User:Ben Mills/Sandpit

Ideal gas
$$ pV = nRT \,\!$$

van der Waals' equation
$$p = \frac{nRT}{V-nb} - a \left ( \frac{n}{V} \right )^2 $$

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

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

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

$$\mbox{p}^6 = \mbox{p}_x^2 \quad \mbox{p}_y^2 \quad \mbox{p}_z^2 \,\!$$

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

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, HbX, 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
$$ v = f \lambda \,\!$$

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

$$ E = h f \,\!$$

$$ E = h \nu \,\!$$

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

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

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

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

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

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

$$ p = mv \,\!$$

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

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

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