Equation Sheet

• Mole, Avogadro (6): N_A ~ 6.022 • 10²³ , 1 mole of carbon atoms 12g
• Length of a C-C single bond: ~ 1.54Å (from 1.4Å in benzene,shorter for double bonds)
• Kinetic energy = ½ mv²
• Energy units: J ≡ kg m² s^-2 ; 1 cal = 4.184 J; Cal ≡ kcal;
• 1eV = 1.6 10^-19 J (corresponds to 1240 nm). E=mc² ; c ≈ 3×10⁸ m/s , E= hf (f is cycles per second, or hertz [hertz] ≡ [sec]^-1
• 1 eV of one particle translates into ≈ 100 kJ of N_A of particles
• Force, Newton’s 2nd law, F =m a , Units: N ≡ kg m s^-2,
• 1 kg-force (or kgf) ≡ 1 kg m s^-2 = 9.8 N; 1 pound-force ≈ 4.45 N
• 1 psi ≡ pound-force-per-square-inch = 6894.76 Pa ≈ 0.068 atm ; 1 atm ≈ 14.7 psi;
• unit of psig (g for gauge) means in addition to the external pressure.
• Momentum P =m v , Δp = F Δt
• Celsius /Kelvin conversion °K = 273.15 + °C
• Gas constant R 8.314 J/(K•mol) 1.986 (≈2) cal/(K•mol) ; 0.08205746 L atm / (K mol) ; 5.189×10¹⁹ eV/(K mol)
• RT at room temperature of 302K is ≈ 0.6 kcal/mol or 2.5 kJ/mol
• Boltzmann constant (one molecule): k_B = R/N_A
• Mean translational movement energy molecules in gas: E = (1/2)mv² = (3/2)RT
• Graham's Law (effusion rate vs mol. mass): Rate₁ / Rate₂ = (M₂ /M₁ )^½
• Gas Law: PV = nRT , n = m/M , eg mass (m) in [g] and molar mass M in [g/mol]
• Density = m/V = PM/(RT)
• Pressure units: 1 Pa= 1 kg·m⁻¹ · s⁻² = 1N/m² ; 1 atm ≈ 10m of water = 760mmHg ≈ 1bar = 100,000 Pa; 1atm = 101,325 Pa ; 1bar= 14.5 psi.
• The Barometric Formula: P_h = P₀ • Exp( -M gh/RT ), M - molecular mass in [kg/mol]; g = 9.80665 m/s² ; h - height/altitude in [m].
• Half-pressure altitude: h_½ ~ 1/M , M - molecular mass (SI: kg)
• 1st law of thermodynamics: ΔU = q - w
• Enthalpy H = U + PV
• Heat capacity, C ≡ q / ΔT = ΔH / ΔT, at constant P or V
• Molar heat capacity C_m = C/n = q/(n ΔT ); Specific heat c=C/m = q/(m ΔT ), m-mass
• C_P ≡ ∂H /∂T, C_V ≡ ∂U / ∂T, C_P ~ C_V for liquids and solids ; C_P ~ C_V +N_moles • R for gases ( C_V +R for 1mole )
• Molar: C_V = N_{excited_vibr_modes} • R + N_{free_translation_rotation} • 0.5R (the second term is vanishing for large molecules)
• Dulong-Petit law (modern version): for any solid substance the high T limit of C is 3R • N_atoms (3 vibrations per atom)
• Water: specific heat = 4.184 J/(g K), or 1 cal/(g K) or 1 kcal/(kg K), 1 ml weighs 1g, 1L weighs 1kg
• Enthalpy at T, Kirchhoff: H_T2 = H_T1 + C_P (T₂ – T₁ )
• Calorimetry: ΔH = s m ΔT, m is mass, s is specific heat.
• Entropy measurement and classical definition: ΔS = q/T
• Entropy at T, (C_P ≈ const) : S_T2 = S_T1 + C_P ln (T₂ / T₁ )
• Boltzmann for entropy: S=k_B ln(N_total ), S_m = Rln( n₁ ) (n₁ is the number of states of ONE molecule)
• Gibbs free energy: G = H – TS = U + PV – TS; dG = PdV – SdT
• Estimate of ligand entropic cost of binding -T × ΔS_{conformational} ≈ 0.6n_{rotatable_bonds} [kcal/mol]
• Phase equilibrium: G_phase1 =G_phase2; ΔH_trs =T_trs ΔS_trs
• Trouton’s rule: ΔS_vap ~ 85-89 J/(K mol) for many liquids (except water), hence ΔH_trs is proportional to T_trs
• Clausius-Clapeyron equation: ln( P₂ / P₁ ) = (Δ H_vap / R)• ( 1/T₁ - 1/T₂ )
• Hess: ΔH_reaction = ΔH_f (products) - ΔH_f (reactants) ; same for ΔS and ΔG

• Fundamental equation dG of a mixture: dG = VdP-SdT+μ_A dn_A + μ_B dn_B + ....
• Molar fraction: x_A = n_A / Σn_i , n_i is the number of moles of component i
• Dalton's Gas Mixture Law: P_A = x_A • P_total , x_A is a molar fraction
• Chemical potential changes from the reference state 0 (value at partial pressure P₀ or V₀ or C₀ ):
  • μ_A (V) = μ_A0 − RT•ln(V_A /V₀ ) (gas)
  • μ_A (P) = μ_A0 + RT•ln(P_A /P₀ ) (gas)
  • μ_A (C) = μ_A0 + RT•ln(x_A /x₀ ) (solution)
• Chemical equilibrium: ln K = − ΔG⁰ _reaction / RT , or ΔG⁰ = -RTlnK , or K = exp( − ΔG⁰ /RT )
• K ≈ C_A ^νA • C_B ^νB • .. , ν_i - stoichiometry coefficients, negative for reactants. Use activities instead of concentrations.
• K ≈ ( [C]^c [D]^d ) /( [A]^a [B]^b ) for aA + bB ⇔ cC + dD . Stoichiometry coeff. a,b,c,d define the unit of K
• van't Hoff : ln( K₁ /K₂ ) = ( Δ_r H⁰ / R)(1/T₂ − 1/T₁ )
• van't Hoff : log₁₀ ( K ) = − ( Δ_r H⁰ / 2.3 RT ) + ( Δ_r S⁰ / 2.3 R )

Colligative Properties

• Entropy of binary mixing (x + y = 1 mole): ΔS_mix = –R (x ln x + y ln y); 0.7R for 0.5:0.5
• μ of water in solution: μ_w = μ_{w_pure} + RT ln x_w ≈ μ_{w_pure} - RT x_solute (x_solute << 1)
• Raoult's law: ΔP_{w_vapor} = x_solute • P_{w_pure} (non-volatile solute)
• Henry's law: P_{A_vapor} = x_A • K_{H_A} (A is a volatile solute or dissolvable gas)
  • oxygen (O₂ ) : K_H = 769.2 L·atm/mol
  • carbon dioxide (CO₂ ) : K_H = 29.41 L·atm/mol
  • hydrogen (H₂ ) : K_H = 1282.1 L·atm/mol
• Freezing: ΔT_f = K_f • c_solute . K_{f_water} ≈ 1.853 K·L/mol.
• Boiling : ΔT_b = K_b • c_solute . K_{b_water} ≈ 0.512 K·L/mol.
• x_A and c_A take van't Hoff factors i (or dissociation) into account. i=x₁ + 2•x₂ + 3•x₃ ..
• Molarity and Osmolarity (per L of solution): M ≡ mole/L , Osm ≡ osmole/L .
• Plasma osmolarity: 285 ± 10 mosmoles/L (mOsm)
• Simple osmosis rules: 1) water follows stuff; 2) pressure is proportional to the total molar concentration difference
• Pressure: Δ_osm P = ΔM_solute • RT ≈ 25.4 • ΔM_solute [atm], or 25.7 ..[bar] (C_solute in Osm, T_body = 309.75K ) , M → iM
• Water: 55.5 mol/L ( 1000g L^-1 / 18g/mol )

Binding reaction (dissociation and association)

• K_eq = exp( -ΔG₀ /RT ) ; pK_d ≈ 0.7 * ΔG₀ [kcal/mole]; ΔG₀ ≈ 1.4 pK_d
• PL ↔ P + L; K_d = [P][L]/[PL] ; K_a = 1/K_d ; ΔG_a = -ΔG_d ; ΔG_a = RT Ln K_d
• x = [PL]; (P₀ -x)(L₀ -x) = x • K_d
• x = ½ • ( P₀ + L₀ + K_d - ((P₀ + L₀ + K_d )² -4P₀ L_d )^½ )
• Non-abundant protein target P₀ << K_d : [PL]/[P] ≈ L₀ /K_d (50% inhibition @ L₀ = K_d )
• Fraction_of_receptor_bound ≈ L₀ /(K_d + L₀ ) ,where L₀ is total ligand concentration.
• Protein in excess [PL] << P₀ : [PL]/[L] ≈ P₀ / K_d , fraction_drug_bound ≈ P₀ / (P₀ +K_d )

Quantum Mechanics, Waves, Radioactivity, Fluorescence

• E = hν = ℏω , ν - cycles per second, ω - radians per second, E=mc²
• Planck constant: h=6.626 10^-34 J.s , ℏ ≈ 10^-34 J.s
• wavelength λ ; λ = c/ν ; E = h c/ λ (c = 3.•10⁸ m/s )
• E=mc² The speed of light: ≈ 3.•10⁸ meters per second (299,792,458 m·s⁻¹ ).
• 1 eV = 1.6 • 10^-19 J ; 1 J = 1 J = 6.2415 • 10¹⁵ keV
• 1eV corresponds to 1240 nm (nano meter,infrared) λ = 1240 nm/ E[eV]
• charge e = 1.60217 • 10⁻¹⁹ C (coulomb or s·A)
• hν = mc²

Interaction Energetics

• U = (1/4πε₀ )(q₁ q₂ /(εd) ) , d distance, ε is dielectric constant
• Units: U [kcal/mole] = C ( q₁ q₂ /(εd) ) , if C=332, d in Å q in A.U.
• Formal_charge = N_valence_electrons – N_electrons_in_lone_pairs - N_bonds
• Solvation energies: U_m-w ≈ C (q² /2r_q )( 1/ε_w - 1/ε_m ); U_w-m = Const × Z² /r_q
• Van der Waals : U_vw = A/d¹² - B/d⁶
• Hydrophobic effect U_hp = σ A_area
• Polar surface area (PSA): PSA < 140 Ų ; PSA_BBB < 75 Ų
• Conformational entropy loss estimate: ΔN_states ≈ 3^ΔNtor , ΔS_conf ≈ R • ΔN_tor • ln(3)

Acid-Base equilibrium, LogP

• LogP ≡ Log₁₀ (C_oct / C_water )
• pH ≡ -Log₁₀ ( [H+] ) ; pOH = 14 − pH
• pKa = -Log₁₀ ( [A-][H+]/[AH] ), Ka = [A-][H+]/[AH]
• pKa_base = -Log₁₀ ( [B][H+]/[BH+] ), Ka_base = [B][H+]/[BH+]
• Henderson-Hasselbalch (Log₁₀ .. ):
  • Log([A-]/[AH]) = pH − pKa
  • Log([B]/[BH+]) = pH − pKa
• logD = log ( all_forms_in_octanol / all forms in water )
• acids: logD = logP - log10(1 + 10^pH-pKa ) ≈ logP – (pH – pKa) (for pH > pKa+1, charged form dominates)
• bases: logD = logP - log10(1 + 10^-(pH-pKa) ) ≈ logP + (pH – pKa) (for pH < pKa-1, charged form dominates)

Drug solutions, pH, solubility

• Strong acid: pH = -log c ; strong base: pH ~ 14+log C ( C >> 10^-7 M )
• Weak acid: pH = ½ pKa − ½log(C) , ( C >> 10^-7 M )
• Weak base: pH = 7 + ½ pKa + ½log(C) , ( C >> 10^-7 M )
• More accurate (acid): (c-x)K_A = x(x+10^-7 ), solve for x ≡ [A-]
• Solubility: μ_{0_aq} + RT ln S_w = μ_{0_crystal}
• Solubility w.acid: S₀ ≡[AH], [A-]=S-S₀ ; Log((S - S₀ )/S₀ )=pH-pK_A ; S = S₀ (1+10^pH-pKa )
• Solubility w.base: S₀ ≡[B], [BH+]=S-S₀ ; Log((S - S₀ )/S₀ )=pK_A -pH ; S = S₀ (1+10^pKa-pH )
• LogSw ≡ Log₁₀ (S)

Kinetics. Rates

• d[A]/dt = k×([A]^x ×[B]^y ,...), reaction order is x + y + .., e.g.
• Zero order: d[A]/dt = -k ; [A] = [A₀ ]-kt ; t_½ = [A₀ ]/2k
• First order: d[A]/dt = -k[A] ; [A] = [A₀ ]× exp( -kt ) ; t_½ = ln(2)/k
• Second order: d[A]/dt = –k[A]² ; 1/[A] = 1/[A₀ ] + kt
• k ~ exp( -ΔG_act /RT )
• reversible reactions: A ↔ B : K = [B]/[A] = k_f / k_r ;
• binding reactions: Kd = k_off / k_on = [A][B]/[AB] ;
• Absorption: Amount absorbed: ∝ Area × ( ΔC ) × Time
• Dissolution rate (R_D ) ∝ Area × ( C_saturated - C )

Methods

• X-ray: distances ( d ) to reflection angles ( θ ) : 2d sin( θ ) = nλ
• Mass.Spec: F = q( E + v ⊗ B ) , R = (m/q) × ( v/B )
• Chem.shift: ΔE ∝ ν ∝ γ B ; B = f( B₀ and electron shielding)
• Units of B: Tesla (T) = 1 Newton/(Ampere ∙ meter) = kg/(A ∙ s² )
• For a 21.1 tesla magnet, the resonant frequency of H+ is about 900 MHz. Earth: 5 to 65 μT

Target binding free energies, enthalpies and entropic contributions for 9 HIV drugs

Hydrogen bonding

H-Bond enthalpies

Type	Enthalpy kJ/mol	kcal/mol
O−H…:N	29	6.9
O−H…:O	21	5.0
N−H…:N	13	3.1
N−H…:O	8	1.9

Heats of Solution of various ions in water

Periodic Table and some masses

N El Mass ≈ Isotopes 1 H 1.00794 1. 1 H, 2 H (0.015%), 3 H* (12.32y) 2 He 4.002602 4. 3 Li 6.941 7. 5 B 10.811 11. 6 C 12.011 12. 11 C(20min), 13 C (1.1%), 14 C* (5700y) 7 N 14.00674 14. 8 O 15.9994 16. 9 F 18.9984 19. 18 F(110min) 10 Ne 20.18 20.

N El Mass ≈ Isotopes 11 Na 22.98977 23. 12 Mg 24.305 24.3 15 P 30.97376 31. 16 S 32.066 32. 17 Cl 35.4527 35.5 35 Cl (75%), 37 Cl (25%), 36 Cl* (308,000y) 19 K 39.0983 39. 20 Ca 40.078 40. 26 Fe 55.847 55.5 30 Zn 65.39 65.4

N El Mass ≈ Isotopes 35 Br 79.9 80. 79 Br (50.7%) 81 Br (49.3%) 53 I 126.9 127. 127 I , 129 I (15.7M y)

Radionuclides