User:Ben Mills/Sandpit

Gas laws

Ideal gas

$pV=nRT\,\!$

van der Waals' equation

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

Electron configs

$1{\mbox{s}}^{2}2{\mbox{s}}^{2}2{\mbox{p}}^{6}3{\mbox{s}}^{2}3{\mbox{p}}^{6}3{\mbox{d}}^{10}4{\mbox{s}}^{2}4{\mbox{p}}^{6}4{\mbox{d}}^{10}4{\mbox{f}}^{14}\,\!$

$1{\mbox{s}}^{2}\,2{\mbox{s}}^{2}\,2{\mbox{p}}^{6}\,3{\mbox{s}}^{2}\,3{\mbox{p}}^{6}\,3{\mbox{d}}^{10}\,4{\mbox{s}}^{2}\,4{\mbox{p}}^{6}\,4{\mbox{d}}^{10}\,4{\mbox{f}}^{14}\,\!$

$[{\mbox{He}}]\,2{\mbox{s}}^{2}\,2{\mbox{p}}_{x}^{2}\,2{\mbox{p}}_{y}^{2}\,2{\mbox{p}}_{z}^{2}\,3{\mbox{s}}^{2}\,3{\mbox{p}}^{6}\,3{\mbox{d}}_{z^{2}}^{2}\,3{\mbox{d}}_{x^{2}-y^{2}}^{2}\,3{\mbox{d}}_{xy}^{2}\,3{\mbox{d}}_{xz}^{2}\,3{\mbox{d}}_{yz}^{2}\,\!$

${\mbox{p}}^{6}={\mbox{p}}_{x}^{2}\quad {\mbox{p}}_{y}^{2}\quad {\mbox{p}}_{z}^{2}\,\!$

${\mbox{d}}^{10}={\mbox{d}}_{z^{2}}^{2}\quad {\mbox{d}}_{x^{2}-y^{2}}^{2}\quad {\mbox{d}}_{xy}^{2}\quad {\mbox{d}}_{xz}^{2}\quad {\mbox{d}}_{yz}^{2}\,\!$

Stoichiometry

Reactions of acids with bases

acid + base → salt + water

AH + BOH → AB + H₂O

HCl + NaOH → NaCl + H₂O

$a{\mbox{H}}_{b}{\mbox{X}}+b{\mbox{M(OH)}}_{a}\to {\mbox{M}}_{b}{\mbox{X}}_{a}+{\mbox{H}}_{2}{\mbox{O}}\,\!$

Reactions of acids with metals

acid + metal → salt + hydrogen

2AH + 2M → 2MA + H₂

2HCl + 2Na → 2NaCl + H₂

2AH + M → MA₂ + H₂

2HCl + Mg → MgCl₂ + H₂

6AH + 2M → 2MA₃ + 3H₂

6HCl + 2Al → 2AlCl₃ + 3H₂

For any acid, H_bX, and any metal, M, with valency a, the stoichiometry of the metal-acid reaction is as follows:

$a{\mbox{H}}_{b}{\mbox{X}}+b{\mbox{M}}\to {\mbox{M}}_{b}{\mbox{X}}_{a}+{\frac {ab}{2}}{\mbox{H}}_{2}\,\!$

Reactions of acids with carbonates

acid + carbonate → salt + water + carbon dioxide

2AH + M₂CO₃ → 2MA + H₂O + CO₂

2HCl + Na₂CO₃ → 2NaCl + H₂O + CO₂

2AH + MCO₃ → MA₂ + H₂O + CO₂

2HCl + CaCO₃ → CaCl₂ + H₂O + CO₂

${\frac {2}{b}}{\mbox{H}}_{b}{\mbox{X}}+{\mbox{M}}_{\frac {2}{a}}{\mbox{CO}}_{3}\to {\mbox{MX}}_{\frac {a}{b}}+{\mbox{CO}}_{2}+{\mbox{H}}_{2}{\mbox{O}}\,\!$

Precipitation reactions

$b{\mbox{M}}^{a+}{\mbox{(aq)}}+a{\mbox{X}}^{b-}{\mbox{(aq)}}\to {\mbox{M}}_{b}{\mbox{X}}_{a}{\mbox{(s)}}\,\!$

Quantum mechanics

${\hat {H}}\psi =E\psi \,\!$

${\hat {H}}=-{\frac {\hbar ^{2}}{2m_{e}}}\nabla ^{2}-{\frac {e^{2}}{4\pi \epsilon _{0}r}}\,\!$

$\nabla ^{2}=\left({\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}+{\frac {\partial ^{2}}{\partial z^{2}}}\right)\,\!$

Schrödinger equation for a one-electron system

$\psi _{n,l,m}(x,y,z)\,\!$

$E_{n}=-R_{h}{\frac {Z^{2}}{n^{2}}}\,\!$

$R_{h}={\frac {m_{e}e^{4}}{8\epsilon _{0}^{2}h^{2}}}\,\!$

GCSE maths

$x^{a}\times x^{b}=x^{a+b}\,\!$

${\frac {x^{a}}{x^{b}}}=x^{a-b}\,\!$

$(x^{a})^{b}=x^{ab}\,\!$

$x^{-1}={\frac {1}{x}}\,\!$

$x^{-n}={\frac {1}{x^{n}}}\,\!$

$x^{\frac {1}{2}}={\sqrt {x}}\,\!$

$x^{\frac {1}{n}}={\sqrt[{n}]{x}}\,\!$

$x^{-{\frac {1}{n}}}={\frac {1}{\sqrt[{n}]{x}}}\,\!$

Thermodynamics

$S=k\ln w\,\!$

$\Delta G=\Delta H-T\Delta S\,\!$

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

$C_{p}=\left({\frac {\partial H}{\partial T}}\right)_{p}$

$C_{V}=\left({\frac {\partial U}{\partial T}}\right)_{V}$

Physics

$v=f\lambda \,\!$

$c=\lambda \nu \,\!$

$E=hf\,\!$

$E=h\nu \,\!$

$t_{\frac {1}{2}}={\frac {\ln {2}}{\lambda }}$

$\lambda ={\frac {\ln {2}}{t_{\frac {1}{2}}}}$

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

${\frac {dN}{dt}}=-\lambda t$

$I(x)=I_{0}e^{-\mu x}\,\!$

$\lambda ={\frac {h}{p}}$

$p=mv\,\!$

$\mathbf {p} =m\mathbf {v} \,\!$

$\mathbf {F} =m{\frac {d^{2}\mathbf {r} }{dt^{2}}}\,\!$

$\mathbf {F} ={\frac {d\mathbf {p} }{dt}}\,\!$