化学公式Chemistry formulas by discipline
304 条公式 · 12 个学科 · 24 个主题304 formulas · 12 disciplines · 24 topics
普通化学General chemistry
化学计量与基本定律 Stoichiometry & basic laws
物质的量是联系宏观质量、体积与微观粒子数的桥梁。本节公式是全部定量计算的起点。
The amount of substance links macroscopic mass and volume to particle counts. Every quantitative calculation starts here.
- 质量守恒定律Law of conservation of mass\(\sum m_{\text{反应物}} = \sum m_{\text{生成物}}\)\(\sum m_{\text{reactants}} = \sum m_{\text{products}}\)
- 相对原子质量Relative atomic mass\(A_r(\mathrm{X}) = \frac{m(\mathrm{X})}{\tfrac{1}{12}\,m(^{12}\mathrm{C})}\)
- 溶质质量分数Mass fraction (percentage by mass)\(w = \frac{m_{\text{溶质}}}{m_{\text{溶液}}}\times 100\% = \frac{m_{\text{溶质}}}{m_{\text{溶质}} + m_{\text{溶剂}}}\times 100\%\)\(w = \frac{m_{\text{solute}}}{m_{\text{solution}}}\times 100\% = \frac{m_{\text{solute}}}{m_{\text{solute}} + m_{\text{solvent}}}\times 100\%\)
- 物质的量与质量Amount of substance from mass\(n = \frac{m}{M}\)
- 物质的量与粒子数Amount of substance from number of particles\(n = \frac{N}{N_A}, \qquad N_A = 6.02214076\times10^{23}\ \mathrm{mol^{-1}}\)
- 气体摩尔体积Molar volume of a gas\(n = \frac{V}{V_m}, \qquad V_m \approx 22.4\ \mathrm{L\,mol^{-1}}\ (0\,{}^\circ\mathrm{C},\ 101.325\,\mathrm{kPa})\)
- 物质的量浓度Molar concentration (molarity)\(c_{\mathrm{B}} = \frac{n_{\mathrm{B}}}{V}\)
- 稀释公式Dilution equation\(c_1 V_1 = c_2 V_2\)
- 物质的量浓度与质量分数换算Molarity–mass fraction conversion\(c = \frac{1000\,\rho\, w}{M}\)
- 平均摩尔质量Average molar mass of a mixture\(\bar M = \frac{m_{\text{总}}}{n_{\text{总}}} = \sum_i x_i M_i\)\(\bar M = \frac{m_{\text{total}}}{n_{\text{total}}} = \sum_i x_i M_i\)
- 产率与转化率Percent yield and conversion\(\text{产率} = \frac{m_{\text{实际}}}{m_{\text{理论}}}\times100\%,\qquad \alpha = \frac{n_{\text{已反应}}}{n_{\text{起始}}}\times 100\%\)\(\text{yield} = \frac{m_{\text{actual}}}{m_{\text{theoretical}}}\times100\%,\qquad \alpha = \frac{n_{\text{reacted}}}{n_{\text{initial}}}\times 100\%\)
- 原子经济性Atom economy\(\mathrm{AE} = \frac{M_{\text{目标产物}}}{\sum \nu_i M_{\text{反应物},i}}\times 100\%\)\(\mathrm{AE} = \frac{M_{\text{desired product}}}{\sum \nu_i M_{\text{reactants},i}}\times 100\%\)
- 滴定计量关系Titration stoichiometry\(\frac{c_\mathrm{A} V_\mathrm{A}}{\nu_\mathrm{A}} = \frac{c_\mathrm{B} V_\mathrm{B}}{\nu_\mathrm{B}}\)
原子结构与量子化起源 Atomic structure & the origins of quantisation
光子、物质波、不确定性原理、玻尔模型与氢光谱、莫塞莱定律、屏蔽与有效核电荷，以及质能关系。
Photons, matter waves, the uncertainty principle, the Bohr model and hydrogen spectrum, Moseley’s law, shielding and effective nuclear charge, and mass–energy equivalence.
- 光的波长、频率与波数Wavelength, frequency and wavenumber of light\(c = \lambda\nu,\qquad \tilde\nu = \frac1\lambda = \frac\nu c\)
- 光子能量（普朗克–爱因斯坦关系）Photon energy (Planck–Einstein relation)\(E = h\nu = \frac{hc}{\lambda} = hc\tilde\nu,\qquad E/\mathrm{eV} \approx \frac{1239.84}{\lambda/\mathrm{nm}}\)
- 光电效应方程Photoelectric equation\(E_{k,\max} = \tfrac12 m_ev_{\max}^2 = h\nu - \Phi,\qquad \nu_0 = \frac\Phi h\)
- 德布罗意波长de Broglie wavelength\(\lambda = \frac hp = \frac{h}{mv} = \frac{h}{\sqrt{2mE_k}}\)
- 海森堡不确定关系Heisenberg uncertainty principle\(\Delta x\,\Delta p_x \ge \frac\hbar2,\qquad \Delta E\,\Delta t \gtrsim \frac\hbar2\)
- 玻尔模型的能级Energy levels of the Bohr model\(E_n = -\frac{m_e e^4}{8\varepsilon_0^2h^2}\cdot\frac{Z^2}{n^2} = -13.6\ \mathrm{eV}\cdot\frac{Z^2}{n^2},\qquad r_n = \frac{n^2a_0}{Z}\)
- 玻尔半径Bohr radius\(a_0 = \frac{4\pi\varepsilon_0\hbar^2}{m_e e^2} = 5.29177\times10^{-11}\ \mathrm m\)
- 里德伯公式（氢原子光谱）Rydberg formula (hydrogen spectrum)\(\tilde\nu = \frac1\lambda = R_\mathrm HZ^2\left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right),\qquad R_\mathrm H = 1.09678\times10^{7}\ \mathrm m^{-1}\)
- 莫塞莱定律Moseley's law\(\sqrt\nu = k\,(Z - \sigma),\qquad \nu_{K\alpha} \approx \frac34\,cR_\infty(Z - 1)^2\)
- 斯莱特规则与有效核电荷Slater's rules and effective nuclear charge\(Z_\text{eff} = Z - S,\qquad E_n \approx -13.6\ \mathrm{eV}\left(\frac{Z_\text{eff}}{n^*}\right)^2,\quad r \approx \frac{n^{*2}a_0}{Z_\text{eff}}\)
- 电离能Ionisation energy\(\mathrm{M(g)} \to \mathrm{M^+(g)} + e^-,\quad \Delta H \approx I_1;\qquad I \approx 13.6\ \mathrm{eV}\left(\frac{Z_\text{eff}}{n^*}\right)^2\)
- 质能关系Mass–energy equivalence\(E = mc^2,\qquad \Delta E = \Delta m\,c^2\)
化学键与分子结构 Chemical bonding & molecular structure
形式电荷、键级、价层电子对、偶极矩、电负性标度、键能估算与玻恩–哈伯循环。
Formal charge, bond order, VSEPR electron pairs, dipole moments, electronegativity scales, bond-energy estimates and the Born–Haber cycle.
- 形式电荷Formal charge\(\mathrm{FC} = V - N - \frac{B}{2}\)
- 键级（分子轨道理论）Bond order (molecular orbital theory)\(\text{键级} = \frac{n_\text{成键} - n_\text{反键}}{2}\)\(\text{bond order} = \frac{n_\text{bonding} - n_\text{antibonding}}{2}\)
- 价层电子对数（VSEPR）Steric number (VSEPR)\(\mathrm{SN} = \frac{V + L - q}{2}\ \ (\text{对 } \mathrm{AX}_m\text{，} V \text{ 为中心原子价电子数}),\qquad \mathrm{SN} = n_\sigma + n_\text{孤对}\)\(\mathrm{SN} = \frac{V + L - q}{2}\ \ (\text{for} \mathrm{AX}_m\text{, } V \text{valence e⁻ of central atom}),\qquad \mathrm{SN} = n_\sigma + n_\text{lone pairs}\)
- 偶极矩Dipole moment\(\boldsymbol\mu = \sum_i q_i\mathbf r_i,\qquad \mu = q\,d\ (\text{两点电荷}),\qquad 1\ \mathrm D = 3.336\times10^{-30}\ \mathrm{C\,m}\)\(\boldsymbol\mu = \sum_i q_i\mathbf r_i,\qquad \mu = q\,d\ (\text{two point charges}),\qquad 1\ \mathrm D = 3.336\times10^{-30}\ \mathrm{C\,m}\)
- 键的离子性百分数Percent ionic character\(\%\text{离子性} = \frac{\mu_\text{实测}}{e\,d}\times100\%,\qquad \%\text{离子性}_\text{鲍林} = \left(1 - e^{-(\Delta\chi)^2/4}\right)\times100\%\)\(\%\text{ionic character} = \frac{\mu_\text{measured}}{e\,d}\times100\%,\qquad \%\text{ionic character}_\text{Pauling} = \left(1 - e^{-(\Delta\chi)^2/4}\right)\times100\%\)
- 鲍林电负性Pauling electronegativity\(|\chi_\mathrm A - \chi_\mathrm B| = \sqrt{\frac{\Delta}{\mathrm{eV}}},\qquad \Delta = D(\mathrm{A{-}B}) - \sqrt{D(\mathrm{A{-}A})\,D(\mathrm{B{-}B})}\)
- 马利肯电负性Mulliken electronegativity\(\chi_\mathrm M = \frac{I + A}{2},\qquad \chi_\mathrm P \approx 1.35\sqrt{\chi_\mathrm M/\mathrm{eV}} - 1.37\)
- 阿莱–罗乔电负性Allred–Rochow electronegativity\(\chi_\mathrm{AR} = 0.744 + 0.359\,\frac{Z_\text{eff}}{(r_\text{cov}/\text{Å})^2}\)
- 由键焓估算反应焓Reaction enthalpy from bond enthalpies\(\Delta_rH \approx \sum D(\text{断开的键}) - \sum D(\text{形成的键})\)\(\Delta_rH \approx \sum D(\text{bonds broken}) - \sum D(\text{bonds formed})\)
- 玻恩–哈伯循环Born–Haber cycle\(\Delta_fH^\circ(\mathrm{MX}) = \Delta_\text{sub}H(\mathrm M) + I(\mathrm M) + \tfrac12D(\mathrm X_2) - E_\text{ea}(\mathrm X) + U_L\)
- 伦纳德–琼斯势Lennard-Jones potential\(V(r) = 4\varepsilon\left[\left(\frac{\sigma}{r}\right)^{12} - \left(\frac{\sigma}{r}\right)^6\right],\qquad r_\text{min} = 2^{1/6}\sigma,\quad V(r_\text{min}) = -\varepsilon\)
物理化学Physical chemistry
气体 Gases
从经验气体定律到理想气体状态方程，再到分子运动论和真实气体修正。
From the empirical gas laws to the ideal-gas equation, kinetic theory and real-gas corrections.
- 玻意耳定律Boyle's law\(p_1V_1 = p_2V_2 \quad (T, n\ \text{一定})\)\(p_1V_1 = p_2V_2 \quad (T, n\ \text{const.})\)
- 查理定律与盖-吕萨克定律Charles's and Gay-Lussac's laws\(\frac{V_1}{T_1} = \frac{V_2}{T_2}\ (p\ \text{一定}), \qquad \frac{p_1}{T_1} = \frac{p_2}{T_2}\ (V\ \text{一定})\)\(\frac{V_1}{T_1} = \frac{V_2}{T_2}\ (p\ \text{const.}), \qquad \frac{p_1}{T_1} = \frac{p_2}{T_2}\ (V\ \text{const.})\)
- 理想气体状态方程Ideal gas law\(pV = nRT\)
- 道尔顿分压定律Dalton's law of partial pressures\(p_{\text{总}} = \sum_i p_i, \qquad p_i = x_i\, p_{\text{总}}\)\(p_{\text{total}} = \sum_i p_i, \qquad p_i = x_i\, p_{\text{total}}\)
- 格雷姆扩散定律Graham's law of effusion\(\frac{r_1}{r_2} = \sqrt{\frac{M_2}{M_1}}\)
- 气体分子运动论压强公式Kinetic-theory pressure\(p = \frac{1}{3}\frac{N}{V} m \overline{u^2}, \qquad \frac{1}{2}m\overline{u^2} = \frac{3}{2}k_\mathrm{B}T\)
- 分子的三种特征速率Characteristic molecular speeds\(v_p = \sqrt{\frac{2RT}{M}},\quad \bar v = \sqrt{\frac{8RT}{\pi M}},\quad u_{\mathrm{rms}} = \sqrt{\frac{3RT}{M}}\)
- 麦克斯韦–玻尔兹曼速率分布Maxwell–Boltzmann speed distribution\(f(v) = 4\pi \left(\frac{m}{2\pi k_\mathrm{B}T}\right)^{3/2} v^2 \exp\!\left(-\frac{mv^2}{2k_\mathrm{B}T}\right)\)
- 平均自由程与碰撞频率Mean free path and collision frequency\(\lambda = \frac{k_\mathrm{B}T}{\sqrt2\,\sigma\, p}, \qquad z = \frac{\sqrt2\,\sigma\,\bar v\, p}{k_\mathrm{B}T}, \qquad \sigma = \pi d^2\)
- 气压公式（玻尔兹曼高度分布）Barometric formula\(p(h) = p_0 \exp\!\left(-\frac{Mgh}{RT}\right)\)
- 范德华方程van der Waals equation\(\left(p + \frac{an^2}{V^2}\right)(V - nb) = nRT\)
- 压缩因子Compressibility factor\(Z = \frac{pV_m}{RT}\)
- 维里方程Virial equation of state\(Z = \frac{pV_m}{RT} = 1 + \frac{B(T)}{V_m} + \frac{C(T)}{V_m^2} + \cdots\)
溶液与依数性 Solutions & colligative properties
浓度的各种表示、理想溶液定律，以及只与溶质粒子数有关的依数性。
Concentration scales, ideal-solution laws and the colligative properties that depend only on the number of solute particles.
- 摩尔分数Mole fraction\(x_i = \frac{n_i}{\sum_j n_j}, \qquad \sum_i x_i = 1\)
- 质量摩尔浓度Molality\(b_\mathrm{B} = \frac{n_\mathrm{B}}{m_\mathrm{A}}\)
- 拉乌尔定律Raoult's law\(p_\mathrm{A} = x_\mathrm{A}\, p_\mathrm{A}^*\)
- 亨利定律Henry's law\(p_\mathrm{B} = k_{x}\, x_\mathrm{B} \quad\text{或}\quad c_\mathrm{B} = k_\mathrm{H}\, p_\mathrm{B}\)\(p_\mathrm{B} = k_{x}\, x_\mathrm{B} \quad\text{or}\quad c_\mathrm{B} = k_\mathrm{H}\, p_\mathrm{B}\)
- 沸点升高Boiling-point elevation\(\Delta T_b = i\,K_b\, b_\mathrm{B}, \qquad K_b = \frac{R\,T_b^{*2} M_\mathrm{A}}{\Delta_\mathrm{vap}H_m}\)
- 凝固点降低Freezing-point depression\(\Delta T_f = i\,K_f\, b_\mathrm{B}, \qquad K_f = \frac{R\,T_f^{*2} M_\mathrm{A}}{\Delta_\mathrm{fus}H_m}\)
- 渗透压（范特霍夫公式）Osmotic pressure (van 't Hoff equation)\(\Pi = i\,c_\mathrm{B}RT\)
- 范特霍夫因子van 't Hoff factor\(i = 1 + \alpha(\nu - 1)\)
- 离子强度Ionic strength\(I = \frac12 \sum_i c_i z_i^2\)
- 德拜–休克尔极限定律Debye–Hückel limiting law\(\lg \gamma_\pm = -A\,|z_+ z_-|\sqrt{I}\)
- 活度与活度系数Activity and activity coefficient\(a_i = \gamma_i \frac{c_i}{c^\circ}, \qquad \mu_i = \mu_i^\circ + RT\ln a_i\)
- 能斯特分配定律Nernst distribution law\(K_D = \frac{c_\mathrm{B}(\text{有机相})}{c_\mathrm{B}(\text{水相})}\)\(K_D = \frac{c_\mathrm{B}(\text{organic})}{c_\mathrm{B}(\text{aqueous})}\)
化学热力学 Chemical thermodynamics
热和功、热力学第一定律与焓、热化学、熵与第二、第三定律、吉布斯能与亥姆霍兹能、麦克斯韦关系、化学势与相平衡。
Heat and work, the first law and enthalpy, thermochemistry, entropy and the second and third laws, Gibbs and Helmholtz energies, Maxwell relations, chemical potential and phase equilibria.
- 热量计算Heat and heat capacity\(Q = c\,m\,\Delta T = C\,\Delta T = nC_m\Delta T\)
- 热力学第一定律First law of thermodynamics\(\Delta U = q + w,\qquad dU = \delta q + \delta w\)
- 体积功Pressure–volume work\(w = -\int_{V_1}^{V_2}p_\text{ex}\,dV,\qquad w = -p_\text{ex}\Delta V\ (\text{恒外压})\)\(w = -\int_{V_1}^{V_2}p_\text{ex}\,dV,\qquad w = -p_\text{ex}\Delta V\ (\text{const. external p})\)
- 焓Enthalpy\(H = U + pV,\qquad \Delta H = q_p\ (\text{恒压、只做体积功})\)\(H = U + pV,\qquad \Delta H = q_p\ (\text{const. p , pV work only})\)
- ΔH 与 ΔU 的关系Relation between ΔH and ΔU\(\Delta_rH = \Delta_rU + \Delta n_gRT,\qquad C_{p,m} - C_{V,m} = R\ (\text{理想气体})\)\(\Delta_rH = \Delta_rU + \Delta n_gRT,\qquad C_{p,m} - C_{V,m} = R\ (\text{ideal gas})\)
- 盖斯定律Hess's law\(\Delta_rH = \sum_i \Delta H_i\ (\text{各步之和，与路径无关})\)\(\Delta_rH = \sum_i \Delta H_i\ (\text{sum of steps , path-independent})\)
- 由标准生成焓求反应焓Reaction enthalpy from standard enthalpies of formation\(\Delta_rH^\circ = \sum\nu\,\Delta_fH^\circ(\text{产物}) - \sum\nu\,\Delta_fH^\circ(\text{反应物})\)\(\Delta_rH^\circ = \sum\nu\,\Delta_fH^\circ(\text{products}) - \sum\nu\,\Delta_fH^\circ(\text{reactants})\)
- 基尔霍夫定律Kirchhoff's law of thermochemistry\(\Delta_rH(T_2) = \Delta_rH(T_1) + \int_{T_1}^{T_2}\Delta_rC_p\,dT,\qquad \left(\frac{\partial\Delta_rH}{\partial T}\right)_p = \Delta_rC_p\)
- 理想气体可逆等温过程Reversible isothermal process of an ideal gas\(w = -nRT\ln\frac{V_2}{V_1} = -nRT\ln\frac{p_1}{p_2},\qquad \Delta U = 0,\quad q = -w,\quad \Delta S = nR\ln\frac{V_2}{V_1}\)
- 理想气体可逆绝热过程（泊松方程）Reversible adiabatic process (Poisson's equations)\(pV^\gamma = \text{常数},\qquad TV^{\gamma-1} = \text{常数},\qquad \gamma = \frac{C_{p,m}}{C_{V,m}},\qquad w = \Delta U = nC_{V,m}(T_2 - T_1)\)\(pV^\gamma = \text{const.},\qquad TV^{\gamma-1} = \text{const.},\qquad \gamma = \frac{C_{p,m}}{C_{V,m}},\qquad w = \Delta U = nC_{V,m}(T_2 - T_1)\)
- 卡诺热机效率Carnot efficiency\(\eta = \frac{-w}{q_h} = 1 - \frac{T_c}{T_h},\qquad \frac{q_h}{T_h} + \frac{q_c}{T_c} = 0\)
- 熵与克劳修斯不等式（第二定律）Entropy and the Clausius inequality (second law)\(dS = \frac{\delta q_\text{rev}}{T},\qquad dS \ge \frac{\delta q}{T},\qquad \Delta S_\text{孤立} \ge 0\)\(dS = \frac{\delta q_\text{rev}}{T},\qquad dS \ge \frac{\delta q}{T},\qquad \Delta S_\text{isolated} \ge 0\)
- 变温过程的熵变Entropy change on heating\(\Delta S = \int_{T_1}^{T_2}\frac{C_p}{T}dT = C_p\ln\frac{T_2}{T_1}\ (\text{恒压，} C_p\text{ 恒定}),\qquad \Delta S = nC_{V,m}\ln\frac{T_2}{T_1} + nR\ln\frac{V_2}{V_1}\)\(\Delta S = \int_{T_1}^{T_2}\frac{C_p}{T}dT = C_p\ln\frac{T_2}{T_1}\ (\text{const. p ,} C_p\text{const.}),\qquad \Delta S = nC_{V,m}\ln\frac{T_2}{T_1} + nR\ln\frac{V_2}{V_1}\)
- 相变熵Entropy of phase transition\(\Delta_\text{trs}S = \frac{\Delta_\text{trs}H}{T_\text{trs}}\)
- 玻尔兹曼熵公式Boltzmann entropy formula\(S = k_\mathrm B\ln W\)
- 热力学第三定律与规定熵Third law and absolute entropy\(\lim_{T\to0}S = 0\ (\text{完美晶体}),\qquad S(T) = \int_0^T\frac{C_p}{T}dT + \sum\frac{\Delta_\text{trs}H}{T_\text{trs}},\qquad C_p \approx aT^3\ (\text{德拜外推})\)\(\lim_{T\to0}S = 0\ (\text{perfect crystal}),\qquad S(T) = \int_0^T\frac{C_p}{T}dT + \sum\frac{\Delta_\text{trs}H}{T_\text{trs}},\qquad C_p \approx aT^3\ (\text{Debye extrapolation})\)
- 吉布斯自由能（吉布斯–亥姆霍兹关系）Gibbs free energy\(G = H - TS,\qquad \Delta G = \Delta H - T\Delta S\ (\text{恒温}),\qquad \Delta G_{T,p} \le 0\ (\text{自发})\)\(G = H - TS,\qquad \Delta G = \Delta H - T\Delta S\ (\text{const. T}),\qquad \Delta G_{T,p} \le 0\ (\text{spontaneous})\)
- 亥姆霍兹自由能Helmholtz free energy\(A = U - TS,\qquad \Delta A_{T,V} \le 0\ (\text{自发}),\qquad w_\text{max} = \Delta A\ (\text{恒温})\)\(A = U - TS,\qquad \Delta A_{T,V} \le 0\ (\text{spontaneous}),\qquad w_\text{max} = \Delta A\ (\text{const. T})\)
- 吉布斯–亥姆霍兹方程Gibbs–Helmholtz equation\(\left(\frac{\partial(G/T)}{\partial T}\right)_p = -\frac{H}{T^2},\qquad \left(\frac{\partial(\Delta G/T)}{\partial T}\right)_p = -\frac{\Delta H}{T^2}\)
- 热力学基本方程Fundamental thermodynamic equations\(dU = T\,dS - p\,dV,\quad dH = T\,dS + V\,dp,\quad dA = -S\,dT - p\,dV,\quad dG = -S\,dT + V\,dp\)
- 麦克斯韦关系式Maxwell relations\(\left(\frac{\partial T}{\partial V}\right)_S = -\left(\frac{\partial p}{\partial S}\right)_V,\ \left(\frac{\partial T}{\partial p}\right)_S = \left(\frac{\partial V}{\partial S}\right)_p,\ \left(\frac{\partial S}{\partial V}\right)_T = \left(\frac{\partial p}{\partial T}\right)_V,\ \left(\frac{\partial S}{\partial p}\right)_T = -\left(\frac{\partial V}{\partial T}\right)_p\)
- 化学势Chemical potential\(\mu_i = \left(\frac{\partial G}{\partial n_i}\right)_{T,p,n_{j\ne i}},\qquad \mu_i = \mu_i^\circ + RT\ln a_i\)
- 吉布斯–杜亥姆方程Gibbs–Duhem equation\(\sum_i n_i\,d\mu_i = -S\,dT + V\,dp \xrightarrow{T,p\ \text{恒定}} \sum_i x_i\,d\mu_i = 0\)\(\sum_i n_i\,d\mu_i = -S\,dT + V\,dp \xrightarrow{T,p\ \text{const.}} \sum_i x_i\,d\mu_i = 0\)
- 克拉珀龙方程Clapeyron equation\(\frac{dp}{dT} = \frac{\Delta_\text{trs}H}{T\,\Delta_\text{trs}V}\)
- 克劳修斯–克拉珀龙方程Clausius–Clapeyron equation\(\frac{d\ln p}{dT} = \frac{\Delta_\text{vap}H}{RT^2},\qquad \ln\frac{p_2}{p_1} = -\frac{\Delta_\text{vap}H}{R}\left(\frac{1}{T_2} - \frac{1}{T_1}\right)\)
- 特鲁顿规则Trouton's rule\(\Delta_\text{vap}S = \frac{\Delta_\text{vap}H}{T_b} \approx 85\text{–}88\ \mathrm{J\,mol^{-1}\,K^{-1}}\)
- 吉布斯相律Gibbs phase rule\(F = C - P + 2\)
- 杠杆规则Lever rule\(\frac{n_\alpha}{n_\beta} = \frac{x_\beta - x_0}{x_0 - x_\alpha}\)
- 焦耳–汤姆孙效应Joule–Thomson effect\(\mu_\mathrm{JT} = \left(\frac{\partial T}{\partial p}\right)_H = \frac{V}{C_p}(T\alpha - 1),\qquad T_\text{inv} \approx \frac{2a}{Rb}\ (\text{范德华气体，低压})\)\(\mu_\mathrm{JT} = \left(\frac{\partial T}{\partial p}\right)_H = \frac{V}{C_p}(T\alpha - 1),\qquad T_\text{inv} \approx \frac{2a}{Rb}\ (\text{van der Waals gas , low pressure})\)
化学平衡 Chemical equilibrium
平衡常数、标准吉布斯能与平衡常数的关系、反应等温式、温度对平衡的影响、沉淀溶解平衡与勒夏特列原理的定量形式。
Equilibrium constants, their link to standard Gibbs energy, the reaction isotherm, temperature effects, solubility equilibria and the quantitative form of Le Chatelier’s principle.
- 平衡常数Equilibrium constant\(a\mathrm A + b\mathrm B \rightleftharpoons c\mathrm C + d\mathrm D:\qquad K_c = \frac{[\mathrm C]^c[\mathrm D]^d}{[\mathrm A]^a[\mathrm B]^b},\qquad K^\circ = \prod_\mathrm J a_\mathrm J^{\nu_\mathrm J}\)
- 标准吉布斯能与平衡常数Standard Gibbs energy and the equilibrium constant\(\Delta_rG^\circ = -RT\ln K^\circ,\qquad K^\circ = e^{-\Delta_rG^\circ/RT}\)
- 范特霍夫等温式（反应商）Reaction isotherm (reaction quotient)\(\Delta_rG = \Delta_rG^\circ + RT\ln Q = RT\ln\frac QK\)
- 范特霍夫方程van 't Hoff equation\(\frac{d\ln K}{dT} = \frac{\Delta_rH^\circ}{RT^2},\qquad \ln\frac{K_2}{K_1} = -\frac{\Delta_rH^\circ}{R}\left(\frac{1}{T_2} - \frac{1}{T_1}\right)\)
- Kp 与 Kc 的关系Relation between Kp and Kc\(K_p = K_c\,(RT)^{\Delta n_g},\qquad \Delta n_g = \sum\nu_\text{气态产物} - \sum\nu_\text{气态反应物}\)\(K_p = K_c\,(RT)^{\Delta n_g},\qquad \Delta n_g = \sum\nu_\text{gaseous products} - \sum\nu_\text{gaseous reactants}\)
- 溶度积Solubility product\(\mathrm M_m\mathrm X_n(s) \rightleftharpoons m\mathrm M^{n+} + n\mathrm X^{m-}:\quad K_{sp} = [\mathrm M^{n+}]^m[\mathrm X^{m-}]^n,\qquad s = \left(\frac{K_{sp}}{m^mn^n}\right)^{1/(m+n)}\)
- 同离子效应Common-ion effect\(\mathrm{AgCl\ 在\ } c\ \mathrm{mol/L\ Cl^-\ 中}:\quad s = \frac{K_{sp}}{c + s} \approx \frac{K_{sp}}{c}\)\(\mathrm{AgCl\ \text{ in }\ } c\ \mathrm{mol/L\ Cl^-\ \text{ in }}:\quad s = \frac{K_{sp}}{c + s} \approx \frac{K_{sp}}{c}\)
- 勒夏特列原理的定量形式Le Chatelier's principle, quantitatively\(\left(\frac{\partial\ln K_x}{\partial p}\right)_T = -\frac{\Delta n_g}{p},\qquad \left(\frac{\partial\ln K}{\partial T}\right)_p = \frac{\Delta_rH^\circ}{RT^2}\)
- 气相解离度Degree of dissociation in the gas phase\(\mathrm{A_2} \rightleftharpoons 2\mathrm A:\quad K_p = \frac{4\alpha^2}{1 - \alpha^2}\cdot\frac{p}{p^\circ},\qquad \alpha = \sqrt{\frac{K_p}{K_p + 4p/p^\circ}}\)
电化学 Electrochemistry
法拉第定律、电池电动势与能斯特方程、电导与离子迁移，以及电极动力学（巴特勒–福尔默、塔费尔）和双电层。
Faraday’s laws, cell EMF and the Nernst equation, conductance and ion transport, electrode kinetics (Butler–Volmer, Tafel) and the double layer.
- 法拉第电解定律Faraday's laws of electrolysis\(m = \frac{QM}{zF} = \frac{ItM}{zF},\qquad n_e = \frac{Q}{F}\)
- 电池电动势Cell potential (EMF)\(E^\circ_\text{cell} = E^\circ_\text{正极(还原)} - E^\circ_\text{负极(氧化)} = E^\circ_\text{cathode} - E^\circ_\text{anode}\)\(E^\circ_\text{cell} = E^\circ_\text{cathode ( reduction )} - E^\circ_\text{anode ( oxidation )} = E^\circ_\text{cathode} - E^\circ_\text{anode}\)
- 能斯特方程Nernst equation\(E = E^\circ - \frac{RT}{nF}\ln Q = E^\circ - \frac{0.05916\ \mathrm V}{n}\lg Q\ \ (25\,{}^\circ\mathrm C)\)
- 吉布斯能与电动势Gibbs energy and EMF\(\Delta_rG = -nFE,\qquad \Delta_rS = nF\left(\frac{\partial E}{\partial T}\right)_p,\qquad \Delta_rH = -nFE + nFT\left(\frac{\partial E}{\partial T}\right)_p\)
- 标准电动势与平衡常数Standard EMF and the equilibrium constant\(\ln K = \frac{nFE^\circ}{RT},\qquad \lg K = \frac{nE^\circ}{0.05916\ \mathrm V}\ (25\,{}^\circ\mathrm C)\)
- 拉蒂默图中电势的组合Combining potentials (Latimer diagrams)\(E^\circ_{13} = \frac{n_1E^\circ_{12} + n_2E^\circ_{23}}{n_1 + n_2}\)
- 电导率与摩尔电导率Conductivity and molar conductivity\(\kappa = \frac{1}{\rho} = \frac{K_\text{cell}}{R},\qquad \Lambda_m = \frac{\kappa}{c}\)
- 科尔劳施定律Kohlrausch's laws\(\Lambda_m = \Lambda_m^\circ - \mathcal K\sqrt c\ (\text{强电解质}),\qquad \Lambda_m^\circ = \nu_+\lambda_+^\circ + \nu_-\lambda_-^\circ\ (\text{离子独立移动})\)\(\Lambda_m = \Lambda_m^\circ - \mathcal K\sqrt c\ (\text{strong electrolyte}),\qquad \Lambda_m^\circ = \nu_+\lambda_+^\circ + \nu_-\lambda_-^\circ\ (\text{independent ion migration})\)
- 奥斯特瓦尔德稀释定律Ostwald's dilution law\(K_a = \frac{\alpha^2c}{1 - \alpha},\qquad \alpha = \frac{\Lambda_m}{\Lambda_m^\circ},\qquad \frac{1}{\Lambda_m} = \frac{1}{\Lambda_m^\circ} + \frac{c\Lambda_m}{K_a(\Lambda_m^\circ)^2}\)
- 离子迁移数Transport (transference) number\(t_+ = \frac{I_+}{I} = \frac{u_+}{u_+ + u_-} = \frac{\nu_+\lambda_+}{\Lambda_m},\qquad t_+ + t_- = 1\)
- 巴特勒–福尔默方程Butler–Volmer equation\(j = j_0\left[\exp\!\left(\frac{\alpha_aF\eta}{RT}\right) - \exp\!\left(-\frac{\alpha_cF\eta}{RT}\right)\right],\qquad \eta = E - E_\text{eq}\)
- 塔费尔方程Tafel equation\(\eta = a + b\lg|j|,\qquad b = \frac{2.303RT}{\alpha F}\ (\approx 118\ \mathrm{mV/dec}\ \text{当 } \alpha = 0.5)\)\(\eta = a + b\lg|j|,\qquad b = \frac{2.303RT}{\alpha F}\ (\approx 118\ \mathrm{mV/dec}\ \text{when} \alpha = 0.5)\)
- 德拜长度与双电层Debye length and the electrical double layer\(\kappa^{-1} = \sqrt{\frac{\varepsilon_r\varepsilon_0k_\mathrm BT}{2N_Ae^2I}} \approx \frac{0.304\ \mathrm{nm}}{\sqrt{I/(\mathrm{mol\,L^{-1}})}}\ (25\,{}^\circ\mathrm C\ \text{水}),\qquad \psi(x) = \psi_0e^{-\kappa x}\)\(\kappa^{-1} = \sqrt{\frac{\varepsilon_r\varepsilon_0k_\mathrm BT}{2N_Ae^2I}} \approx \frac{0.304\ \mathrm{nm}}{\sqrt{I/(\mathrm{mol\,L^{-1}})}}\ (25\,{}^\circ\mathrm C\ \text{water}),\qquad \psi(x) = \psi_0e^{-\kappa x}\)
化学动力学 Chemical kinetics
速率与速率方程、各级反应的积分式、温度效应（阿伦尼乌斯、艾林）、复杂反应的近似方法，以及光化学猝灭和电子转移理论。
Rates and rate laws, integrated laws, temperature dependence (Arrhenius, Eyring), approximations for complex mechanisms, photochemical quenching and electron-transfer theory.
- 化学反应速率Rate of reaction\(v = \frac{1}{\nu_\mathrm B}\frac{d[\mathrm B]}{dt},\qquad \text{平均速率 } \bar v = \frac{\Delta c}{\Delta t}\)\(v = \frac{1}{\nu_\mathrm B}\frac{d[\mathrm B]}{dt},\qquad \text{mean rate} \bar v = \frac{\Delta c}{\Delta t}\)
- 速率方程与反应级数Rate law and reaction order\(v = k[\mathrm A]^m[\mathrm B]^n,\qquad \text{总级数} = m + n\)\(v = k[\mathrm A]^m[\mathrm B]^n,\qquad \text{overall order} = m + n\)
- 零级反应Zero-order reaction\([\mathrm A] = [\mathrm A]_0 - kt,\qquad t_{1/2} = \frac{[\mathrm A]_0}{2k}\)
- 一级反应First-order reaction\(\ln\frac{[\mathrm A]_0}{[\mathrm A]} = kt,\qquad [\mathrm A] = [\mathrm A]_0e^{-kt},\qquad t_{1/2} = \frac{\ln2}{k}\)
- 二级反应Second-order reaction\(\frac{1}{[\mathrm A]} = \frac{1}{[\mathrm A]_0} + kt,\qquad t_{1/2} = \frac{1}{k[\mathrm A]_0}\)
- 各级反应的半衰期Half-lives for different orders\(t_{1/2} = \frac{2^{n-1} - 1}{(n-1)k[\mathrm A]_0^{\,n-1}}\ (n \ne 1),\qquad t_{1/2} = \frac{\ln2}{k}\ (n = 1)\)
- 阿伦尼乌斯方程Arrhenius equation\(k = A\,e^{-E_a/RT},\qquad \ln k = \ln A - \frac{E_a}{RT},\qquad \ln\frac{k_2}{k_1} = \frac{E_a}{R}\left(\frac{1}{T_1} - \frac{1}{T_2}\right)\)
- 碰撞理论速率常数Collision-theory rate constant\(k = P\,\sigma\left(\frac{8k_\mathrm BT}{\pi\mu}\right)^{1/2}N_A\,e^{-E_a/RT}\)
- 艾林方程（过渡态理论）Eyring equation (transition-state theory)\(k = \kappa\frac{k_\mathrm BT}{h}\,e^{-\Delta^\ddagger G^\circ/RT} = \kappa\frac{k_\mathrm BT}{h}\,e^{\Delta^\ddagger S^\circ/R}\,e^{-\Delta^\ddagger H^\circ/RT}\)
- 稳态近似Steady-state approximation\(\frac{d[\mathrm I]}{dt} \approx 0\)
- 预平衡近似Pre-equilibrium approximation\(\mathrm A + \mathrm B \underset{k_{-1}}{\overset{k_1}{\rightleftharpoons}} \mathrm I \xrightarrow{k_2} \mathrm P:\qquad v = k_2K_1[\mathrm A][\mathrm B],\quad K_1 = \frac{k_1}{k_{-1}}\)
- 连续反应Consecutive reactions\([\mathrm B] = \frac{k_1[\mathrm A]_0}{k_2 - k_1}\left(e^{-k_1t} - e^{-k_2t}\right),\qquad t_\text{max} = \frac{\ln(k_2/k_1)}{k_2 - k_1}\)
- 平行反应Parallel (competing) reactions\(\mathrm A \xrightarrow{k_1} \mathrm B,\ \ \mathrm A \xrightarrow{k_2} \mathrm C:\qquad [\mathrm A] = [\mathrm A]_0e^{-(k_1+k_2)t},\qquad \frac{[\mathrm B]}{[\mathrm C]} = \frac{k_1}{k_2}\)
- 一级对峙反应（弛豫）Reversible first-order reaction (relaxation)\([\mathrm A] - [\mathrm A]_\text{eq} = ([\mathrm A]_0 - [\mathrm A]_\text{eq})\,e^{-(k_1 + k_{-1})t},\qquad K = \frac{k_1}{k_{-1}},\qquad \tau = \frac{1}{k_1 + k_{-1}}\)
- 林德曼–欣谢尔伍德机理Lindemann–Hinshelwood mechanism\(k_\text{uni} = \frac{k_1k_2[\mathrm M]}{k_{-1}[\mathrm M] + k_2}\)
- 链反应与链长Chain reactions and chain length\(\text{链长 } \nu = \frac{\text{链传递速率}}{\text{链引发速率}},\qquad \mathrm{H_2 + Br_2}:\ v = \frac{k[\mathrm H_2][\mathrm{Br_2}]^{1/2}}{1 + k'[\mathrm{HBr}]/[\mathrm{Br_2}]}\)\(\text{chain length} \nu = \frac{\text{propagation rate}}{\text{initiation rate}},\qquad \mathrm{H_2 + Br_2}:\ v = \frac{k[\mathrm H_2][\mathrm{Br_2}]^{1/2}}{1 + k'[\mathrm{HBr}]/[\mathrm{Br_2}]}\)
- 斯特恩–福尔默方程Stern–Volmer equation\(\frac{I_0}{I} = \frac{\tau_0}{\tau} = 1 + k_q\tau_0[\mathrm Q] = 1 + K_\mathrm{SV}[\mathrm Q]\)
- 马库斯电子转移理论Marcus theory of electron transfer\(\Delta G^\ddagger = \frac{(\lambda + \Delta G^\circ)^2}{4\lambda},\qquad k_\mathrm{ET} = A\exp\!\left[-\frac{(\lambda + \Delta G^\circ)^2}{4\lambda k_\mathrm BT}\right]\)
表面、胶体与传递过程 Surfaces, colloids & transport
吸附等温式、表面张力与弯曲界面、润湿，以及扩散、沉降和黏性流动。
Adsorption isotherms, surface tension and curved interfaces, wetting, and diffusion, sedimentation and viscous flow.
- 朗缪尔吸附等温式Langmuir adsorption isotherm\(\theta = \frac{Kp}{1 + Kp},\qquad \frac{p}{V} = \frac{1}{KV_m} + \frac{p}{V_m}\)
- BET 多层吸附等温式BET isotherm\(\frac{p}{V(p_0 - p)} = \frac{1}{V_mc} + \frac{c - 1}{V_mc}\cdot\frac{p}{p_0},\qquad S = \frac{V_mN_A\sigma}{V_\text{mol}}\)
- 弗罗因德利希等温式Freundlich isotherm\(\frac{x}{m} = K_F\,c^{1/n},\qquad \lg\frac xm = \lg K_F + \frac1n\lg c\)
- 杨–拉普拉斯方程Young–Laplace equation\(\Delta p = \gamma\left(\frac{1}{R_1} + \frac{1}{R_2}\right) \xrightarrow{\text{球面}} \frac{2\gamma}{r},\qquad h = \frac{2\gamma\cos\theta}{\rho g r}\)\(\Delta p = \gamma\left(\frac{1}{R_1} + \frac{1}{R_2}\right) \xrightarrow{\text{sphere}} \frac{2\gamma}{r},\qquad h = \frac{2\gamma\cos\theta}{\rho g r}\)
- 开尔文方程Kelvin equation\(\ln\frac{p_r}{p_0} = \frac{2\gamma V_m}{rRT}\)
- 杨氏方程（接触角）Young's equation (contact angle)\(\gamma_{sg} = \gamma_{sl} + \gamma_{lg}\cos\theta,\qquad W_a = \gamma_{lg}(1 + \cos\theta)\)
- 吉布斯吸附等温式Gibbs adsorption isotherm\(\Gamma_2 = -\frac{1}{RT}\left(\frac{\partial\gamma}{\partial\ln a_2}\right)_T \approx -\frac{c}{RT}\frac{d\gamma}{dc}\)
- 菲克第一定律Fick's first law\(J = -D\frac{\partial c}{\partial x}\)
- 菲克第二定律（扩散方程）Fick's second law (diffusion equation)\(\frac{\partial c}{\partial t} = D\frac{\partial^2 c}{\partial x^2},\qquad c(x, t) = \frac{n_0}{A\sqrt{4\pi Dt}}\,e^{-x^2/4Dt}\)
- 爱因斯坦–斯莫卢霍夫斯基关系（均方位移）Einstein–Smoluchowski relation (mean-square displacement)\(\langle x^2\rangle = 2Dt\ (\text{一维}),\qquad \langle r^2\rangle = 6Dt\ (\text{三维}),\qquad D = \frac{\lambda^2}{2\tau}\)\(\langle x^2\rangle = 2Dt\ (\text{1D}),\qquad \langle r^2\rangle = 6Dt\ (\text{3D}),\qquad D = \frac{\lambda^2}{2\tau}\)
- 斯托克斯–爱因斯坦方程Stokes–Einstein equation\(D = \frac{k_\mathrm BT}{6\pi\eta r}\)
- 斯托克斯定律与沉降速度Stokes' law and sedimentation velocity\(F_d = 6\pi\eta rv,\qquad v_t = \frac{2r^2(\rho_p - \rho_f)g}{9\eta}\)
- 哈根–泊肃叶定律Hagen–Poiseuille law\(Q = \frac{\pi r^4\,\Delta p}{8\eta L}\)
量子化学与结构化学Quantum & structural chemistry
量子力学基础与模型体系 Quantum mechanics & model systems
薛定谔方程、可精确求解的模型（箱中粒子、谐振子、刚性转子、氢原子）以及近似方法（变分、微扰、休克尔、哈特里–福克）。
The Schrödinger equation, exactly solvable models (particle in a box, harmonic oscillator, rigid rotor, hydrogen atom) and approximation methods (variational, perturbation, Hückel, Hartree–Fock).
- 薛定谔方程Schrödinger equation\(i\hbar\frac{\partial\Psi}{\partial t} = \hat H\Psi,\qquad \hat H\psi = E\psi,\qquad \hat H = -\frac{\hbar^2}{2m}\nabla^2 + V(\mathbf r)\)
- 一维箱中粒子Particle in a one-dimensional box\(E_n = \frac{n^2h^2}{8mL^2},\qquad \psi_n(x) = \sqrt{\frac2L}\sin\frac{n\pi x}{L},\quad n = 1, 2, 3, \ldots\)
- 环上粒子Particle on a ring\(E_{m_l} = \frac{m_l^2\hbar^2}{2mR^2},\qquad \psi_{m_l}(\phi) = \frac{1}{\sqrt{2\pi}}e^{im_l\phi},\quad m_l = 0, \pm1, \pm2, \ldots\)
- 量子谐振子Quantum harmonic oscillator\(E_v = \left(v + \tfrac12\right)h\nu,\qquad \nu = \frac{1}{2\pi}\sqrt{\frac{k}{\mu}},\qquad \psi_v(y) = N_vH_v(y)e^{-y^2/2},\ \ y = \sqrt{\frac{\mu\omega}{\hbar}}\,x\)
- 刚性转子Rigid rotor\(E_J = \frac{\hbar^2}{2I}J(J+1) = hcB\,J(J+1),\qquad B = \frac{h}{8\pi^2cI},\quad I = \mu r^2\)
- 氢原子（类氢离子）能级与波函数Hydrogen atom energy levels and wavefunctions\(E_n = -\frac{\mu e^4 Z^2}{8\varepsilon_0^2h^2n^2} = -\frac{13.6\ \mathrm{eV}\cdot Z^2}{n^2},\qquad \psi_{nlm} = R_{nl}(r)\,Y_l^m(\theta,\phi)\)
- 氢原子径向分布函数Radial distribution function of hydrogen\(P(r) = r^2R_{nl}^2(r),\qquad R_{10} = 2\left(\frac{Z}{a_0}\right)^{3/2}e^{-Zr/a_0},\qquad r_\text{max}(1s) = \frac{a_0}{Z}\)
- 原子轨道的角度部分（实球谐函数）Angular parts of atomic orbitals (real spherical harmonics)\(s:\ \tfrac{1}{\sqrt{4\pi}};\quad p_z:\ \sqrt{\tfrac{3}{4\pi}}\cos\theta;\quad d_{z^2}:\ \sqrt{\tfrac{5}{16\pi}}(3\cos^2\theta - 1);\quad d_{xz}:\ \sqrt{\tfrac{15}{4\pi}}\sin\theta\cos\theta\cos\phi\)
- 量子隧穿（矩形势垒）Quantum tunnelling through a rectangular barrier\(T \approx 16\frac{E}{V_0}\left(1 - \frac{E}{V_0}\right)e^{-2\kappa a},\qquad \kappa = \frac{\sqrt{2m(V_0 - E)}}{\hbar}\)
- 变分原理Variational principle\(E[\tilde\psi] = \frac{\langle\tilde\psi|\hat H|\tilde\psi\rangle}{\langle\tilde\psi|\tilde\psi\rangle} \ge E_0\)
- 休克尔分子轨道法Hückel molecular orbital theory\(\det\left|\mathbf H - E\mathbf 1\right| = 0,\quad H_{ii} = \alpha,\ \ H_{ij} = \beta\ (\text{相邻}),\ \ S_{ij} = \delta_{ij}\)\(\det\left|\mathbf H - E\mathbf 1\right| = 0,\quad H_{ii} = \alpha,\ \ H_{ij} = \beta\ (\text{adjacent}),\ \ S_{ij} = \delta_{ij}\)
- 玻恩–奥本海默近似Born–Oppenheimer approximation\(\Psi(\mathbf r, \mathbf R) \approx \psi_\text{el}(\mathbf r; \mathbf R)\,\chi_\text{nuc}(\mathbf R),\qquad \hat H_\text{el}\psi_\text{el} = E_\text{el}(\mathbf R)\psi_\text{el}\)
- 泡利原理与斯莱特行列式Pauli principle and the Slater determinant\(\Psi(1,2,\ldots,N) = \frac{1}{\sqrt{N!}}\begin{vmatrix}\chi_1(1) & \chi_2(1) & \cdots & \chi_N(1)\\ \chi_1(2) & \chi_2(2) & \cdots & \chi_N(2)\\ \vdots & & \ddots & \vdots\\ \chi_1(N) & \chi_2(N) & \cdots & \chi_N(N)\end{vmatrix}\)
- 哈特里–福克方程与罗特汉方程Hartree–Fock and Roothaan equations\(\hat f\chi_i = \varepsilon_i\chi_i,\qquad \hat f = \hat h + \sum_j\left(\hat J_j - \hat K_j\right),\qquad \mathbf F\mathbf C = \mathbf S\mathbf C\boldsymbol\varepsilon\)
- 定态微扰理论Time-independent perturbation theory\(E_n \approx E_n^{(0)} + \langle n|\hat H'|n\rangle + \sum_{k\ne n}\frac{|\langle k|\hat H'|n\rangle|^2}{E_n^{(0)} - E_k^{(0)}}\)
分子光谱与结构测定 Molecular spectroscopy & structure determination
吸收定律、振动与转动光谱、核磁共振参数、X 射线衍射和荧光等结构测定中最常用的关系式。
The absorption law, vibrational and rotational spectra, NMR parameters, X-ray diffraction and fluorescence — the relations used most in structure determination.
- 朗伯–比尔定律Beer–Lambert law\(A = \lg\frac{I_0}{I} = -\lg T = \varepsilon\, b\, c\)
- 振动频率（胡克定律）Vibrational wavenumber from Hooke's law\(\tilde\nu = \frac{1}{2\pi c}\sqrt{\frac{k}{\mu}},\qquad \mu = \frac{m_1m_2}{m_1 + m_2}\)
- 约化质量Reduced mass\(\mu = \frac{m_1 m_2}{m_1 + m_2},\qquad \frac{1}{\mu} = \frac{1}{m_1} + \frac{1}{m_2}\)
- 转动常数与键长Rotational constant and bond length\(B = \frac{h}{8\pi^2 c\, I} = \frac{h}{8\pi^2 c\,\mu r_e^2},\qquad \tilde\nu_{J\to J+1} = 2B(J+1)\)
- 转动能级的玻尔兹曼布居Boltzmann populations of rotational levels\(\frac{N_J}{N_0} = (2J+1)\exp\!\left[-\frac{hcBJ(J+1)}{k_\mathrm{B}T}\right],\qquad J_\text{max} \approx \sqrt{\frac{k_\mathrm{B}T}{2hcB}} - \frac12\)
- 莫尔斯势Morse potential\(V(r) = D_e\left[1 - e^{-a(r - r_e)}\right]^2,\qquad a = \sqrt{\frac{k_e}{2D_e}}\)
- 非谐振动能级与伯奇–斯波纳外推Anharmonic vibrational levels & Birge–Sponer extrapolation\(G(v) = \tilde\omega_e\left(v + \tfrac12\right) - \tilde\omega_ex_e\left(v + \tfrac12\right)^2,\qquad \Delta G_{v+1/2} = \tilde\omega_e - 2\tilde\omega_ex_e(v+1)\)
- 拉莫尔频率与核磁共振条件Larmor frequency and the NMR resonance condition\(\nu_0 = \frac{\gamma B_0}{2\pi},\qquad \Delta E = \gamma\hbar B_0,\qquad \frac{N_\beta}{N_\alpha} = e^{-\gamma\hbar B_0/k_\mathrm{B}T}\)
- 化学位移Chemical shift\(\delta = \frac{\nu - \nu_\text{ref}}{\nu_0}\times10^6\ \text{ppm} = (\sigma_\text{ref} - \sigma)\times10^6\)
- 自旋–自旋裂分（n + 1 规律）Spin–spin splitting (n + 1 rule)\(M = 2nI + 1 \xrightarrow{I = 1/2} n + 1,\qquad \text{强度比} = \binom{n}{k},\ k = 0, \ldots, n\)\(M = 2nI + 1 \xrightarrow{I = 1/2} n + 1,\qquad \text{intensity ratio} = \binom{n}{k},\ k = 0, \ldots, n\)
- 卡普拉斯方程Karplus equation\({}^3J(\phi) = A\cos^2\phi + B\cos\phi + C\)
- 布拉格定律Bragg's law\(n\lambda = 2d\sin\theta\)
- 弗兰克–康登原理Franck–Condon principle\(I_{v'\leftarrow v''} \propto |\mu_{el}|^2\,\left|\langle\chi_{v'}|\chi_{v''}\rangle\right|^2\)
- 荧光量子产率与寿命Fluorescence quantum yield and lifetime\(\Phi_F = \frac{k_r}{k_r + k_{nr}} = k_r\,\tau_F,\qquad \tau_F = \frac{1}{k_r + k_{nr}}\)
- 光谱选择定则Spectroscopic selection rules\(\langle\psi_f|\hat{\boldsymbol\mu}|\psi_i\rangle \ne 0 \iff \Gamma_f\otimes\Gamma_{x,y,z}\otimes\Gamma_i \ni A_1\)
- 拉曼位移Raman shift\(\Delta\tilde\nu = \frac{1}{\lambda_0} - \frac{1}{\lambda_s},\qquad \frac{I_\text{反斯托克斯}}{I_\text{斯托克斯}} = \left(\frac{\tilde\nu_0 + \tilde\nu_m}{\tilde\nu_0 - \tilde\nu_m}\right)^4 e^{-hc\tilde\nu_m/k_\mathrm{B}T}\)\(\Delta\tilde\nu = \frac{1}{\lambda_0} - \frac{1}{\lambda_s},\qquad \frac{I_\text{anti-Stokes}}{I_\text{Stokes}} = \left(\frac{\tilde\nu_0 + \tilde\nu_m}{\tilde\nu_0 - \tilde\nu_m}\right)^4 e^{-hc\tilde\nu_m/k_\mathrm{B}T}\)
- 伍德沃德–菲泽规则Woodward–Fieser rules\(\lambda_\text{max} = \lambda_\text{母体} + \sum \Delta\lambda_i\)\(\lambda_\text{max} = \lambda_\text{parent} + \sum \Delta\lambda_i\)
群论与分子对称性 Group theory & molecular symmetry
特征标的正交关系、约化公式、投影算符与直积，以及由此计算分子振动模式和光谱活性。点群特征标表见“点群”栏目。
Character orthogonality, the reduction formula, projection operators and direct products, and their use for vibrational modes and spectroscopic activity. Character tables are in the Point groups section.
- 广义正交定理与特征标正交关系Great orthogonality theorem & character orthogonality\(\sum_R \Gamma_i(R)_{mn}^{*}\,\Gamma_j(R)_{m'n'} = \frac{h}{\sqrt{l_il_j}}\,\delta_{ij}\delta_{mm'}\delta_{nn'},\qquad \sum_{C} g_C\,\chi_i(C)^{*}\chi_j(C) = h\,\delta_{ij}\)
- 约化公式Reduction formula\(n_i = \frac{1}{h}\sum_C g_C\,\chi_\Gamma(C)\,\chi_i(C)^{*}\)
- 投影算符Projection operator\(\hat P^{\,i} = \frac{l_i}{h}\sum_R \chi_i(R)^{*}\,\hat R\)
- 直积与跃迁积分的对称性Direct products and integral symmetry\(\chi_{\Gamma_a\otimes\Gamma_b}(R) = \chi_a(R)\,\chi_b(R),\qquad \int\psi_a\hat O\psi_b\,d\tau \ne 0 \Rightarrow \Gamma_a\otimes\Gamma_O\otimes\Gamma_b \ni A_1\)
- 振动自由度与振动模式对称性Vibrational degrees of freedom and mode symmetries\(N_\text{vib} = 3N - 6\ (\text{非线形}),\quad 3N - 5\ (\text{线形}),\qquad \Gamma_\text{vib} = \Gamma_{3N} - \Gamma_\text{trans} - \Gamma_\text{rot}\)\(N_\text{vib} = 3N - 6\ (\text{non-linear}),\quad 3N - 5\ (\text{linear}),\qquad \Gamma_\text{vib} = \Gamma_{3N} - \Gamma_\text{trans} - \Gamma_\text{rot}\)
统计热力学Statistical thermodynamics
统计热力学 Statistical thermodynamics
从分子能级出发计算宏观热力学量：玻尔兹曼分布、配分函数、能量均分与固体热容。
Computing macroscopic thermodynamics from molecular energy levels: the Boltzmann distribution, partition functions, equipartition and heat capacities of solids.
- 玻尔兹曼分布Boltzmann distribution\(\frac{N_i}{N} = \frac{g_i\,e^{-\varepsilon_i/k_\mathrm{B}T}}{q},\qquad \frac{N_j}{N_i} = \frac{g_j}{g_i}\,e^{-(\varepsilon_j - \varepsilon_i)/k_\mathrm{B}T}\)
- 分子配分函数Molecular partition function\(q = \sum_i g_i\,e^{-\varepsilon_i/k_\mathrm{B}T},\qquad q = q_\text{trans}\,q_\text{rot}\,q_\text{vib}\,q_\text{el}\)
- 平动配分函数Translational partition function\(q_\text{trans} = \frac{V}{\Lambda^3},\qquad \Lambda = \frac{h}{\sqrt{2\pi m k_\mathrm{B}T}}\)
- 转动配分函数Rotational partition function\(q_\text{rot} \approx \frac{k_\mathrm{B}T}{\sigma hcB} = \frac{T}{\sigma\Theta_\text{rot}}\ (\text{线形分子}),\qquad q_\text{rot} = \frac{\sqrt\pi}{\sigma}\left(\frac{k_\mathrm{B}T}{hc}\right)^{3/2}\frac{1}{\sqrt{ABC}}\ (\text{非线形})\)\(q_\text{rot} \approx \frac{k_\mathrm{B}T}{\sigma hcB} = \frac{T}{\sigma\Theta_\text{rot}}\ (\text{linear molecule}),\qquad q_\text{rot} = \frac{\sqrt\pi}{\sigma}\left(\frac{k_\mathrm{B}T}{hc}\right)^{3/2}\frac{1}{\sqrt{ABC}}\ (\text{non-linear})\)
- 振动配分函数Vibrational partition function\(q_\text{vib} = \frac{1}{1 - e^{-\Theta_\text{vib}/T}}\ (\text{从 } v = 0 \text{ 算起}),\qquad \Theta_\text{vib} = \frac{hc\tilde\nu}{k_\mathrm{B}}\)\(q_\text{vib} = \frac{1}{1 - e^{-\Theta_\text{vib}/T}}\ (\text{from} v = 0 \text{}),\qquad \Theta_\text{vib} = \frac{hc\tilde\nu}{k_\mathrm{B}}\)
- 电子配分函数Electronic partition function\(q_\text{el} = g_0 + g_1e^{-\varepsilon_1/k_\mathrm{B}T} + \cdots \approx g_0\)
- 由配分函数求热力学函数Thermodynamic functions from the partition function\(U - U_0 = k_\mathrm{B}T^2\left(\frac{\partial\ln Q}{\partial T}\right)_V,\quad A - A_0 = -k_\mathrm{B}T\ln Q,\quad S = \frac{U - U_0}{T} + k_\mathrm{B}\ln Q\)
- 统计熵Statistical entropy\(S = k_\mathrm{B}\ln W,\qquad S = -k_\mathrm{B}\sum_i p_i\ln p_i\)
- 萨克尔–泰特罗德方程Sackur–Tetrode equation\(S_m = R\ln\left(\frac{V_m}{N_A\Lambda^3}\right) + \frac52R = R\ln\left(\frac{k_\mathrm{B}T}{p\Lambda^3}\right) + \frac52 R\)
- 能量均分定理Equipartition theorem\(\langle\varepsilon\rangle = \frac12k_\mathrm{B}T\ \text{（每个平方项）},\qquad C_{V,m} = \frac{f}{2}R\)\(\langle\varepsilon\rangle = \frac12k_\mathrm{B}T\ \text{( per quadratic term )},\qquad C_{V,m} = \frac{f}{2}R\)
- 爱因斯坦固体热容Einstein heat capacity of solids\(C_{V,m} = 3R\left(\frac{\Theta_E}{T}\right)^2\frac{e^{\Theta_E/T}}{\left(e^{\Theta_E/T} - 1\right)^2}\)
- 德拜固体热容（T³ 定律）Debye heat capacity (T³ law)\(C_{V,m} = 9R\left(\frac{T}{\Theta_D}\right)^3\int_0^{\Theta_D/T}\frac{x^4e^x}{(e^x - 1)^2}dx \xrightarrow{T \ll \Theta_D} \frac{12\pi^4}{5}R\left(\frac{T}{\Theta_D}\right)^3\)
- 由配分函数计算平衡常数Equilibrium constants from partition functions\(K = \prod_\mathrm{J}\left(\frac{q_{\mathrm{J},m}^\circ}{N_A}\right)^{\nu_\mathrm{J}}e^{-\Delta_rE_0/RT}\)
无机化学Inorganic chemistry
配位化学与晶体场 Coordination chemistry & ligand fields
晶体场稳定化能、磁矩、电子计数、配合物稳定常数与电子光谱参数。
Crystal-field stabilisation, magnetic moments, electron counting, stability constants and spectroscopic parameters.
- 八面体晶体场稳定化能Octahedral crystal-field stabilisation energy\(\mathrm{CFSE} = \left(-0.4\,n_{t_{2g}} + 0.6\,n_{e_g}\right)\Delta_\mathrm{o} + m P\)
- 四面体分裂能与 CFSETetrahedral splitting and CFSE\(\Delta_\mathrm{t} \approx \frac49\Delta_\mathrm{o},\qquad \mathrm{CFSE}_\mathrm{t} = \left(-0.6\,n_e + 0.4\,n_{t_2}\right)\Delta_\mathrm{t}\)
- 唯自旋磁矩Spin-only magnetic moment\(\mu_\text{so} = \sqrt{n(n+2)}\ \mu_\mathrm{B} = 2\sqrt{S(S+1)}\ \mu_\mathrm{B}\)
- 18 电子规则18-electron rule\(N_\text{VE} = N_{d}(\mathrm{M}) + \sum_\text{配体} N_\text{给出} \pm q = 18\)\(N_\text{VE} = N_{d}(\mathrm{M}) + \sum_\text{ligand} N_\text{donated} \pm q = 18\)
- 配合物逐级与累积稳定常数Stepwise and overall stability constants\(K_i = \frac{[\mathrm{ML}_i]}{[\mathrm{ML}_{i-1}][\mathrm{L}]},\qquad \beta_n = \prod_{i=1}^n K_i = \frac{[\mathrm{ML}_n]}{[\mathrm{M}][\mathrm{L}]^n}\)
- 螯合效应Chelate effect\(\Delta G^\circ = \Delta H^\circ - T\Delta S^\circ,\qquad [\mathrm{Ni(NH_3)_6}]^{2+} + 3\,\mathrm{en} \rightleftharpoons [\mathrm{Ni(en)_3}]^{2+} + 6\,\mathrm{NH_3}\)
- 分裂能的因子分解与电子云扩展效应Ligand-field splitting factorisation & nephelauxetic effect\(\Delta_\mathrm{o} \approx f_\text{配体}\cdot g_\text{金属}\ (10^3\ \mathrm{cm^{-1}}),\qquad \beta = \frac{B_\text{配合物}}{B_\text{自由离子}} = 1 - h_\text{配体}k_\text{金属}\)\(\Delta_\mathrm{o} \approx f_\text{ligand}\cdot g_\text{metal}\ (10^3\ \mathrm{cm^{-1}}),\qquad \beta = \frac{B_\text{complex}}{B_\text{free ion}} = 1 - h_\text{ligand}k_\text{metal}\)
- d 电子组态谱项能量（拉卡参数）d-configuration term energies (Racah parameters)\(E(^3\!P) - E(^3\!F) = 15B\ \ (d^2, d^8),\qquad E(^4\!P) - E(^4\!F) = 15B\ \ (d^3, d^7)\)
- 姜–泰勒效应Jahn–Teller effect\(E(Q) = \pm F Q + \tfrac12 kQ^2,\qquad E_\mathrm{JT} = -\frac{F^2}{2k}\)
离子晶体与晶格能 Ionic solids & lattice energy
晶格能的计算与估算、马德隆常数、半径比规则、堆积效率与晶体密度。
Calculating and estimating lattice energies, Madelung constants, radius ratios, packing efficiency and crystal density.
- 玻恩–朗德方程Born–Landé equation\(U_L = -\frac{N_A M z_+ z_- e^2}{4\pi\varepsilon_0 r_0}\left(1 - \frac{1}{n}\right)\)
- 马德隆常数Madelung constant\(M = \sum_{j\neq i}\frac{\pm z_j/z}{r_{ij}/r_0}\)
- 卡普斯京斯基方程Kapustinskii equation\(U_L \approx -\frac{1.2025\times10^{-4}\,\nu\,|z_+z_-|}{r_+ + r_-}\left(1 - \frac{3.45\times10^{-11}}{r_+ + r_-}\right)\ \mathrm{J\,mol^{-1}\,m}\)
- 半径比规则Radius ratio rule\(\frac{r_+}{r_-}:\ \ 0.225\ (\mathrm{CN\,4})\ \ 0.414\ (\mathrm{CN\,6})\ \ 0.732\ (\mathrm{CN\,8})\ \ 1\ (\mathrm{CN\,12})\)
- 堆积效率（空间利用率）Packing efficiency\(\varphi = \frac{Z\cdot\tfrac43\pi r^3}{V_\text{晶胞}}\)\(\varphi = \frac{Z\cdot\tfrac43\pi r^3}{V_\text{cell}}\)
- 由晶胞求晶体密度Crystal density from the unit cell\(\rho = \frac{Z\,M}{N_A\,V_\text{晶胞}} = \frac{Z\,M}{N_A\,a^3}\ (\text{立方})\)\(\rho = \frac{Z\,M}{N_A\,V_\text{cell}} = \frac{Z\,M}{N_A\,a^3}\ (\text{cubic})\)
- 戈尔德施密特容忍因子Goldschmidt tolerance factor\(t = \frac{r_\mathrm{A} + r_\mathrm{X}}{\sqrt2\,(r_\mathrm{B} + r_\mathrm{X})}\)
有机化学Organic chemistry
结构、立体化学与物理有机 Structure, stereochemistry & physical organic chemistry
不饱和度、旋光与对映体过量、构象布居，以及描述取代基效应和反应活性的线性自由能关系。
Degrees of unsaturation, optical rotation and ee, conformer populations, and the linear free-energy relationships that quantify substituent effects and reactivity.
- 不饱和度Degree of unsaturation (double-bond equivalents)\(\Omega = \frac{2C + 2 + N - H - X}{2}\)
- 比旋光度Specific rotation\([\alpha]_\lambda^{T} = \frac{\alpha}{l\, c}\)
- 对映体过量Enantiomeric excess\(ee = \frac{|n_R - n_S|}{n_R + n_S}\times 100\% \approx \frac{[\alpha]_{\text{观测}}}{[\alpha]_{\text{纯}}}\times 100\%\)\(ee = \frac{|n_R - n_S|}{n_R + n_S}\times 100\% \approx \frac{[\alpha]_{\text{obs}}}{[\alpha]_{\text{pure}}}\times 100\%\)
- 构象布居（玻尔兹曼）Conformer populations (Boltzmann)\(\frac{n_2}{n_1} = \exp\!\left(-\frac{\Delta G^\circ}{RT}\right),\qquad x_i = \frac{e^{-G_i/RT}}{\sum_j e^{-G_j/RT}}\)
- 柯廷–哈米特原理Curtin–Hammett principle\(\frac{[\mathrm{P_B}]}{[\mathrm{P_A}]} = \frac{k_\mathrm{B}}{k_\mathrm{A}}K = \exp\!\left(-\frac{G^\ddagger_\mathrm{B} - G^\ddagger_\mathrm{A}}{RT}\right)\)
- 哈米特方程Hammett equation\(\lg\frac{k_\mathrm{X}}{k_\mathrm{H}} = \rho\,\sigma_\mathrm{X} \qquad\text{或}\qquad \lg\frac{K_\mathrm{X}}{K_\mathrm{H}} = \rho\,\sigma_\mathrm{X}\)\(\lg\frac{k_\mathrm{X}}{k_\mathrm{H}} = \rho\,\sigma_\mathrm{X} \qquad\text{or}\qquad \lg\frac{K_\mathrm{X}}{K_\mathrm{H}} = \rho\,\sigma_\mathrm{X}\)
- 塔夫特方程Taft equation\(\lg\frac{k}{k_0} = \rho^*\sigma^* + \delta E_s\)
- 布朗斯特催化定律Brønsted catalysis law\(\lg k_{\mathrm{HA}} = \alpha\lg K_a + C_1,\qquad \lg k_{\mathrm{B}} = \beta\lg K_b + C_2\)
- 休克尔 4n+2 规则Hückel's 4n + 2 rule\(N_\pi = 4n + 2\ \ (n = 0, 1, 2, \ldots)\ \Rightarrow\ \text{芳香性}; \qquad N_\pi = 4n\ \Rightarrow\ \text{反芳香性}\)\(N_\pi = 4n + 2\ \ (n = 0, 1, 2, \ldots)\ \Rightarrow\ \text{aromatic}; \qquad N_\pi = 4n\ \Rightarrow\ \text{antiaromatic}\)
- 动力学同位素效应Kinetic isotope effect\(\frac{k_\mathrm{H}}{k_\mathrm{D}} \approx \exp\!\left[\frac{h c\,(\tilde\nu_\mathrm{H} - \tilde\nu_\mathrm{D})}{2k_\mathrm{B}T}\right],\qquad \tilde\nu_\mathrm{D} \approx \frac{\tilde\nu_\mathrm{H}}{\sqrt2}\)
- 迈尔亲电性–亲核性方程Mayr–Patz equation\(\lg k_{20\,{}^\circ\mathrm{C}} = s_N\,(N + E)\)
- 斯温–斯科特方程Swain–Scott equation\(\lg\frac{k}{k_0} = s\,n\)
- 贝尔–埃文斯–波拉尼原理Bell–Evans–Polanyi principle\(E_a = E_0 + \alpha\,\Delta_r H\)
- pKa 与取代基：酸性的定量比较pKa and substituent effects\(\Delta G^\circ_{\text{电离}} = 2.303\,RT\,\mathrm{p}K_a,\qquad \Delta\mathrm{p}K_a = \frac{\delta\Delta G^\circ}{5.71\ \mathrm{kJ\,mol^{-1}}}\ (25\,{}^\circ\mathrm{C})\)\(\Delta G^\circ_{\text{ionisation}} = 2.303\,RT\,\mathrm{p}K_a,\qquad \Delta\mathrm{p}K_a = \frac{\delta\Delta G^\circ}{5.71\ \mathrm{kJ\,mol^{-1}}}\ (25\,{}^\circ\mathrm{C})\)
分析化学Analytical chemistry
酸碱平衡与 pH 计算 Acid–base equilibria & pH
从 pH 的定义到弱酸、缓冲溶液、多元酸和滴定曲线的计算。
From the definition of pH to weak acids, buffers, polyprotic acids and titration curves.
- pH 的定义Definition of pH\(\mathrm{pH} = -\lg a_{\mathrm H^+} \approx -\lg\frac{[\mathrm H^+]}{c^\circ},\qquad \mathrm{pOH} = -\lg[\mathrm{OH^-}]\)
- 水的离子积Ion product of water\(K_w = [\mathrm H^+][\mathrm{OH^-}] = 1.0\times10^{-14}\ (25\,{}^\circ\mathrm C),\qquad \mathrm{pH} + \mathrm{pOH} = \mathrm pK_w = 14.00\)
- 共轭酸碱对的 Ka 与 KbKa and Kb of a conjugate pair\(K_a\,K_b = K_w,\qquad \mathrm pK_a + \mathrm pK_b = \mathrm pK_w\)
- 一元弱酸溶液的 pHpH of a weak monoprotic acid\([\mathrm H^+] \approx \sqrt{K_a c}\quad(c/K_a \ge 500,\ K_ac \ge 20K_w),\qquad \mathrm{pH} = \tfrac12(\mathrm pK_a - \lg c)\)
- 一元弱碱溶液的 pHpH of a weak monoprotic base\([\mathrm{OH^-}] \approx \sqrt{K_bc},\qquad \mathrm{pH} = 14 - \tfrac12(\mathrm pK_b - \lg c)\)
- 亨德森–哈塞尔巴尔赫方程（缓冲溶液 pH）Henderson–Hasselbalch equation\(\mathrm{pH} = \mathrm pK_a + \lg\frac{[\mathrm A^-]}{[\mathrm{HA}]} \approx \mathrm pK_a + \lg\frac{c_\text{共轭碱}}{c_\text{酸}}\)\(\mathrm{pH} = \mathrm pK_a + \lg\frac{[\mathrm A^-]}{[\mathrm{HA}]} \approx \mathrm pK_a + \lg\frac{c_\text{conj. base}}{c_\text{acid}}\)
- 缓冲容量Buffer capacity\(\beta = \frac{dc_b}{d\mathrm{pH}} = 2.303\left([\mathrm H^+] + [\mathrm{OH^-}] + \frac{cK_a[\mathrm H^+]}{(K_a + [\mathrm H^+])^2}\right)\)
- 酸碱滴定曲线Acid–base titration curves\([\mathrm H^+] - [\mathrm{OH^-}] = \frac{c_aV_a - c_bV_b}{V_a + V_b}\ (\text{强酸–强碱}),\qquad \mathrm{pH}_{V_b = V_e/2} = \mathrm pK_a\ (\text{弱酸})\)\([\mathrm H^+] - [\mathrm{OH^-}] = \frac{c_aV_a - c_bV_b}{V_a + V_b}\ (\text{strong acid – strong base}),\qquad \mathrm{pH}_{V_b = V_e/2} = \mathrm pK_a\ (\text{weak acid})\)
- 酸碱分布分数Distribution (speciation) fractions\(\alpha_{\mathrm{HA}} = \frac{[\mathrm H^+]}{[\mathrm H^+] + K_a},\quad \alpha_{\mathrm A^-} = \frac{K_a}{[\mathrm H^+] + K_a};\qquad \alpha_j(\mathrm H_n\mathrm A) = \frac{[\mathrm H^+]^{n-j}\prod_{i\le j}K_{ai}}{\sum_{k=0}^n[\mathrm H^+]^{n-k}\prod_{i\le k}K_{ai}}\)
- 两性物质溶液的 pHpH of an amphiprotic species\([\mathrm H^+] = \sqrt{\frac{K_{a1}(K_{a2}c + K_w)}{K_{a1} + c}} \approx \sqrt{K_{a1}K_{a2}},\qquad \mathrm{pH} \approx \tfrac12(\mathrm pK_{a1} + \mathrm pK_{a2})\)
- 盐溶液的 pH（水解）pH of salt solutions (hydrolysis)\(\text{强碱弱酸盐：}[\mathrm{OH^-}] = \sqrt{\frac{K_w}{K_a}c};\qquad \text{强酸弱碱盐：}[\mathrm H^+] = \sqrt{\frac{K_w}{K_b}c};\qquad \text{弱酸弱碱盐：}[\mathrm H^+] \approx \sqrt{\frac{K_wK_a}{K_b}}\)\(\text{weak-acid salt of strong base :}[\mathrm{OH^-}] = \sqrt{\frac{K_w}{K_a}c};\qquad \text{weak-base salt of strong acid :}[\mathrm H^+] = \sqrt{\frac{K_w}{K_b}c};\qquad \text{weak acid–weak base salt :}[\mathrm H^+] \approx \sqrt{\frac{K_wK_a}{K_b}}\)
- 多元弱酸溶液的 pHpH of a polyprotic acid\([\mathrm H^+] \approx \sqrt{K_{a1}c}\quad(K_{a1} \gg K_{a2},\ c/K_{a1} \ge 500),\qquad [\mathrm A^{2-}] \approx K_{a2}\)
误差统计、校准与色谱 Statistics, calibration & chromatography
测量数据的统计处理、校准与检出限，以及色谱分离的定量描述。
Statistical treatment of measurements, calibration and detection limits, and the quantitative description of chromatographic separations.
- 平均值与标准偏差Mean and standard deviation\(\bar x = \frac1n\sum_{i=1}^n x_i,\qquad s = \sqrt{\frac{\sum_i(x_i - \bar x)^2}{n - 1}},\qquad \mathrm{RSD} = \frac{s}{\bar x}\times100\%\)
- 平均值的置信区间Confidence interval of the mean\(\mu = \bar x \pm \frac{t_{\alpha,\,n-1}\,s}{\sqrt n}\)
- t 检验与 F 检验t-test and F-test\(t = \frac{|\bar x - \mu_0|}{s}\sqrt n,\qquad t = \frac{|\bar x_1 - \bar x_2|}{s_p}\sqrt{\frac{n_1n_2}{n_1 + n_2}},\qquad F = \frac{s_1^2}{s_2^2}\ (s_1 > s_2)\)
- 狄克逊 Q 检验Dixon's Q test\(Q = \frac{|x_\text{可疑} - x_\text{邻近}|}{x_\text{最大} - x_\text{最小}}\)\(Q = \frac{|x_\text{suspect} - x_\text{nearest}|}{x_\text{max} - x_\text{min}}\)
- 误差传递Propagation of uncertainty\(\sigma_y^2 = \sum_i\left(\frac{\partial y}{\partial x_i}\right)^2\sigma_{x_i}^2\)
- 最小二乘线性回归（校准曲线）Least-squares linear regression (calibration)\(b = \frac{\sum(x_i - \bar x)(y_i - \bar y)}{\sum(x_i - \bar x)^2},\quad a = \bar y - b\bar x,\quad r = \frac{\sum(x_i - \bar x)(y_i - \bar y)}{\sqrt{\sum(x_i - \bar x)^2\sum(y_i - \bar y)^2}}\)
- 检出限与定量限Limit of detection and quantitation\(\mathrm{LOD} = \frac{3s_\text{空白}}{b},\qquad \mathrm{LOQ} = \frac{10s_\text{空白}}{b}\)\(\mathrm{LOD} = \frac{3s_\text{blank}}{b},\qquad \mathrm{LOQ} = \frac{10s_\text{blank}}{b}\)
- 保留因子Retention factor\(k = \frac{t_R - t_M}{t_M} = K\,\frac{V_S}{V_M}\)
- 选择性因子Selectivity factor\(\alpha = \frac{k_2}{k_1} = \frac{K_2}{K_1}\ (\alpha > 1)\)
- 理论塔板数与塔板高度Plate number and plate height\(N = 16\left(\frac{t_R}{w_b}\right)^2 = 5.54\left(\frac{t_R}{w_{1/2}}\right)^2,\qquad H = \frac{L}{N}\)
- 色谱分离度（珀内尔方程）Chromatographic resolution (Purnell equation)\(R_s = \frac{2(t_{R2} - t_{R1})}{w_{b1} + w_{b2}} = \frac{\sqrt N}{4}\cdot\frac{\alpha - 1}{\alpha}\cdot\frac{k_2}{1 + k_2}\)
- 范第姆特方程van Deemter equation\(H = A + \frac{B}{u} + C\,u,\qquad u_\text{opt} = \sqrt{\frac BC},\quad H_\text{min} = A + 2\sqrt{BC}\)
- 比移值 R_fRetardation factor R_f\(R_f = \frac{\text{斑点中心移动距离}}{\text{溶剂前沿移动距离}}\)\(R_f = \frac{\text{distance moved by spot}}{\text{distance moved by solvent front}}\)
- 条件稳定常数（EDTA 滴定）Conditional formation constant (EDTA titration)\(\lg K'_\mathrm{MY} = \lg K_\mathrm{MY} - \lg\alpha_{\mathrm Y(\mathrm H)} - \lg\alpha_\mathrm M,\qquad \lg(c_\mathrm M K'_\mathrm{MY}) \ge 6\ \text{可准确滴定}\)\(\lg K'_\mathrm{MY} = \lg K_\mathrm{MY} - \lg\alpha_{\mathrm Y(\mathrm H)} - \lg\alpha_\mathrm M,\qquad \lg(c_\mathrm M K'_\mathrm{MY}) \ge 6\ \text{titratable}\)
- 氧化还原滴定的计量点电位Equivalence-point potential of a redox titration\(E_\text{sp} = \frac{n_1E_1^{\circ\prime} + n_2E_2^{\circ\prime}}{n_1 + n_2}\)
- 重量分析换算因数Gravimetric factor\(w_\text{待测} = \frac{m_\text{称量形} \times F}{m_\text{试样}}\times 100\%,\qquad F = \frac{a\,M_\text{待测}}{b\,M_\text{称量形}}\)\(w_\text{analyte} = \frac{m_\text{weighed form} \times F}{m_\text{sample}}\times 100\%,\qquad F = \frac{a\,M_\text{analyte}}{b\,M_\text{weighed form}}\)
生物化学Biochemistry
酶动力学、结合与生物能学 Enzyme kinetics, binding & bioenergetics
酶促反应速率、抑制类型、配体结合、生化标准态与膜电位等生物化学定量关系。
Quantitative biochemistry: enzyme rates, inhibition, ligand binding, the biochemical standard state and membrane potentials.
- 米氏方程Michaelis–Menten equation\(v_0 = \frac{V_{\max}[\mathrm{S}]}{K_\mathrm{M} + [\mathrm{S}]}\)
- 林–贝氏双倒数作图Lineweaver–Burk plot\(\frac{1}{v_0} = \frac{K_\mathrm{M}}{V_{\max}}\cdot\frac{1}{[\mathrm{S}]} + \frac{1}{V_{\max}}\)
- 竞争性抑制Competitive inhibition\(v_0 = \frac{V_{\max}[\mathrm{S}]}{\alpha K_\mathrm{M} + [\mathrm{S}]},\qquad \alpha = 1 + \frac{[\mathrm{I}]}{K_\mathrm{I}}\)
- 非竞争性（混合型）抑制Noncompetitive / mixed inhibition\(v_0 = \frac{V_{\max}[\mathrm{S}]}{\alpha K_\mathrm{M} + \alpha'[\mathrm{S}]},\qquad \alpha = 1 + \frac{[\mathrm{I}]}{K_\mathrm{I}},\ \ \alpha' = 1 + \frac{[\mathrm{I}]}{K_\mathrm{I}'}\)
- 反竞争性抑制Uncompetitive inhibition\(v_0 = \frac{V_{\max}[\mathrm{S}]}{K_\mathrm{M} + \alpha'[\mathrm{S}]}\)
- 希尔方程Hill equation\(\theta = \frac{[\mathrm{L}]^{n_\mathrm{H}}}{K_{0.5}^{\,n_\mathrm{H}} + [\mathrm{L}]^{n_\mathrm{H}}}\)
- 单位点配体结合Single-site ligand binding\(\theta = \frac{[\mathrm{L}]}{K_d + [\mathrm{L}]},\qquad K_d = \frac{[\mathrm{P}][\mathrm{L}]}{[\mathrm{PL}]}\)
- 斯卡查德作图Scatchard plot\(\frac{\nu}{[\mathrm{L}]} = \frac{n}{K_d} - \frac{\nu}{K_d}\)
- 生化标准态自由能 ΔG°′Biochemical standard free energy ΔG°′\(\Delta G^{\circ\prime} = \Delta G^\circ + \nu_{\mathrm{H^+}}RT\ln 10^{-7},\qquad \Delta G = \Delta G^{\circ\prime} + RT\ln Q'\)
- ATP 水解与反应偶联ATP hydrolysis and reaction coupling\(\Delta G_{\text{总}} = \Delta G_1 + \Delta G_{\mathrm{ATP}} < 0,\qquad \Delta G^{\circ\prime}_{\mathrm{ATP\to ADP + P_i}} \approx -30.5\ \mathrm{kJ\,mol^{-1}}\)\(\Delta G_{\text{total}} = \Delta G_1 + \Delta G_{\mathrm{ATP}} < 0,\qquad \Delta G^{\circ\prime}_{\mathrm{ATP\to ADP + P_i}} \approx -30.5\ \mathrm{kJ\,mol^{-1}}\)
- 氨基酸等电点Isoelectric point of an amino acid\(\mathrm{pI} = \frac{\mathrm{p}K_{a1} + \mathrm{p}K_{a2}}{2}\)
- 膜电位的能斯特方程Nernst equation for membrane potential\(E_{\text{ion}} = \frac{RT}{zF}\ln\frac{[\mathrm{X}]_{\text{外}}}{[\mathrm{X}]_{\text{内}}} \approx \frac{61.5\ \mathrm{mV}}{z}\lg\frac{[\mathrm{X}]_{\text{外}}}{[\mathrm{X}]_{\text{内}}}\ (37\,{}^\circ\mathrm{C})\)\(E_{\text{ion}} = \frac{RT}{zF}\ln\frac{[\mathrm{X}]_{\text{out}}}{[\mathrm{X}]_{\text{in}}} \approx \frac{61.5\ \mathrm{mV}}{z}\lg\frac{[\mathrm{X}]_{\text{out}}}{[\mathrm{X}]_{\text{in}}}\ (37\,{}^\circ\mathrm{C})\)
- 戈德曼–霍奇金–卡茨方程Goldman–Hodgkin–Katz equation\(V_m = \frac{RT}{F}\ln\frac{P_\mathrm{K}[\mathrm{K^+}]_\text{外} + P_\mathrm{Na}[\mathrm{Na^+}]_\text{外} + P_\mathrm{Cl}[\mathrm{Cl^-}]_\text{内}}{P_\mathrm{K}[\mathrm{K^+}]_\text{内} + P_\mathrm{Na}[\mathrm{Na^+}]_\text{内} + P_\mathrm{Cl}[\mathrm{Cl^-}]_\text{外}}\)\(V_m = \frac{RT}{F}\ln\frac{P_\mathrm{K}[\mathrm{K^+}]_\text{out} + P_\mathrm{Na}[\mathrm{Na^+}]_\text{out} + P_\mathrm{Cl}[\mathrm{Cl^-}]_\text{in}}{P_\mathrm{K}[\mathrm{K^+}]_\text{in} + P_\mathrm{Na}[\mathrm{Na^+}]_\text{in} + P_\mathrm{Cl}[\mathrm{Cl^-}]_\text{out}}\)
- DNA 解链温度估算DNA melting temperature estimates\(T_m \approx 2(\mathrm{A}+\mathrm{T}) + 4(\mathrm{G}+\mathrm{C})\ {}^\circ\mathrm{C}\quad(\text{< 14 nt，华莱士规则})\)\(T_m \approx 2(\mathrm{A}+\mathrm{T}) + 4(\mathrm{G}+\mathrm{C})\ {}^\circ\mathrm{C}\quad(\text{< 14 nt, Wallace rule})\)
- 斯韦德贝里方程Svedberg equation\(s = \frac{v}{\omega^2 r} = \frac{M(1 - \bar v\rho)}{N_A f} = \frac{M D (1 - \bar v\rho)}{RT}\)
- 蛋白质 A₂₈₀ 浓度测定Protein concentration from A₂₈₀\(c = \frac{A_{280}}{\varepsilon_{280}\, l},\qquad \varepsilon_{280} \approx 5500\,n_\mathrm{W} + 1490\,n_\mathrm{Y} + 125\,n_\mathrm{SS}\ \ \mathrm{L\,mol^{-1}\,cm^{-1}}\)
高分子化学Polymer chemistry
聚合反应与高分子物理 Polymerisation & polymer physics
平均分子量与分布、逐步聚合与链式聚合动力学、共聚组成、高分子溶液和玻璃化转变。
Molar-mass averages and distributions, step-growth and chain-growth kinetics, copolymer composition, polymer solutions and the glass transition.
- 数均分子量Number-average molar mass\(\bar M_n = \frac{\sum_i N_iM_i}{\sum_i N_i} = \sum_i x_iM_i\)
- 重均分子量Weight-average molar mass\(\bar M_w = \frac{\sum_i N_iM_i^2}{\sum_i N_iM_i} = \sum_i w_iM_i\)
- 分散度（多分散指数）Dispersity (polydispersity index)\(Đ = \frac{\bar M_w}{\bar M_n} \ge 1\)
- 卡罗瑟斯方程Carothers equation\(\bar X_n = \frac{1}{1 - p},\qquad \bar X_n = \frac{1 + r}{1 + r - 2rp}\ (\text{计量比 } r = N_A/N_B \le 1)\)\(\bar X_n = \frac{1}{1 - p},\qquad \bar X_n = \frac{1 + r}{1 + r - 2rp}\ (\text{stoichiometric ratio} r = N_A/N_B \le 1)\)
- 弗洛里最概然分布Flory (most probable) distribution\(x_x = (1-p)p^{x-1},\qquad w_x = x(1-p)^2p^{x-1}\)
- 自由基聚合速率Rate of free-radical polymerisation\(R_p = k_p[\mathrm M]\left(\frac{fk_d[\mathrm I]}{k_t}\right)^{1/2}\)
- 动力学链长Kinetic chain length\(\nu = \frac{R_p}{R_i} = \frac{k_p[\mathrm M]}{2(fk_dk_t[\mathrm I])^{1/2}},\qquad \bar X_n = 2\nu\ (\text{偶合终止}),\ \ \nu\ (\text{歧化终止})\)\(\nu = \frac{R_p}{R_i} = \frac{k_p[\mathrm M]}{2(fk_dk_t[\mathrm I])^{1/2}},\qquad \bar X_n = 2\nu\ (\text{combination}),\ \ \nu\ (\text{disproportionation})\)
- 梅奥–刘易斯共聚方程Mayo–Lewis copolymerisation equation\(\frac{d[\mathrm M_1]}{d[\mathrm M_2]} = \frac{[\mathrm M_1]\left(r_1[\mathrm M_1] + [\mathrm M_2]\right)}{[\mathrm M_2]\left([\mathrm M_1] + r_2[\mathrm M_2]\right)},\qquad F_1 = \frac{r_1f_1^2 + f_1f_2}{r_1f_1^2 + 2f_1f_2 + r_2f_2^2}\)
- 马克–霍温克方程Mark–Houwink equation\([\eta] = K\,\bar M_v^{\,a},\qquad [\eta] = \lim_{c\to0}\frac{\eta_\text{sp}}{c},\ \ \eta_\text{sp} = \frac{\eta - \eta_0}{\eta_0}\)
- 弗洛里–哈金斯混合自由能Flory–Huggins free energy of mixing\(\frac{\Delta G_\text{mix}}{RT} = n_1\ln\phi_1 + n_2\ln\phi_2 + \chi\,n_1\phi_2\)
- 福克斯方程（共混物玻璃化温度）Fox equation for blend Tg\(\frac{1}{T_g} = \frac{w_1}{T_{g,1}} + \frac{w_2}{T_{g,2}}\)
- WLF 方程Williams–Landel–Ferry equation\(\lg a_T = -\frac{C_1(T - T_\text{ref})}{C_2 + T - T_\text{ref}},\qquad C_1 \approx 17.44,\ C_2 \approx 51.6\ \mathrm K\ (T_\text{ref} = T_g)\)
- 高分子链均方末端距Mean-square end-to-end distance\(\langle R^2\rangle_0 = Nb^2\ (\text{自由连接链}),\qquad \langle R^2\rangle_0 = Nl^2\frac{1 - \cos\theta}{1 + \cos\theta}\ (\text{自由旋转链}),\qquad \langle R^2\rangle_0 = C_\infty Nl^2\)\(\langle R^2\rangle_0 = Nb^2\ (\text{freely jointed chain}),\qquad \langle R^2\rangle_0 = Nl^2\frac{1 - \cos\theta}{1 + \cos\theta}\ (\text{freely rotating chain}),\qquad \langle R^2\rangle_0 = C_\infty Nl^2\)
- 回转半径Radius of gyration\(R_g^2 = \frac1N\sum_i\left|\mathbf r_i - \mathbf r_\text{cm}\right|^2,\qquad \langle R_g^2\rangle_0 = \frac{\langle R^2\rangle_0}{6}\ (\text{理想线形链})\)\(R_g^2 = \frac1N\sum_i\left|\mathbf r_i - \mathbf r_\text{cm}\right|^2,\qquad \langle R_g^2\rangle_0 = \frac{\langle R^2\rangle_0}{6}\ (\text{ideal linear chain})\)
材料与固体化学Materials & solid-state chemistry
半导体、晶体学与材料表征 Semiconductors, crystallography & materials characterisation
能带与载流子统计、晶面间距、晶粒尺寸、霍尔效应与合金晶格常数。
Bands and carrier statistics, interplanar spacings, crystallite size, the Hall effect and alloy lattice constants.
- 带隙与吸收边波长Band gap and absorption-edge wavelength\(\lambda_\text{边} = \frac{hc}{E_g} \approx \frac{1240\ \mathrm{nm\cdot eV}}{E_g}\)\(\lambda_\text{edge} = \frac{hc}{E_g} \approx \frac{1240\ \mathrm{nm\cdot eV}}{E_g}\)
- 费米–狄拉克分布Fermi–Dirac distribution\(f(E) = \frac{1}{e^{(E - E_F)/k_\mathrm BT} + 1}\)
- 本征载流子浓度Intrinsic carrier concentration\(n_i = \sqrt{N_cN_v}\,e^{-E_g/2k_\mathrm BT},\qquad np = n_i^2,\qquad N_c = 2\left(\frac{2\pi m_e^*k_\mathrm BT}{h^2}\right)^{3/2}\)
- 晶面间距（各晶系）Interplanar spacing for crystal systems\(\frac{1}{d^2} = \frac{h^2 + k^2 + l^2}{a^2}\ (\text{立方}),\quad \frac{h^2 + k^2}{a^2} + \frac{l^2}{c^2}\ (\text{四方}),\quad \frac{h^2}{a^2} + \frac{k^2}{b^2} + \frac{l^2}{c^2}\ (\text{正交})\)\(\frac{1}{d^2} = \frac{h^2 + k^2 + l^2}{a^2}\ (\text{cubic}),\quad \frac{h^2 + k^2}{a^2} + \frac{l^2}{c^2}\ (\text{tetragonal}),\quad \frac{h^2}{a^2} + \frac{k^2}{b^2} + \frac{l^2}{c^2}\ (\text{orthorhombic})\)
- 谢乐公式Scherrer equation\(\tau = \frac{K\lambda}{\beta\cos\theta}\)
- 霍尔效应Hall effect\(V_H = \frac{IB}{nqt},\qquad R_H = \frac{1}{nq},\qquad \mu = \frac{|R_H|}{\rho}\)
- 韦加德定律Vegard's law\(a_{\mathrm A_{1-x}\mathrm B_x} = (1 - x)\,a_\mathrm A + x\,a_\mathrm B\)
- 布洛赫定理Bloch's theorem\(\psi_{n\mathbf k}(\mathbf r) = e^{i\mathbf k\cdot\mathbf r}u_{n\mathbf k}(\mathbf r),\qquad u_{n\mathbf k}(\mathbf r + \mathbf R) = u_{n\mathbf k}(\mathbf r)\)
核化学Nuclear chemistry
放射性衰变与核结构 Radioactive decay & nuclear structure
衰变定律、半衰期与活度、放射性测年、衰变链平衡和原子核结合能。
The decay law, half-life and activity, radiometric dating, decay-chain equilibria and nuclear binding energy.
- 放射性衰变定律Radioactive decay law\(N(t) = N_0\,e^{-\lambda t},\qquad -\frac{dN}{dt} = \lambda N\)
- 半衰期Half-life\(t_{1/2} = \frac{\ln 2}{\lambda} \approx \frac{0.693}{\lambda},\qquad N = N_0\left(\frac12\right)^{t/t_{1/2}}\)
- 放射性活度Radioactivity (activity)\(A = -\frac{dN}{dt} = \lambda N = \frac{\ln2}{t_{1/2}}\cdot\frac{m N_A}{M},\qquad A(t) = A_0e^{-\lambda t}\)
- 放射性碳测年Radiocarbon dating\(t = \frac{t_{1/2}}{\ln2}\ln\frac{A_0}{A} = 8267\ \mathrm{a}\times\ln\frac{A_0}{A}\quad(t_{1/2} = 5730\ \mathrm a)\)
- 衰变链与放射平衡Decay chains and radioactive equilibrium\(N_2(t) = \frac{\lambda_1}{\lambda_2 - \lambda_1}N_1(0)\left(e^{-\lambda_1t} - e^{-\lambda_2t}\right) + N_2(0)e^{-\lambda_2t}\)
- 原子核结合能与质量亏损Nuclear binding energy and mass defect\(E_B = \Delta m\,c^2 = \left[Z m_\mathrm{H} + (A - Z)m_n - m_\text{原子}\right]c^2,\qquad 1\ \mathrm u\cdot c^2 = 931.494\ \mathrm{MeV}\)\(E_B = \Delta m\,c^2 = \left[Z m_\mathrm{H} + (A - Z)m_n - m_\text{atoms}\right]c^2,\qquad 1\ \mathrm u\cdot c^2 = 931.494\ \mathrm{MeV}\)
- 魏茨泽克半经验质量公式Semi-empirical mass formula (Weizsäcker)\(E_B = a_VA - a_SA^{2/3} - a_C\frac{Z(Z-1)}{A^{1/3}} - a_A\frac{(A - 2Z)^2}{A} + \delta(A, Z)\)
- 吸收剂量与当量剂量Absorbed dose and equivalent dose\(D = \frac{dE}{dm}\ (\mathrm{Gy} = \mathrm{J/kg}),\qquad H = \sum_R w_R D_R\ (\mathrm{Sv}),\qquad E = \sum_T w_T H_T\)
环境与化工Environmental & chemical engineering
环境化学与化学反应工程 Environmental chemistry & reaction engineering
大气臭氧、气液分配、大气消光、水质需氧量，以及物料衡算与理想反应器设计。
Stratospheric ozone, air–water partitioning, atmospheric extinction, oxygen demand in water, and material balances and ideal-reactor design.
- 查普曼臭氧机理Chapman mechanism for stratospheric ozone\([\mathrm O_3]_\text{ss} = [\mathrm O_2]\sqrt{\frac{j_1k_2[\mathrm M]}{j_3k_4}}\)
- 气–水分配（无量纲亨利常数）Air–water partitioning (dimensionless Henry constant)\(K_\text{aw} = \frac{c_\text{air}}{c_\text{water}} = \frac{H}{RT},\qquad H = \frac{p}{c_\text{water}}\ (\mathrm{Pa\,m^3\,mol^{-1}})\)
- 大气消光与光学厚度Atmospheric extinction and optical depth\(I = I_0\,e^{-\tau m},\qquad \tau = \int\sigma\,n\,dz,\qquad m \approx \frac{1}{\cos\theta_z}\)
- 生化需氧量的一级动力学First-order kinetics of biochemical oxygen demand\(\mathrm{BOD}_t = L_0\left(1 - e^{-k_1t}\right),\qquad \mathrm{BOD}_5 \approx 0.68\,L_0\ (k_1 = 0.23\ \mathrm d^{-1})\)
- 物料衡算Material balance\(\text{输入} - \text{输出} + \text{生成} - \text{消耗} = \text{累积},\qquad \frac{dn_i}{dt} = F_{i,0} - F_i + \nu_i\,r\,V\)\(\text{input} - \text{output} + \text{generation} - \text{consumption} = \text{accumulation},\qquad \frac{dn_i}{dt} = F_{i,0} - F_i + \nu_i\,r\,V\)
- 理想反应器设计方程（CSTR 与 PFR）Ideal reactor design equations (CSTR and PFR)\(\text{CSTR: } \tau = \frac{V}{Q} = \frac{c_{A0}X}{-r_A(X)},\qquad \text{PFR: } \tau = c_{A0}\int_0^X\frac{dX}{-r_A},\qquad \text{一级：}\ X_\text{CSTR} = \frac{k\tau}{1 + k\tau},\ X_\text{PFR} = 1 - e^{-k\tau}\)\(\text{CSTR: } \tau = \frac{V}{Q} = \frac{c_{A0}X}{-r_A(X)},\qquad \text{PFR: } \tau = c_{A0}\int_0^X\frac{dX}{-r_A},\qquad \text{first order :}\ X_\text{CSTR} = \frac{k\tau}{1 + k\tau},\ X_\text{PFR} = 1 - e^{-k\tau}\)
- 放热反应器的热稳定性（谢苗诺夫判据）Thermal stability of exothermic reactors (Semenov criterion)\(\text{放热 } Q_\text{gen} = (-\Delta H)\,V\,k_0e^{-E_a/RT}c,\quad \text{移热 } Q_\text{rem} = UA(T - T_c);\qquad \text{临界：}\ \psi = \frac{(-\Delta H)Vk(T_c)c\,E_a}{UA\,RT_c^2} = \frac1e\)\(\text{generation} Q_\text{gen} = (-\Delta H)\,V\,k_0e^{-E_a/RT}c,\quad \text{removal} Q_\text{rem} = UA(T - T_c);\qquad \text{critical :}\ \psi = \frac{(-\Delta H)Vk(T_c)c\,E_a}{UA\,RT_c^2} = \frac1e\)