of energy is required to break these bonds between atoms
• Solubility. Giant covalent are insoluble in non-polar and polar solvents. The strong covalent bonds that exists throughout the lattice are not disrupted by either solvent type.
• Conductivity. Generally they are nonconductors; as no charged particles are free to move and carry current. Graphite is the exception; delocalised electrons can move freely parallel to the layers.
• Giant covalent examples: Diamond, Graphite, Silica (Silicon Dioxide).
• Summary:
• Simple Molecular: low melting point, soluble in non-polar solvents, non-conductors.
• Giant Covalent: High melting point, insoluble, non-conductors of electricity (except graphite).

### Kinetics

• Rates of reaction vary depending on different factors.
• Collision theory, which main factors affect the rate of a chemical process: Concentration, pressure, temperature and surface area of solid reactants.
• Concentration: More particles are present at higher concentrations and hence more successful collisions occur, leading to increase in reaction rate.
• Pressure: If pressure increases, particles are forced closer together, which increases the probability of successful collision and therefore increases the reaction rate.
• Temperature: At higher temperatures the energies and speeds of particles increase leading to more successful collisions in a given period and hence increases the reaction rate.
• Surface area of solid reactants: When surface area is increased (by for example grinding up a solid), more of the reactant is available to combine with other reactants, this increases the number of successful collisions in a given time and increases the rate.
• Enthalpy profile diagrams. Catalysts provide alternative reaction pathways of lower activation energy. This can be seen on an enthalpy profile diagram.
• Distribution of molecular energies. A Maxwell-boltzmann distribution shows the number of gaseous molecules plotted against the energy they have at a fixed temperature.
• AKA: molecular energy distribution graph.
• It should roughly follow a normal distribution.
• Activation energy can be highlighted as an $x$-value on a Maxwell-Boltzmann distribution. All particles above this value have enough energy to undergo a successful collision.
• A catalyst could lower this critical value and increase the number of particles able to undergo a successful collision.
• Increasing temperature of reactant molecules will change the shape of the distribution. Temperature generally affects the distribution as follows:
• Lower temp distributions are moved to the left with higher peaks.
• Higher temp distributions are moved to the right with lower peaks.
• The curves should always start at the origin and should end up being asymptotic to the energy axis at higher energy values.
• Small increases in temperature can cause a large increase in reaction rate as the number of reactant particles may increase significantly.
• Maxwell-boltzmann distributions change with different concentrations.
• The curve will retain the same basic shape but the peak will increase as there are now more reactant particles.
• The overall area of the curve will increase.

### Chemical equilibria and Le Chatelier’s principle

• Some chemical reactions are reversible: E.g. $N_2 (g) + 3H_2 (g) \rightleftharpoons 2NH_3 (g)$
• For a general reaction $A (g) + B (g) \rightleftharpoons AB (g)$, after starting with only particles of type A and B, particles of type AB are formed. After a period of time a reaction is described as being in equilibrium if number of A, B and AB particles remain the same. However these won’t be all the same particles, they are constantly being formed and broken up. This is called a dynamic equilibrium in which you have a steady state because rates of the forward reaction and reverse reaction are the same.
• Dynamic equilibrium definition:
• the concentrations of the reactants and products remain constant, and
• the forward and reverse reactions are proceeding at equal rates.
• Homogeneous reactions are ones where are all the reactants and products are in the same physical state.
• Heterogeneous reactions are ones where are the reactants and products are in the different state.
• Positions of equilibrium is a notional measure of how far to the left-hand side or right-hand side a reaction is.
• Le Chatelier’s principle states that:

if a factor is changed which affects a system in equilibrium, the position of equilibrium will move in a direction so as to oppose this change.

• Pressure: An increase in pressure shifts the position of equilibrium to the side with the smallest gas volume, hence this side will increase in yield. A decrease in pressure shifts the position of equilibrium to the side with the largest gas volume hence this side will increase in yield.
• If the amount in moles is the same on both sides of a reaction, pressure will have no effect on the position of equilibrium. E.g. $2HI \rightleftharpoons H_2(g)+I_2(g)$
• Temperature: Changing temperature affects the position of equilibrium but it depends on whether the reaction is exothermic or endothermic.
• For a forwardly exothermic reaction: Increasing temp shifts position of equilibrium in the reverse endothermic direction, from right to left.
• For a forwardly exothermic reaction: Decreasing temp shifts position of equilibrium in the forward exothermic direction, from left to right.
• For a forwardly endothermic reaction: Increasing temp shifts position of equilibrium in the forward endothermic direction, from left to right.
• For a forwardly endothermic reaction: Decreasing temp shifts position of equilibrium in the reverse exothermic direction, from right to left.
• E.g. For the forwardly exothermic reaction: $N_2(g)+3H_2(g) \rightleftharpoons 2NH_3(g); \Delta H=-92\ kJ mol^{-1}$, increasing temperature would decrease the yield of ammonia.
• Concentration: If concentrations of reactants and products are changed, the equilibrium will adjust to replace and substance that has been removed or added.
• Examples: Haemoglobin and oxygen (mountain climbers), coloured $Co(II)$ complexes.
• Catalyst: A catalyst has not effect on the position of equilibrium but allows the reaction to get there faster. Forward and reverse reactions move more quickly, but there is no change to the e position.

### Equilibrium constant $K_c$, for homogeneous systems

• The law of chemical equilibrium: The direction taken by a reaction is dependent not only on the mass of the various components of the reaction, but also their concentration.
• $K_c$ is an equilibrium constant calculated from concentrations of reactants and products (in $mol$ $dm^{-3}$)
• $K_c$ can be calculated for reactions in solution or homogeneous gaseous reactions, as the concentration of a solution or a gas can be calculated as the number of moles in a certain volume (in $dm{3}$).
• $K_c$ is only affected by temperature, if temperature is constant it remains constant. All other factors, e.g. pressure, will not affect $K_c$.

For reaction: $aA+bB \rightleftharpoons cC+dD$,
$K_c=\frac{[C]^c [D]^d}{[A]^a[B]^b}$, where $[ C ]$ is the concentration of $C$ in mol dm^-3 in the mixture and c is the balancing number for $C$ and so on for $D,A$ and $B$. Units of $K_c=\frac{(mol \; dm^{-3})^{(c+d)}}{(mol \; dm^{-3})^{(a+b)}}$.

• The equilibrium constant for the reverse reaction is ${K_c}^{-1}$ and the units are given by $(mol^y \; dm^{x})^{-1}$.
• When a reaction is forwardly exothermic, an increase in temperature shifts the position of equilibrium to the left as it absorbs heat. This lowers concentrations of products and increases concentrations of reactants so $K_c$ decreases.
• A change in concentration or pressure will not change $K_c$ for homogeneous gaseous reactions.

### Oxidation, reduction and redox equations

• Redox reactions are reactions where oxidation and reduction occur simultaneously. They are important for everyday life.
• E.g: batteries use redox reactions, natural gas combustion, biological respiration.
• Oxidation state: the hypothetical charge on an atom assuming the bonding is completely ionic.
• E.g. the oxidation state of $Fe^{3+}=+3$, $O^{2-}=-2$, $Na^{+}=+1$.
Oxidation Reduction
Gain of oxygen Loss of oxygen
Loss of hydrogen Gain of hydrogen
Loss of electrons Gain of electrons
• In general:
• The total sum of oxidation states in a compound must add up to zero.
• The total sum of oxidation states of all elements in a molecular ion must add up to the charge of the ion.
• The maximum oxidation state of an element is ‘+ group number’, e.g. nitrogen is in Group 5 so its maximum oxidation state is +5.
• The minimum oxidation state of an element is ‘group number -8’, e.g. nitrogen is in Group 5 so its maximum oxidation state is -5.
• Rules for determining oxidation state:
• Oxygen has an oxidation state of -2 in all compounds except in peroxides where it has an oxidation state of -1 and in $OF_2$ where it has an oxidation state of +2.
• Hydrogen has an oxidation state of +1 in all compounds except in metallic hydrides where it has an oxidation state of -1.
• Group 1 elements have an oxidation state of +1 in all compounds.
• Group 2 elements have an oxidation state of +2 in all compounds.
• The oxidation states of transition metals and p block elements vary.
• In simple binary compounds, the more electronegative element has the negative oxidation state.
• d block elements where the charge on the ion is give e.g copper(II), $Cu^{2+}$, copper has an oxidation state of +2; iron(III), oxidation state of +3.
• Names and formulae of ions and compounds…
• All molecular ions which end in –te contain d or p block element atoms and some oxygen atoms.
• E.g. carbonate ion, manganate ion, dichromate ion, nitrite ion…
• Also: hydrogen carbonate, dihydrogen carbonate, hydrogen phosphate ions…
• An oxidising agent is a chemical which causes oxidation in another species.
• Oxidising agents accept electrons, they cause another substance to lose electrons, thus oxidising it.
• A reducing agent is a chemical which causes reduction in another species.
• Reducing agents lose electrons, causing another substance to gain electrons, thus reducing it.
• Half equations include electrons, helping to show oxidation and reduction processes in reactions.
• E.g. $Sn^{2+}+2e^- \rightarrow Sn$. Here tin(II) ions gain electrons, reducing to tin.
• E.g. $Mg \rightarrow Mg^{2+}+2e^-$. Here magnesium atoms lose electrons, oxidising to magnesium ions.
• Organic oxidation and reduction reactions. Organic chemicals such as primary and secondary alcohols are oxidised by reagents such as acidified potassium dichromate(VI) solution.
• Aldehydes, ketones and carboxylic acid can be reduced using lithat ($LiAlH_4$).
• $[O]$ is used to represent the oxidising agent in reactions where organic chemicals are oxidised.
• E.g. The oxidation of ethanol to ethanal: $CH_3CH_2OH + [O]\rightarrow CH_3CHO+H_2O$.
• $[H]$ is used to represent the reducing agent in equations showing the reduction of organic chemicals.
• E.g. The reduction of ethanoic acid to ethanol: $CH_3COOH+4[H]\rightarrow CH_3CH_2OH+H_2O$.
• Half equations can be combined by balancing numbers of electrons and adding them together.
• E.g. The half equations $Al \rightarrow Al^{3+}+3e^{-}$ (OXIDATION) and $F_2+2e^{-}\rightarrow 2F^-$ (REDUCTION) can be balanced (by balancing the electrons) and combined, removing the electrons, into the ionic equation: $Al+3F_2 \rightarrow 2Al^{3+}+6F^{-}$.

### Periodicity and Group 2

• Group 2 metals are known as the ‘alkaline earth metals’. (Their oxides and hydroxides are alkaline. ‘Earth’ was used by the first chemists to describe non-metallic substances which are insoluble in water). They are fairly reactive metals.
• Group 1 is also called the ‘alkali metals’. They are soft highly reactive metals.
• Group 7, the Halogens. Very reactive non-metals.
• Group 0, are called the Noble Gases becaused they don’t react. Form few compounds.
• The Periodic table is divided into blocks which correspond to which sub-shell the highest energy electrons are in: s, p, d and f.
• The f block elements are also referred to as the lathanide series and the actinide series. The actinide series is all radioactive and some are not found in nature.

#### Period 3

• The atomic radius decreases across period 3. This is due to the increasing atomic charge that happens across a period. Outer electrons are pulled closer to the nucleus.
• The melting points of the metallic elements increase. The melting point of silicon is very high. The melting points of phosphorus, sulfur, chlorine and argon are low. The melting point of elements after silicon increases from P to S, but decreases again. Argon has a very low melting point due to weak van der Waals’ forces.
• First ionisation energies generally increase. It decreases from magnesium to aluminium.
• This general increase is called by an increase in nuclear charge with no increased shielding.
• The lower than expected value for Al is due to additional shielding due to electron configuration.
• Sulfur also shows this trend.

#### Group 2 - alkaline earth metals

• First ionisation energy decreases
• Melting points generally decrease but increase from magnesium to calcium.
• The strength of the metallic bond is proportional to the number of delocalised electrons per atom, the charge of the metal ion and the size of the ion.
• Berlyllium does not react with water.
• Magnesium reacts very slowly with water and faster with steam. Magnesium hydroxide is produced with water and magnesium oxide with steam (magnesium hydroxide thermally decomposes to the oxide).
• Calcium, strontium and barium all react with water to produce the metal hydroxide and hydrogen.
• Reactivity increases down the group.
• Group 2 hydroxide (*$OH^{-}$) solubility increases down the group.
• Group 2 sulfate (*$SO_4^{2-}$) solubility decreases down the group.
• Barium salts can be used as a test for sulfate ions. A thick white precipitate forms when barium ions are added to sulfate ions.
• $Ba^{2+}(aq)+SO_4^{2-}(aq) \rightarrow BaSO_4(s)$.

#### Selected uses of Group 2 elements and their compounds

• Magnesium. Used in extraction of titanium.
• Magnesium hydroxide. AKA ‘milk of magnesia’, as its slightly alkaline it can be used in the neutralisation of excess acid in the gut but does not irritate the oesophagus.
• Calcium hydroxide. AKA ‘slaked lime’. Used to neutralise acidic soil.
• Calcium oxide. Used to neutralise sulfur dioxide, which may be produced as a by-product of fossil fuel combustion, to prevent acid rain.
• Barium sulfate. Can be eaten as part of a ‘barium meal’. It is good at absorbing X-rays, and so is used for locating the gut. Barium sulfate is insoluble so is harmless to humans.

### Halogens

• Fluorine (poisonous yellow gas), Chlorine (poisonous, dense yello-green gas (chloros = green), Bromine (caustic red-brown volatile liquid), Iodine (shiny, grey black solid which sublimes to a violet vapour), Astatine (apperance unkown, highly unstable)…
• Mean bondy enthalpies generally increase down the group. Fluorine breaks the trend as the closeness of the atoms in one molecule mean the electrons repel each other and make the bond enthalpy closer to iodine.
• Colours become darker
• Reactivity decreases (outer electrons are further away, increased shielding means electrons are gained less easily).
• Atomic radius increases (more electrons in more energy levels increases the atom size).
• Boiling points increase (more electrons lead to greater induced dipole-dipole attractions and hence greater van der Waals’ forces between molecules).
• Electronegativity increases (the bonding electrons in covalent bonds are closer to the nucleus, less shielding means the positive nucleus attracts the bonding pair more strongly and so electronegativity is greater).

#### Redox reactions of halogens

• halate (I) ions (all have oxidation date +1) :
• chlorate (I): $ClO^{-}$
• bromate (I): $BrO^{-}$
• iodate (I): $IO^{-}$
• sodium halate (I) salts are:
• sodium chlorate (I): $NaClO$
• sodium bromate (I): $NaBrO$
• sodium iodate (I): $NaIO$

Chlorine is slightly soluble in water and result pale green solution.

• $HCl$ and $HOCl$, hydrochloric and chloric (I) acid are formed.
• $Cl_2 (l) + H_2O (l) \rightleftharpoons HCl (aq) + HClO (aq)$.
• In bright sunlight: $2HClO(aq) \rightarrow 2HCl(aq) + O_2 (g)$.

Mass medication? Fluoride in drinking water: pros are the prevention of tooth decay, cons are possible other health risks.

• Some halogens react with halides ions (single halogen atoms) in solution.
• Chlorine water: pale green solution
• Bromine water: orange solution
• Iodine water: pale yellow solution (solution with iodide ions forms a brown solution).
• The oxidising ability of the halogens reduces as atomic number increases.
• Less shielding and proximity to the nucleus contribute to this.
• Oxidising agents readily accept electrons.
• Halogens gain electrons into their outer energy level to complete it.

#### Displacement reactions

Displace means to push out…a more reactive element can displace a less reactive element from a compound.

• We expect fluorine to displace all other halides from a solution of a halide compound.
• Chlorine to displace bromide and iodide from a solution.
• Bromine to displace idodide.
• The reducing ability of the halide ions increaseas as the atomic number increases.
• A substances which loses electrons easily is a good reducing agent.
• Electrons which are further from and more shielded by the nucleus are lost more easily due to a weaker nuclear attraction.
• Reaction of solid halides with concentrated sulfuric acid
• $NaF + H_2SO_4 \rightarrow NaHSO_4 + HF$ (NOT REDOX)
• Simplest ionic equation for this reaction: $F^- + H_2SO_4 \rightarrow HSO_4^- + HF$
• $H_2SO_4 + NaCl \rightarrow NaHSO_4 + HCl$ (NOT REDOX)
• $Cl^- + H_2SO_4 \rightarrow HSO_4^- + HCl$

#### Identification of halide ions

• Cl-, Br- and I- can be distinguished by using silver(I) ions. (Silver (I) nitrate is often used ($AgNO_3$))
• $Ag^+(aq) + Cl^- \rightarrow AgCl (s)$, white precipitate
• $Ag^+(aq) + Br^- \rightarrow AgBr (s)$, cream precipitate
• $Ag^+(aq) + I^- \rightarrow AgI (s)$, yellow precipitate
• Furthermore, concentrated ammonia solution can be added. AgCl and AgBr both dissolve forming a colourless solution. AgI does not dissolve.

#### Identification tests

• Precipitate is often abbreviated to ppt.
• If a precipitate is expected then a solution must be added to a solution.

### Organic chemistry introduction

• Organic compounds are those that contain carbon.
• Carbon is a unique element.
• There are approximately $10^7$ compounds containing carbon and hydrogen whose formulae are known by chemists.
• Compounds containing hydrogen and carbon are called hydrocarbons.
• Carbon can form 4 covalent bonds. They can all be single like in methane or be a mixture of single and double, as in ethene.
• carbon-carbon and carbon-hydrogen bonds are relatively strong and non-polar.
• A functional group: is a group of atoms which responsible for the characteristic reactions of a compound. Molecules with the same functional groups belong to the same chemical family.
• A homologous series is a group of compounds with the same general formula. All members of a homologous series:
• have the same general formula,
• show a gradation in physical properties,
• have the same functional group,
• differ from successive members by a $CH_2$ group.
• Molecular formula show the actual number of atoms of each element, e.g. $C_3H_6$ (propene), $C_4H_9Br$ (1-Bromobutane),…
• Displayed formula show all atoms and all covalent bonds for a compound.
• Structural formula of a compounds show the arrangement of atoms, carbon by carbon with the attached hydrogens and functional groups without showing bonds.
• Empirical formula show the lowest whole number ratio of atoms of each element for a compound.
• Skeletal formula. These show the structure of a compound without drawing individual carbon and hydrogen atoms, these are show with lines.
• General formula represents the composition of any member of a homologous series. E.g. \$C_nH_{2n}.
• Isomerism. Chemicals that have the same molecular formula but different structures are called structural isomers.
• Positional isomers have the same carbon chain and the same functional group but it is attached at different points along the carbon chain.
• Functional group isomerism, compounds with the same moleulcar formula but have different functional groups.
• Steroisomers are molecules which have the same molecular and structural formula but a different arrangement of atoms in 3D space.
• There are two types E-Z and optical.

#### Useful tables for organic nomenclature

Root name indictating the longest chain of carbon atoms:

Root name # of carbon atons
meth 1
eth 2
prop 3
but 4
pent 5
hex 6

Prefixes used to name side groups or substituents:

Prefix Side chain group or substituent group
methyl -$CH_3$
ethyl -$CH_2CH_3$
propyl -$CH_2CH_3CH_3$
fluoro -$F$
chloro -$Cl$
bromo -$Br$
iodo -$I$

Prefixes for the # of side chains or groups:

Prefix # of same side chains or groups present
di two
tri three
tetra four
penta five
hex six

Functional groups:

Homologous series Functional group Suffix Example
carboxylic acid OH-C=O -oic acid propanoic acid
ester -OH-C=O -oate ethyl ethanoate
acyl chloride Cl-C=O -oly chloride butanoyl chloride
nitrile -C$\equiv$N -nitrile propanenitrile
aldehyde H-C=O -al ethanal
ketone -C=O -one propanone
alcohol -OH -ol butanol
amine -$NH_2$ -amine ethylamine
alkene >C=C< -ene propene
halogenoalkane -C-X (X=halogen) named as a substituted hydrocarbon 1-bromobutane

### Alkanes ($C_nH_{2n+2}$)

A homologous series is a series of carbon-based compounds which have the same general formula and functional group. They have similar chemical properties and show a gradation in physical properties with successive members differing by a $CH_2$ group. Alkanes are an example of a homologous series. All members of the series have the general formula $C_nH_{2n+2}$ While they do not strictly have a functional group, all alkanes are saturated hydrocarbons containing only C-C or C-H single bonds. They all react with halogens in UV light to form the hydrogen halide and a halogenoalkane. As the series is descended the physical properties change slightly; the first four members of the alkanes are gases at room temperature, C5–C17 liquids and the remaining members are waxy solids with increasing melting points.

• Alkanes are a homologous series of saturated (contain only single carbon bonds) hydrocarbons.
• They are non-polar and experience van der Waals’ forces.
• Their general formula is $C_nH_{2n+2}$.
• Crude oil (petroleum): contains > 150 carbon-based compounds.
• Fraction distillation is the continuous evaporation and condensation of a mixture causing components to seperate due to differences in their boiling points.
• A fraction is a group of compounds with similar boiling points and are removed at the same point of the distillation chamber.
• Industrial cracking of hydrocarbons:
• Thermal cracking: long chain alkanes are heated at very high temperatures (between 1000 and 1200K) and pressures (70atm) for short periods to produce short chain alkanes and alkenes.
• Catalytic cracking: Heating under pressure in the presence of a zeolite catalyst.
• Zeolite is an acidic mineral with a honeycomb structure. This structure gives a large surface area which increases the rate of the reaction;.
• This method is mainly used to produce branched alkanes, cycloalkanes and aromatic compounds; fuels for road vehicles.
• Alkanes combust completely in a plentiful supply of oxygen to produce carbon dioxide and water only.
• $CH_4(g)+2O_2(g)\rightarrow CO_2(g)+2H_2O(g)$ - the complete combustion of methane.
• In a limited supply of air, alkanes burn to form water and carbon dioxide.
• $CH_4(g)+1\frac{1}{2}O_2(g)\rightarrow CO(g)+2H_2O(g)$ - the incomplete combustion of methane.
• If oxygen supply is further limited solid carbon particles are formed (soot).
• $CH_4(g)+O_2(g)\rightarrow C(g)+2H_2O(g)$

### Halogenoalkanes

• Homologous series of saturated carbon compounds containing one or more halogen atoms.
• They are used as refrigerants, propellants, solvents, flame retardants, anaesthetics and pharmaceuticals.
• They have been proven to cause pollution which has led to the depletion of the ozone layer.
• The C-X bond in halogenoalkanes is polar because the halogen atom is more electronegative than the carbon atom.
• The polarity decreases down the group as the halogens become less electronegative down the group. Fluorine is extremely polar whereas iodine is almost non-polar.
• Boiling points increase as the chain length increases for the chloroalkanes, bromoalkanes and the iodoalkanes due to van der Waal’s forces. Boiling points also decrease when the halogen is changed due to permanent dipole-dipole interactions being greater the more polar the carbon-halogen bond is. The van der Waal’s forces have a greater effect.
• They are insoluble in water despite the polar nature of the carbon-halogen bond or only slightly soluble.

#### Synthesis and reactions

• Chloroalkanes can be synthesised by chlorinating the alkanes. Chlorine reacts with methane in the presence of sunlight to form a chloroalkane and $HCl$ gas.
• The mechanism for the syntheiss of halogenoalkanes:
• Step 1: initiation - $Cl_2 \rightarrow Cl\cdot Cl\cdot$ UV light breaks down the diatomic Cl molecules intro single chlorine atoms each with an unpaired electron in it’s outer shell. They are free radicals.
• Step 2: propagation - $Cl\cdot + CH_4 \rightarrow HCl+\cdot CH_3$, $\cdot CH_3 +Cl_2 \rightarrow CH_3Cl + Cl \cdot$. This is a continuing process of free radical production is a chain reaction.
• Step 3: termination - For the reaction the free radicals must collide to form molecules. In this case there are three possible combinations; $Cl_2$ (diatomic chlorine), $CH_3Cl$ (chlorometane), $C_2H_6$ (ethane).
• Nucleophilic substitution reactions. The $\delta^+$ carbon-halogen bond is susceptible to nucleophilic attack.
• By cyanide ion: $C\equiv N$.
• By hydroxide ion: $OH^-$.
• By ammonia molecule: $NH_3$.
• The rate of nucleophilic substitution/hydrolysis reactions depends on the strength of the carbon-halogen bond. The stronger the bond the slower the rate of reaction.
• Amines (ammonia derivatives with a spair pair of electrons) are primary (1 hydrogen, two functional groups), secondary (2 functional) and tertiary (3 functional).
• Elimination reactions: An aqueous solution of potassium hydroxide contains hydroxide ions. The aqueous hydroxide ions will react as nucleophiles with halogenoalkanes, forming an alcohol in a nucleophilic substitution reaction. However, when dissolved in ethanol the hydroxide ions act as a base and accept a proton (a hydrogen ion) to form water. The consequence is that the halogenoalkane molecule loses a hydrogen atom and a halogen atom.
• Concurrent nucleophilic substitution reactions: In practice, both elimination and substitution reactions occur simultaneously.

#### Reaction summary

Reagent Conditions Mechanism Product Effects
Potassium hydroxide Aqueous reagent, dissolve halogenoalkane in a little ethanol, reflux Nucleophilic substitution alcohol -
Potassium cyanide Aqueous reagent, dissolve halogenoalkane in a little ethanol, reflux Nucleophilic substitution nitrile carbon chain increases by one
Concentrated ammonia Excess ammonia, in a sealed tube, under pressure Nucleophilic substitution amine excess ammonia encourages a high yield of primary amine; discourages production of further substituted amines
Potassium hydroxide Dissolve reagent and reactants in ethanol, no water present Elimination alkene product may be a mix of position isomers

### Alkenes ($C_nH_{2n}$)

• Ethene, $C_2H_4$, is the first in the homogolous series of alkenes.
• Worldwide production in the excess of 100 million tonnes.
• The carbon carbon double bond is a centre of high electron density.
• But-2-ene can has two E-Z isomers. Z-but-2-ene (Z from zusammen, meaning altogether), E-but-2-ene (E from entgegen, meaning opposite).
• Addition polymerisation is the reaction whereby alkenes react with other alkene molecules to form polymers (commonly used as plastics).
• Polythene is formed when ethene molecules add on to each other to form a chain of carbona atoms. The double bond is broken to form a monomer which in a chain is called a polymer.
• Poly(propene), a thermoplastic polymer used in a wide variety of applications.
• Poly(chloroethene), commonly called PVC.
• Poly(phenylethene), polystyrene from phenylethene (styrene).
• Addition polymers are unreactive aklane molecules due to their lack of carbon-carbon double bond.
• Useful insulators, as packagin and for making containers.
• Poly(ethene) exists as HDPE and LDPE; HD is less flexible and LD is more flexible.
Polymer recycling code Uses
Poly(ethene) (PE) plastic bags, film wrapping, kitchenware
Poly(propene) (PP) ropes, thinsulate clothing, carpets, crates, furniture
Poly(chloroethene)(PVC) wellington boots, raincoats, drainpipes, frames, door frames, electrical wire insulation
Poly(phenylethene) (PS) expanded polystyrene is used for insulation in houses and packing, unexpanded polystyrene is used for toys and containers

#### Reactions of the alkenes

• The high electron density of the carbon-carbon double bond makes them susceptible to attack from electrophiles.
• $H^{\delta+}-Br^{\delta-}$, hydrogen bromide, is an example of an electrophile.
• e.g. $CH_2=CH_2 + HBr \rightarrow CH_3CH_2Br$
• e.g. $CH_2=CH_2 + Br_2 \rightarrow CH_2BrCH_2Br$
• Mechanism: Alkene + electrophile (A-B) $\rightarrow$ carbocation + nucleophile $\rightarrow$ product
• The mechanism is similar for neutral electrophiles.
• Heterolytic fission has occured in the A-B bond, producing the nucleophile.
• The product in saturated.
• $Br_2$ (as bromine water, an orange solution) + alkene $\rightarrow$ bromoalkane (colourless solution).

Electrophilic addition of polar electrophiles to an unsymmetrical alkene produces two different products, one of which may be present in a greater proportion (the major product) than the other (the minor product).

• E.g. the reaction between but-1-ene and hydrogen bromide produces 2-bromobutane (major product) and 1-bromobutane (minor product).
• The prevalence of the major product is due to the relative stability of the different carbocations.
• Carbocations are classified as, primary, secondary and tertiary depending on the number of alkyl groups, –R, attached.
• Alkyl groups (R groups) such as – $CH_3$ have an electron donating effect; the opposite effect to that of an electronegative atom such as chlorine.
• The relative stability of some carbocations means they stay in the reaction mixture for longer increasing their chance of reacting with an anion.
• The industrial production of ethanol from ethene (reacted with steam):
• $CH_2=CH_2 + H_2O(g) \rightleftharpoons CH_3CH_2OH$. (Hydration)

### Alcohols ($C_nH_{2n+1}OH$)

• Ethanol ($C_2H_5OH$) is part of homologous series of alcohols. It is commonly referred to as ‘alcohol’.
• It is present in the complex mixtures produced by the fermentation process.

#### Physical properties of the alcohols

• They have a greater boiling point than alkanes of a similar $M_r$ due to hydrogen bonding.
• Hydrogen bonding is stronger than van der Waals’. Both hydrogen bonding and van der Waals’ occur in alcohols.
• Short chain alcohols such as ethanol and methanol are soluble in water, the –O-H group strongly contributes to this.
• Alkanes without the –O-H group are insoluble in water.
• Aldehydes, ketones and carboxylic acids are further examples of homologous series.
• aldehydes and ketones contain C=O group.

#### Classification

Type of alcohol General structure Example Position of OH group
primary alcohol, 1 R group $RCH_2(OH)$ butan-1-ol, $CH_3CH_2CH_2CH_2(OH)$ –OH is at the end of the chain
secondary alcohol, 2 R groups $CR_2H(OH)$ butan-2-ol, $CH_3CH_2CH(OH)CH_3$ –OH is positioned along the length of the chain
tertiary alcohol, 3 R groups $CR_3(OH)$ 2-methylpropan-2-ol, $CH_3C(OH)(CH_3)CH_3$ –OH is positioned at a branch in the chain
• Methanol is considered a primary alcohol.

#### Reactions

Oxidation:

• Primary alcohols are easily oxidised (in the presence of oxidising agents) to form aldehydes which can be further oxidised to carboxylic acids
• Secondary alcohols can be oxidised to ketones, which do not undergo further oxidation. ($R_2HC(OH) \rightarrow CR_2=O$)
• Tertiary alcohols are not easily oxidised as they do not have hydrogen atoms directly bonded to the carbon bonded to the OH group.
• They can be oxidised using hot nitric acid. It’s harder because it requires the breaking of a carbon-carbon bond.
• The oxidation of ethanol to produce ethanoic acid (carboxylic acid): $CH_3CH_2OH \rightarrow 2[O] + CH_3COOH + H_2O$
• the carboxylic acid can be removed via distillation.
• To produce a carboxylic acid the oxidising agent is added in excess and refluxed gently.
• Refluxing prevents the alcohol evaporating before reacting. It is immediately returned to the flask via condensation.
• To produce the aldehyde, the primary alcohol must be in excess and the product distilled off immediately. This is to ensure the oxidation is only partial and the carboxylic acid is not formed.
• $CH_3CH_2OH + [O] \rightarrow CH_3CHO + H_2O$
• Secondary must be gently refluxed with an excess of the oxidising element to produce a ketone.
• $CH_3CH(OH)CH_3 \rightarrow CH_3CHO + H_2O$

Acidified potassium or sodium dichromate(VI) solution is a suitable oxidising agent for the oxidation of primary and secondary alcohols. The dichromate(VI) ion, Cr2O72− , is orange in aqueous solution and is reduced to the green chromium(III) ion, Cr3+, as the alcohol is oxidised.

Testing for aldehydes and ketones:

• Tollens’ reagent (ammoniacal silver nitrate ($Ag(NH_3)_2$)) - the silver mirror test.
• When warmed in the presence of an aldehyde silver ions in the Tollens reagent are reduced to silver atoms:
• $Ag^+(aq) + e^-\rightarrow Ag(s)$
• Oxidation of ethanal with Tollens’: $CH_3CHO + [O] \rightarrow CH_3COOH$ to form ethanoic acid.
• Ketones will not react with Tollens’ reagent.
• Fehling’s solution ($Cu^{2+}$) is a mild oxidising agent:
• $C^{2+} + e^- \rightarrow Cu^+$
• When gently warmed with an aldehyde the blue colour disspears and an orange red precipitate forms.
• The solution remains blue when warmed with a ketone.
Test Observations with aldehyde Ketone
warm gently with Tollens’ reagent silver mirror forms on sides of test tube solution remains colourless
warm getnly with Fehling’s solution orange-red precipitate forms solution remains blue

Elimination reactions with alcohols:

• can also be described as a dehydration reaction (where a molecule of water is eliminated).
• E.g. propan-1-ol -> propene + $H_2O$
• Alcohols can be dehydrated using concentrated sulfuric acid as a catalyst at a temperature of 170 °C or by passing the alcohol vapour over a heated aluminium oxide catalyst, Al2O3, at 600 °C.

Initially one of the lone pairs of electrons on oxygen picks up a hydrogen ion from the sulfuric acid and is protonated. A negative hydrogen sulfate ion is produced ($HSO_4^-$). A water molecule is lost from the protonated alcohol forming a carbocation. Finally a hydrogen sulfate ion removes an H+ ion from the carbocation and a double bond forms between two carbon atoms.

• Elimination in a symmetrical alcohol leads to only one product.
• However for an unsymmetrical alcohol a mixture of isomeric products are formed.

#### Industrial production of ethanol

• Ethanol is manufactured industrially via the fermentation of carbohydrates and by reaction of steam with ethene.
• By fermentation: yeast converts sugars such as glucose to ethanol and carbon dioxide. Conditions for the reaction:
• in the presence of yeast. Yeast produces enzymes that convert sugars into methanol
• under anaerobic conditions, to prevent the oxidation of ethanol to ethanoic acid (vinegar)
• at a temperature of 35 degrees, it is too slow below 25 and the enzymes are denatured above 40
• in a neutral aqueous solution
• Yeast is killed in ethanol above concentrations of 14%.
• The ethanol is removed via fractional distillation.
• It can be used to make biofuels.
• By hydration of ethene: ethene + steam $\rightleftharpoons$ ethanol; conditions:
• catalyst of concentrated phosporic acid absorbed on a solid silica surface
• 60 atm of pressure
• 600K temp
• excess ethane to give a high yield

Bioethanol:

• Theoretically ethanol produced via fermentation is carbon-neutral, however due to fossil fuel energy production this is not currently the case.

### Organic analysis

Homologous series Reagent Observations before test After test
alkenes bromine water orange solution colourless solution
primary / secondary alcohols acidified potassium dichromate orange solution green solution
tertiary alcohols acidified potassium dichromate orange solution remains orange
aldehydes Tollens’ reagent colourless solution silver precipiate
aldehydes Fehling’s solution blue solution orange-red ppt formed
carboxylic acids solid sodium carbonate white solid solid dissapears, bubbles of gas evolved (effervescence). Gas turns lime water from colourless to cloudy

Last edited :- Sat, 16 Apr 2022 21:08:59 BST