Pharmaceutical Equations

Cheatsheet Content

### Introduction This cheatsheet covers essential pharmaceutical equations for exams like GPAT, NIPER, and Drug Inspector. Focus on understanding the purpose and recognizing the equations rather than exact memorization for MCQ-based exams. ### Henderson-Hasselbalch Equation - **Purpose:** Determines the percentage of ionized and unionized drug based on pH and pKa. - **Weak Acid:** $pH = pKa + \log \frac{[Ionized \ Form]}{[Unionized \ Form]}$ - Ionized Form = Salt - Unionized Form = Acid - **Weak Base:** $pH = pKa + \log \frac{[Unionized \ Form]}{[Ionized \ Form]}$ - Unionized Form = Base - Ionized Form = Salt - **Percentage Ionization (for Weak Acid):** $\frac{10^{(pH - pKa)}}{1 + 10^{(pH - pKa)}} \times 100$ - **Percentage Ionization (for Weak Base):** $\frac{10^{(pKa - pH)}}{1 + 10^{(pKa - pH)}} \times 100$ - **Related Concepts:** - $pH = -\log[H^+]$ (Sorensen) - $pKa = -\log[Ka]$ ### Noyes-Whitney Equation - **Purpose:** Describes the rate of drug dissolution. - **Rate-Controlling Step:** Diffusion - **Equation:** $\frac{dC}{dt} = \frac{DA}{Vh}(Cs - C)$ - D: Diffusibility/Diffusion Coefficient - A: Surface Area - V: Volume - h: Thickness of stagnant layer - Cs: Steady-state concentration - C: Final concentration - **Alternative Form:** $\frac{dm}{dt} = KA(Cs - C)$ (where $K = D/h$) - **Origin:** Derived from Fick's Law. - **Dissolution:** Mass transfer process where solid converts to liquid. Not needed for injectables, chewable tablets, liquids. ### Fick's Laws - **Fick's First Law:** - **Purpose:** Describes the rate of diffusion. - **Principle:** Drug molecules diffuse from higher to lower concentration. - **Equation:** Rate of diffusion $\propto$ Concentration gradient (implicitly discussed). - **Fick's Second Law:** Incorporates Fick's First Law and modifies Noyes-Whitney. ### Hixson-Crowell Cube Root Law - **Purpose:** Describes dissolution rate of powders where particle size decreases over time. - **Equation:** $W_0^{(1/3)} - W^{(1/3)} = Kt$ - $W_0$: Initial weight of powder - W: Weight of powder after time t - K: Dissolution rate constant - t: Time ### Dankwert's Model (Surface Renewal Theory) - **Purpose:** Related to penetration and surface renewal. - **Equation (Less important for exams):** $\frac{vdC}{dt} = \frac{dm}{dt} = A\sqrt{\frac{D}{\gamma}}(Cs-Cb)$ ### Bragg's Equation - **Purpose:** Used in X-ray crystallography to determine crystal structure. - **Equation:** $n\lambda = 2d\sin\theta$ ### Stokes' Law - **Purpose:** Related to sedimentation for particle size determination (e.g., Anderson pipette). - **Equation for Diameter:** $Diameter = \sqrt{\frac{18\eta H}{(\rho_s - \rho_l)gT}}$ - $\eta$: Viscosity of medium - H: Height - $\rho_s$: Density of solid particle - $\rho_l$: Density of liquid - g: Acceleration due to gravity - T: Time ### BET Equation (Brunauer–Emmett–Teller) - **Purpose:** Related to adsorption methods. - **Adsorption:** Surface phenomenon (molecules adhere to surface). - **Absorption:** Molecules penetrate and are distributed throughout the bulk. ### Kozeny-Carman Equation - **Purpose:** Used in air permeability methods (based on Poiseuille equation). - **Also used in:** Filtration. ### Angle of Repose - **Purpose:** Measures flow properties of powders and granules. - **Equation:** $\tan\theta = \frac{h}{r}$ - h: Height of the pile - r: Radius of the pile - **Related:** Call's Index, Compressibility Ratio, Hausner Ratio also measure flow properties. ### MacLeod Equation - **Purpose:** Describes relationship between surface tension and density of liquids. ### Gibbs Adsorption Equation - **Purpose:** Estimates the number of molecules per unit area that have adsorbed. ### Reynolds Number - **Purpose:** Indicates the flow pattern of fluids. - **Equation:** $Re = \frac{D \cdot V \cdot \rho}{\eta}$ - D: Diameter of the pipe - V: Velocity of the fluid - $\rho$: Density of the fluid - $\eta$: Dynamic viscosity - **Flow Regimes:** - $Re 4000$: Turbulent flow ### Bernoulli's Theorem - **Purpose:** Used in pharmaceutical engineering for measuring fluid flow rate. - **Application:** Principle behind orifice meters and Venturi meters. ### Fanning Equation - **Purpose:** Related to friction losses (appeared in GPAT 2023). ### Hagen-Poiseuille Equation - **Purpose:** Also related to friction losses. ### Fourier's Law - **Purpose:** Related to heat conduction. ### Stefan-Boltzmann Law - **Purpose:** Related to blackbody radiation. ### Laws Governing Size Reduction (KBR) - **Purpose:** Determine energy required for size reduction. - **Rittinger's Theory:** Energy related to new surface area created. - **Bond's Theory:** Another law for size reduction. - **Kick's Theory:** Another law for size reduction. - **Analogy:** KBR also used in IR spectroscopy for sampling. ### Raoult's Law - **Purpose:** Relates partial pressure of a gas in a mixture to its mole fraction, important in distillation. ### Dalton's Law of Partial Pressures - **Purpose:** Total pressure of non-reacting gas mixture is sum of individual partial pressures. - **Equation:** $P_{total} = P_1 + P_2 + P_3 + ...$ - **Related:** Dalton also associated with atomic theory and molecular weight. ### Filtration Equations - **Poiseuille Equation:** Describes theory of filtration (involves pressure difference, pore radius, medium length, viscosity). - **Darcy's Equation:** Describes flow of fluid through porous medium. - **Kozeny-Carman Equation:** Also related to filtration. - **Mnemonic:** "Darpak" (Darcy, Poiseuille, Kozeny-Carman). ### Ideal Gas Law - **Purpose:** Describes behavior of ideal gases. - **Equation:** $PV = nRT$ - P: Pressure - V: Volume - n: Number of moles - R: Gas constant - T: Temperature ### Other Gas Laws (Derived from Ideal Gas Law) - **Gay-Lussac's Law:** $\frac{P_1}{T_1} = \frac{P_2}{T_2}$ (Pressure and Temperature) - **Boyle's Law:** $P \propto \frac{1}{V}$ (Pressure and Volume) - **Charles's Law:** $V \propto T$ (Volume and Temperature) - **Avogadro's Law:** $\frac{V_1}{n_1} = \frac{V_2}{n_2}$ or $V \propto n$ (Volume and Number of Moles) ### Van der Waals Equation - **Purpose:** Describes relationship between P, V, T, and amount of real gases. - **Related:** Van der Waals forces are weak intermolecular forces. ### Graham's Law of Diffusion - **Purpose:** Rate of gas diffusion is inversely proportional to square root of its molecular mass. ### Clausius-Clapeyron Equation - **Purpose:** Describes relationship between vapor pressure and absolute temperature. - **Equation:** $\log\left(\frac{P_2}{P_1}\right) = \frac{\Delta H_{vap}}{2.303R} \left(\frac{T_2 - T_1}{T_1T_2}\right)$ - $\Delta H_{vap}$: Enthalpy of vaporization - R: Gas constant ### Adsorption Isotherms - **Freundlich Isotherm:** Describes relationship between gas pressure and temperature. - **Langmuir Isotherm:** Adsorption isotherm for monomolecular layer formation. ### Arrhenius Equation - **Purpose:** Describes effect of temperature on reaction rates, specifically on liquid viscosity. - **Equation:** $\eta = A \cdot e^{(-E_a/RT)}$ - $\eta$: Viscosity - A: Pre-exponential factor - $E_a$: Activation energy - R: Gas constant - T: Temperature