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

$v = f \lambda \,\!$

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

$E = h f \,\!$

$E = h \nu \,\!$

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

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

$N(t) = N_0 e^{- \lambda t} \,\!$

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

$I(x) = I_0 e^{- \mu x} \,\!$

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

$p = mv \,\!$

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

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

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