Portal:Chemistry/Useful equations and links

Note: I am developing this page as a study aid for myself. I may later find a home for it at the chemistry portal or at Wikibooks or Wikiversity. Shalom (Hello • Peace) 00:18, 4 November 2007 (UTC)

Chemical elements

• Periodic table
• IUPAC nomenclature for organic and inorganic compounds
• Stoichiometry

Ions

• Ionization energy
• Ionic radius

Gases

Main article: Gas laws

Ideal gas law

• Combined gas law

${\frac {P_{1}V_{1}}{T_{1}}}={\frac {P_{2}V_{2}}{T_{2}}}.$

• Ideal gas law

$PV=nRT\,$,

where

P is the pressure (SI unit: pascal)

V is the volume (SI unit: cubic metre)

n is the number of moles of gas

R is the ideal gas constant (SI: 8.3145 J/(mol K))

T is the thermodynamic temperature (SI unit: kelvin).

An equivalent formulation of this law is:

$PV=NkT\,$

where

N is the number of molecules

k is the Boltzmann constant.

Graham's Law

• Graham's law of effusion

${{\mbox{Rate}}_{1} \over {\mbox{Rate}}_{2}}={\sqrt {M_{2} \over M_{1}}}$

where:

Rate₁ is the rate of effusion of the first gas.

Rate₂ is the rate of effusion for the second gas.

M₁ is the molar mass of gas 1

M₂ is the molar mass of gas 2.

Kinetic theory

• Kinetic theory of gases

$P={Nmv_{rms}^{2} \over 3V}$

Also, as Nm is the total mass of the gas, and mass divided by volume is density

$P={1 \over 3}\rho \ v_{rms}^{2}$

where ρ is the density of the gas.

This result is interesting and significant, because it relates pressure, a macroscopic property, to the average (translational) kinetic energy per molecule (1/2mv_rms²), which is a microscopic property.

The root mean square velocity of a molecule is

$v_{rms}^{2}={\frac {3RT}{\mbox{molar mass}}}$

with v in m/s, T in kelvins, and R is the gas constant. The molar mass is given as kg/mol.

Dalton's Law

• Dalton's law of partial pressures

The pressure of a mixture of gases can be defined as the summation

$P_{total}=\sum _{i=1}^{n}{p_{i}}$       or      $P_{total}=p_{1}+p_{2}+\cdots +p_{n}$

where $p_{1},\ p_{2},\ p_{n}$ represent the partial pressure of each component.

It is assumed that the gases do not react with each other.

$\ P_{i}=P_{total}m_{i}$

where $m_{i}\ =$ the mole fraction of the i-the component in the total mixture of m components .

Van der Waals equation

• Van der Waals equation of state

$\left(p+{\frac {n^{2}a}{V^{2}}}\right)\left(V-nb\right)=nRT$,

where

p is the pressure of the fluid

V is the total volume of the container containing the fluid

a is a measure of the attraction between the particles $a=N_{\mathrm {A} }^{2}a'$

b is the volume excluded by a mole of particles $\,b=N_{\mathrm {A} }b'$

n is the number of moles

R is the gas constant, $\,R=N_{\mathrm {A} }k$

Solutions

• Solution (chemistry)
• Concentration
  • Molarity (moles per liter)
  • Molality (moles of solute per mass of solvent)
  • Normality (chemistry) (equivalents in a chemical reaction)
• Acids and bases
  • Bronsted-Lowry acid-base theory
  • Lewis acids and bases
• pH
• Distribution coefficient, also called the partition coefficient

The partition coefficient is the ratio of concentrations of un-ionized compound between the two solutions. To measure the partition coefficient of ionizable solutes, the pH of the aqueous phase is adjusted such that the predominant form of the compound is un-ionized. The logarithm of the ratio of the concentrations of the un-ionized solute in the solvents is called log P:

• $log\ P_{oct/wat}=log{\Bigg (}{\frac {{\big [}solute{\big ]}_{octanol}}{{\big [}solute{\big ]}_{water}^{un-ionized}}}{\Bigg )}$

Equilibrium

• Chemical equilibrium
  • Equilibrium constant (K_eq)
  • Acid dissociation constant (K_a)
  • Solubility product (K_sp)

Electrochemistry

• Electrochemistry
• Electrolysis
• Galvanic cell
• Half reaction
• Redox reaction
  • Oxidation
  • Reduction
• Oxidizing agent
• Reducing agent

Isomerism

• Isomerism
• cis-trans isomerism
• structural isomerism
• diastereomer
• linkage isomer

Chirality

• Chirality (chemistry)
• Cahn Ingold Prelog priority rules for R/S isomerism
• Enantiomer
• Axial chirality

Analytical chemistry

• Primary standard

Spectroscopy

• IR spectroscopy
• UV-vis spectroscopy
• NMR spectroscopy
  • Proton NMR
  • Carbon-13 NMR

Nuclear chemistry

• Nuclear chemistry
• Nuclear decay
• Alpha particle
• Beta particle

Radiocarbon dating

• Carbon-14 dating

The radioactive decay of carbon-14 follows an exponential decay. A quantity is said to be subject to exponential decay if it decreases at a rate proportional to its value. Symbolically, this can be expressed as the following differential equation, where N is the quantity and λ is a positive number called the decay constant:

${\frac {dN}{dt}}=-\lambda N.$

The solution to this equation is:

$N=N_{0}e^{-\lambda t}\,$,

where, for a given sample of carbonaceous matter:

$N_{0}$ = number of radiocarbon atoms at $t=0$, i.e. the origin of the disintegration time,

$N$ = number of radiocarbon atoms remaining after radioactive decay during the time $t$,

${\lambda }=$radiocarbon decay or disintegration constant.

Two related times can be defined:

• mean- or average-life: mean or average time each radiocarbon atom spends in a given sample until it decays.
• half-life: time lapsed for half the number of radiocarbon atoms in a given sample, to decay,

It can be shown that:

$t_{avg}\,$ = ${\frac {1}{\lambda }}$ = radiocarbon mean- or average-life = 8033 years (Libby value)

$t_{\frac {1}{2}}\,$ = $t_{avg}\cdot \ln 2$ = radiocarbon half-life = 5568 years (Libby value)

Notice that dates are customarily given in years BP which implies t(BP) = -t because the time arrow for dates runs in reverse direction from the time arrow for the corresponding ages. From these considerations and the above equation, it results:

For a raw radiocarbon date:

$t(BP)={\frac {1}{\lambda }}{\ln {\frac {N}{N_{0}}}}$

and for a raw radiocarbon age:

$t(BP)=-{\frac {1}{\lambda }}{\ln {\frac {N}{N_{0}}}}$

Thermodynamics

Laws of thermodynamics

• Laws of thermodynamics

• Zeroth law of thermodynamics

  $A\sim B\wedge B\sim C\Rightarrow A\sim C$

• First law of thermodynamics

  $\mathrm {d} U=\delta Q-\delta W\,$

• Second law of thermodynamics

  $\oint {\frac {\delta Q}{T}}\geq 0$

• Third law of thermodynamics

  $T\rightarrow 0,S\rightarrow C$

Identities

• Work (thermodynamics)

Chemical thermodynamics studies PV work, which occurs when the volume of a fluid changes. PV work is represented by the following differential equation:

$dW=-PdV\,$

where:

• W = work done on the system
• P = external pressure
• V = volume

Therefore, we have:

$W=-\int _{V_{i}}^{V_{f}}P\,dV$

• Entropy
  • Entropy (classical thermodynamics)

Clausius defined the change in entropy ds of a thermodynamic system, during a reversible process, as

$dS={\frac {\delta Q}{T}}\!$

where

δQ is a small amount of heat introduced to the system,

T is a constant absolute temperature

Note that the small amount $\delta Q$ of energy transferred by heating is denoted by $\delta Q$ rather than $dQ$, because Q is not a state function while the entropy is.

• Enthalpy

The function H was introduced by the Dutch physicist Heike Kamerlingh Onnes in early 20th century in the following form:

$H=E+pV,\,$

where E represents the energy of the system. In the absence of an external field, the enthalpy may be defined, as it is generally known, by:

$H=U+pV,\,$

• Internal energy

The internal energy is essentially defined by the first law of thermodynamics which states that energy is conserved:

$\Delta U=Q+W+W'\,$

where

ΔU is the change in internal energy of a system during a process.

Q is heat added to a system (measured in joules in SI); that is, a positive value for Q represents heat flow into a system while a negative value denotes heat flow out of a system.

W is the mechanical work done on a system (measured in joules in SI)

W' is energy added by all other processes

Although the internal energy is not exactly measurable, it may be expressed in terms of other similarly unmeasurable quantities. Using the above two equations in the first law of thermodynamics to construct one possible expression for the internal energy of a closed system gives:

$\mathrm {d} U=\delta Q-dW=T\mathrm {d} S-p\mathrm {d} V\,$

• Gibbs free energy

$\Delta G=\Delta H-T\Delta S\,$ for constant temperature

$\Delta G^{\circ }=-RT\ln K\,$

$\Delta G=\Delta G^{\circ }+RT\ln Q\,$

$\Delta G=-nF\Delta E\,$

and rearranging gives

$nF\Delta E^{\circ }=RT\ln K\,$

$nF\Delta E=nF\Delta E^{\circ }-RT\ln Q\,\,$

$\Delta E=\Delta E^{\circ }-{\frac {RT}{nF}}\ln Q\,\,$

which relates the electrical potential of a reaction to the equilibrium coefficient for that reaction.

where

ΔG = change in Gibbs free energy, ΔH = change in enthalpy, T = absolute temperature, ΔS = change in entropy, R = gas constant, ln = natural logarithm, K = equilibrium constant, Q = reaction quotient, n = number of electrons per mole product, F = Faraday constant (coulombs per mole), and ΔE = electrical potential of the reaction. Moreover, we also have:

$K_{eq}=e^{-{\frac {\Delta G^{\circ }}{RT}}}$

$\Delta G^{\circ }=-RT(\ln K_{eq})=-2.303RT(\log K_{eq})$

which relates the equilibrium constant with Gibbs free energy.

• Helmholtz free energy

The Helmholtz energy is defined as:

$A\equiv U-TS\,$

From the first law of thermodynamics we have:

${\rm {d}}U=\delta Q-\delta W\,$

where $U$ is the internal energy, $\delta Q$ is the energy added by heating and $\delta W=p{\rm {d}}V$ is the work done by the system. From the second law of thermodynamics, for a reversible process we may say that $\delta Q=T{\rm {d}}S$. Differentiating the expression for $A$  we have:

${\rm {d}}A={\rm {d}}U-(T{\rm {d}}S+S{\rm {d}}T)\,$

$=(T{\rm {d}}S-p\,{\rm {d}}V)-T{\rm {d}}S-S{\rm {d}}T\,$

$=-p\,{\rm {d}}V-S{\rm {d}}T\,$

Maxwell relations

The four most common Maxwell relations are the equalities of the second derivatives of each of the four thermodynamic potentials, with respect to their thermal natural variable (temperature T  or entropy S ) and their mechanical natural variable (pressure p  or volume V ):

$\left({\frac {\partial T}{\partial V}}\right)_{S}=-\left({\frac {\partial p}{\partial S}}\right)_{V}\qquad ={\frac {\partial ^{2}U}{\partial S\partial V}}$

$\left({\frac {\partial T}{\partial p}}\right)_{S}=+\left({\frac {\partial V}{\partial S}}\right)_{p}\qquad ={\frac {\partial ^{2}H}{\partial S\partial p}}$

$\left({\frac {\partial S}{\partial V}}\right)_{T}=+\left({\frac {\partial p}{\partial T}}\right)_{V}\qquad =-{\frac {\partial ^{2}A}{\partial T\partial V}}$

$\left({\frac {\partial S}{\partial p}}\right)_{T}=-\left({\frac {\partial V}{\partial T}}\right)_{p}\qquad ={\frac {\partial ^{2}G}{\partial T\partial P}}$

where the potentials as functions of their natural thermal and mechanical variables are:

$U(S,V)\,$ - The internal energy

$H(S,p)\,$ - The Enthalpy

$A(T,V)\,$ - The Helmholtz free energy

$G(T,p)\,$ - The Gibbs free energy

Boltzmann distribution

• Boltzmann distribution

In physics, the Boltzmann distribution predicts the distribution function for the fractional number of particles N_i / N occupying a set of states i which each respectively possess energy E_i:

${{N_{i}} \over {N}}={{g_{i}e^{-E_{i}/k_{B}T}} \over {Z(T)}}$

where $k_{B}$ is the Boltzmann constant, T is temperature (assumed to be a sharply well-defined quantity), $g_{i}$ is the degeneracy, or number of states having energy $E_{i}$, N is the total number of particles:

$N=\sum _{i}N_{i}\,$

and Z(T) is called the partition function, which can be seen to be equal to

$Z(T)=\sum _{i}g_{i}e^{-E_{i}/k_{B}T}.$

Chemical kinetics

• Chemical kinetics
• Reaction rate
• Rate law
• Rate equation

	Zero Order	First Order	Second Order	n-th Order
Rate Law	$-{\frac {d[A]}{dt}}=k$	$-{\frac {d[A]}{dt}}=k[A]$	$-{\frac {d[A]}{dt}}=k[A]^{2}$	$-{\frac {d[A]}{dt}}=k[A]^{n}$
Integrated Rate Law	$\ [A]=[A]_{0}-kt$	$\ [A]=[A]_{0}e^{-kt}$	${\frac {1}{[A]}}={\frac {1}{[A]_{0}}}+kt$	${\frac {1}{[A]^{n-1}}}={\frac {1}{{[A]_{0}}^{n-1}}}+(n-1)kt$ [Except first order]
Units of Rate Constant $\ k$	${\frac {M}{s}}$	${\frac {1}{s}}$	${\frac {1}{M\cdot s}}$	${\frac {1}{M^{n-1}\cdot s}}$
Linear Plot to determine $\ k$	$[A]\ {\mbox{vs.}}\ t$	$\ln([A])\ {\mbox{vs.}}\ t$	${\frac {1}{[A]}}\ {\mbox{vs.}}\ t$	${\frac {1}{[A]^{n-1}}}\ {\mbox{vs.}}\ t$ [Except first order]
Half-life	$t_{1/2}={\frac {[A]_{0}}{2k}}$	$t_{1/2}={\frac {\ln(2)}{k}}$	$t_{1/2}={\frac {1}{[A]_{0}k}}$	$t_{1/2}={\frac {2^{n-1}-1}{(n-1)k[A_{0}]^{n-1}}}$ [Except first order]

• Half-life

It can be shown that, for exponential decay, the half-life $t_{1/2}$ obeys this relation:

$t_{1/2}={\frac {\ln(2)}{\lambda }}$

where

• $\ln(2)$ is the natural logarithm of 2 (approximately 0.693), and
• λ is the decay constant, a positive constant used to describe the rate of exponential decay.

The half-life is related to the mean lifetime τ by the following relation:

$t_{1/2}=\ln(2)\cdot \tau$

• Activation energy
• Arrhenius equation

In short, the Arrhenius equation is an expression that shows the dependence of the rate constant k of chemical reactions on the temperature T (in Kelvin) and activation energy E_a, as shown below:

$k=Ae^{{-E_{a}}/{RT}}$

where A is the pre-exponential factor or simply the prefactor and R is the gas constant.

Chromatography

• Chromatography
• van Deemter equation

The Van Deemter equation for the plate height (H) is:

$H=A+{\frac {B}{u}}+C\cdot u$

Where

• A = Eddy-diffusion
• B = Longitudinal diffusion
• C = mass transfer kinetics of the analyte between mobile and stationary phase
• u = Linear Velocity.

A is equal to the multiple paths taken by the chemical compound, in open tubular capillaries this term will be zero as there are no multiple paths. The multiple paths occur in packed columns where several routes through the column packing, which results in band spreading.

B/u is equal to the longitudinal diffusion of the particles of the compound.

Cu is equal to the equilibration point. In a column, there is an interaction between the mobile and stationary phases, Cu accounts for this.

Electronic transitions

• Molecular electronic transition
• Selection rules for infrared (vibrational) spectroscopy
• Spectral lines
• Rydberg formula

${\frac {1}{\lambda _{\mathrm {vac} }}}=R_{\mathrm {H} }\left({\frac {1}{n_{1}^{2}}}-{\frac {1}{n_{2}^{2}}}\right)$

Where

$\lambda _{\mathrm {vac} }$ is the wavelength of the light emitted in vacuum,

$R_{\mathrm {H} }$ is the Rydberg constant for hydrogen,

$n_{1}$ and $n_{2}$ are integers such that $n_{1}<n_{2}$,

By setting $n_{1}$ to 1 and letting $n_{2}$ run from 2 to infinity, the spectral lines known as the Lyman series converging to 91nm are obtained, in the same manner:

$n_{1}$	$n_{2}$	Name	Converge toward
1	$2\rightarrow \infty$	Lyman series	91.13 nm
2	$3\rightarrow \infty$	Balmer series	364.51 nm
3	$4\rightarrow \infty$	Paschen series	820.14 nm
4	$5\rightarrow \infty$	Brackett series	1458.03 nm
5	$6\rightarrow \infty$	Pfund series	2278.17 nm
6	$7\rightarrow \infty$	Humphreys series	3280.56 nm

The Lyman series is in the ultraviolet while the Balmer series is in the visible and the Paschen, Brackett, Pfund, and Humphreys series are in the infrared.

Quantum mechanics

• Quantum mechanics
• Eigenfunction and eigenvalue
• Quantum mechanical operator
• Schrodinger equation
• Particle in a box in three dimensions:

The same separation of variables technique can be applied to the three-dimensional case to give the energy eigenfunctions:

$\psi _{n_{x},n_{y},n_{z}}={\sqrt {\frac {8}{L_{x}L_{y}L_{z}}}}\sin \left({\frac {n_{x}\pi x}{L_{x}}}\right)\sin \left({\frac {n_{y}\pi y}{L_{y}}}\right)\sin \left({\frac {n_{z}\pi z}{L_{z}}}\right)\quad (22)$

$E_{n_{x},n_{y},n_{z}}={\frac {\hbar ^{2}\pi ^{2}}{2m}}\left[\left({\frac {n_{x}}{L_{x}}}\right)^{2}+\left({\frac {n_{y}}{L_{y}}}\right)^{2}+\left({\frac {n_{z}}{L_{z}}}\right)^{2}\right]\quad (23)$

with

$n_{i}=1,2,3,\ldots$

• Rigid rotor

Other

• Oxidation state
• Isoelectronicity
• Crystal structure
• Bond enthalpy or bond energy
• Complex (chemistry)
  • low spin and high spin
• Dipole moment and dielectric constant
• Melting point and boiling point
• Jahn-Teller effect
• Huckel's rule
• Ziegler–Natta catalyst
• Lindemann mechanism and the steady-state approximation
• Chain reaction
• Photoelectric effect
• Adenosine diphosphate and adenosine triphosphate
• Octet rule and 18 electron rule
• Semiconductors
  • n-type semiconductor
  • p-type semiconductor
• Photochemistry
• Franck-Condon principle
• Hydrogen bonding

Molecules

• glyceride
• peptide
• nucleotide
• saccharide
• terpene