PF1009 Kinetics Lecture & Tutorial Slides PDF

Kinetics; a short course for PF1009 Dr. Humphrey Moynihan  Kinetics Kinetics deals with the rates of processes How quickly does a process occur? How long will it take for a system to reach equilibrium? How does that timescale compare with , e.g., the shelf- life or a...

Kinetics; a short course for PF1009 Dr. Humphrey Moynihan  Kinetics Kinetics deals with the rates of processes How quickly does a process occur? How long will it take for a system to reach equilibrium? How does that timescale compare with , e.g., the shelf- life or a drug product or the time for the absorption, distribution and excretion of a drug? 2 Kinetics vs. thermodynamics Thermodynamics determines the position of equilibrium for a process However, the time it takes for the system to reach equilibrium is determined by kinetics Thermodynamics give information about the relative energies of the initial and final states The rate of the process is determined by the energetics of the process pathway This is reflected in the kinetics Kinetics in a pharmaceutical context Stability over time of Drug substances (Active Pharmaceutical Ingredients) Drug products (formulations of drug substances) The rate of key pharmaceutical processes, including Processes for manufacturing drug substances and drug products Dissolution of drug substances and products Pharmacological and biochemical processes 4 Pharmacokinetics Investigates the kinetics of absorption, distribution, metabolism and excretion of drugs Also referred to as pharmacokinetics & drug metabolism Advanced specialised topic Need fundamentals of kinetics first  Concentrations of components The key parameters in kinetics are variations in the concentration of components Process occurs faster if the concentration of one or more of the components is increased Higher concentration results in faster reaction rate 6 Temperature Temperature is a key factor affecting kinetics The rate of a process increases as the temperature is increased Molecules must encounter each other in order to react As molecules move more rapidly they collide more frequently, leading to an increased rate. Hence, higher temperature results in faster reaction rate 7 Consumption and production of components during a process Instantaneous rate at time t 8 The Reaction Rate The reaction rate –the “speed” of a process Rate = decrease in concentration of reactants or increase in concentration of products with respect to time. 𝑑 𝑝𝑟𝑜𝑑𝑢𝑐𝑡𝑠 𝑑 𝑟𝑒𝑎𝑐𝑡𝑎𝑛𝑡𝑠 𝑅𝑎𝑡𝑒 = =− 𝑑𝑡 𝑑𝑡 9 Kinetics: example C4H9Cl(aq) + H2O(l) → C4H9OH(aq) + HCl(aq) Rates normally decrease as the reaction proceeds because the reactants are being consumed (i.e. concentration is decreasing) 10 Showing the data graphically Instantaneous rate can be determined from the slope of a tangent to the curve at the point of interest Note: the initial rate can be approximately measure by rapid monitoring 11 Reaction rates and stoichiometry C4H9Cl(aq) + H2O(l) → C4H9OH(aq) + HCl(aq) 1 mol C4H9OH is being produced for every mol C4H9Cl consumed The rate of disappearance of the reactant is the same as the rate of appearance of the product 𝑑 𝐶4𝐻9𝑂𝐻 𝑑 𝐶4𝐻9𝐶𝑙 𝑅𝑎𝑡𝑒 = =− 𝑑𝑡 𝑑𝑡 12 Reaction rates and stoichiometry 2HI(g) → H2(g) + I2(g) 1 𝑑 𝐻𝐼 𝑑 𝐻2 𝑑 𝐼2 𝑅𝑎𝑡𝑒 = − = = 2 𝑑𝑡 𝑑𝑡 𝑑𝑡 Generally: 2A + 3B → 2C + 4D 1𝑑 𝐴 1𝑑 𝐵 1𝑑 𝐶 1𝑑 𝐷 𝑅𝑎𝑡𝑒 = − =− = = 2 𝑑𝑡 3 𝑑𝑡 2 𝑑𝑡 4 𝑑𝑡 Choice of term to define the rate often depends on which component can be monitored experimentally 13 The Rate Law The rate of a reaction is directly proportional to the concentrations of the reactants aA + bB → cC + dD The general rate law is: Rate = k[A]x[B]y k = rate coefficient (sometimes called the rate constant) which varies with temperature x is the order of reaction with respect to A y is the order of reaction with respect to B 14 Reaction order Rate = k[A]x[B]y x and y are called reaction orders The overall reaction order is x + y When x + y = 0 = zero order reaction = 1 = first order reaction = 2 = second order reaction NB: Reaction orders cannot be inferred from the stoichiometries of the balanced equation and must be determined experimentally 15 Reaction order If the following reaction is second order with respect to A and zero order wrt B. 3A + 2B → 2C + 3D Then the rate law for the reaction is: Rate = k[A]2 NOT Rate = k[A]3 16 Using the initial rates method to obtain the order of reaction Q. The initial rate of a reaction A+B→C was measured at several different starting concentrations of A and B. The results of the experiment are displayed in the table below. Using the data, determine the rate law for the reaction, Exp. no. [A]/molL-1 [B]/molL-1 Initial rate/molL-1s-1 1 0.100 0.100 4.0 x 10-5 2 0.100 0.200 4.0 x 10-5 3 0.200 0.100 16.0 x 10-5 17  Changing the concentration of B has no effect on the rate of the reaction. What does this tell us about the order of reaction with respect to [B]? The reaction is zero order in [B] However doubling [A] has the effect of quadrupoling the rate. What does this tell us about [A]? The reaction is second order with respect to [A] The rate law: Rate = k[A]2[B]0 = k[A]2 18  Zero order kinetics Rate is independent of concentration 12 12 Concentration of reactant 10 10 Reaction rate 8 8 6 6 4 4 2 2 0 0 1 2 3 4 5 6 7 8 9 0 2 4 6 8 10 Time Time The rate is independent of the Concentration falls at a reactant concentration and constant rate until all the remains constant until all reactant is used up reactant is used up, when it falls abruptly to zero 19  Zero order kinetics Rate of reaction independent of concentrations of reactants 12 [A] − [A]o = −kt [A]0 initial concentration of A Concentration of reactant 10 8 [A] = −kt + [A]o 6 4 2 y = mx + c 0 1 2 3 4 5 6 7 8 9 Plot of [A] vs. t should give a Time straight line plot if zero order kinetics (slope = −k) 20 First order Kinetics Rate = k[A]1 Rate = k[A] The rate is directly proportional to the conc. of the reactant A Double A = Double rate Quarter A = Quarter Rate 21 First order kinetics ln [A] − ln [A]0 = −kt ln [A] = −kt + ln [A]0 Plot of ln[A] vs. t should give a straight line plot if first order kinetics (slope = −k) 22 First order kinetics Variation of concentration w.r.t. time – gives a curve Plot of natural logarithm v. time yielding a straight line 23 Second order kinetics Plot of 1/[A] vs. t should give a straight line plot if second order kinetics (slope = k) 24 Rate laws for zero order, first order and second order kinetics Differential form Integrated form Zero order [A] = −kt + [A]o First order ln [A] = −kt + ln [A]0 Second order  Half-life The half-life of a reaction, t½ is the time required for the concentration of a reactant to drop to half of its initial value [A]t½ = ½[A]0 Concentration of A at initial concentration t = t1/2 26 t1/2 for a zero order reaction [A] = -kt +[A]0 At t = t1/2, [A] = ½[A]0 ½[A]0 = -kt1/2 + [A]0 - ½[A]0 = -kt1/2 t1/2 = ½[A]0 k Half-life 27  Half-life for a first order reaction ln[A] = -kt + ln[A]0 At t = t1/2, [A] = ½[A]0 ln [A]0 = -kt1/2 + ln[A]0 2 ln [A]0 - ln[A]0 = -kt1/2 2 ln [A]0 = -kt1/2 2[A]0 ln ½ = -kt1/2 -ln ½ = kt1/2 ln 2 = kt1/2 (note: independent of [A]) ln 2/k = t1/2 28 Half life of a second order process 29 Half-life formulas for zero order, first order and second order kinetics t1/2 = [A]0 ln 2/k = t1/2 2k First order Zero order Second order  Temperature and Rate The rates of most processes increase as the temperature increases The faster rate at higher temperature is due to an increase in the rate coefficient (k) with increasing temperature 31 The Collision Model Increasing the temperature will cause the molecules to move around more quickly which will cause more collisions and the rate will increase In most reactions only a tiny amount of collisions will lead to a reaction- this is governed by two different factors (a) the orientation factor (b) the energy of the collision [Activation Energy (Ea)] 32 The Orientation Factor Molecules must be orientated in a certain way so that when collisions do occur they will lead to reaction 33 Activation Energy In order to react, colliding molecules must have a total energy equal to or greater than some minimum value The minimum energy required to initiate a process is called the Activation Energy (Ea) Every reaction has a different Ea value Multi-step processes have multiple activation energies; the step with the largest Ea is the rate determining step 34 Activation Energy (Ea) 35 Arrhenius Equation For most reactions the increase in rate with increasing temperature is non-linear Because the reaction-rate is affected by: 1. The fraction of molecules possessing Ea 2. The number of collisions occurring per second 3. The fraction of collisions that have the correct orientation 36 Arrhenius Equation k = Ae-Ea/RT k = rate coefficient Ea = activation energy R = gas constant T = temperature A = frequency factor (related to the frequency of collisions and the probability that the collisions are favorably oriented for the reaction) 37 Arrhenius Equation k = Ae-Ea/RT Taking natural log of both sides: lnk = -Ea/RT + ln A A graph of lnk versus 1/T will be a straight line with a slope equal to –Ea/R and a y-intercept equal to lnA 38  Arrhenius Equation Can also evaluate Ea if you know the rate coefficient of a reaction at 2 different temperatures e.g. T1 has a rate coefficient k1 and T2 has a rate coefficient k2 Subtracting k2 from k1 and simplifying: Activation energy for degradation and rates of degradation important to evaluate pharmaceutical stability 39 Multi-step processes Many pharmaceutically important processes involve multiple steps Each step has a corresponding activation energy and rate coefficient, k. E.g., a two-step process with an intermediate product k1 is the rate coefficient for conversion of A to B k2 is the rate coefficient for the conversion of B to C The smaller the rate coefficient, the slower the process step Rate determining steps In such cases, the kinetics are controlled by the rate determining step If the rate of formation of B from A is much slower than the conversion of B to C… ….there is no build up of B. The conversion of A to B is the rate determining step for the overall process 𝑘1 ≪ 𝑘2 𝑑𝐴 𝑅𝑎𝑡𝑒 𝑜𝑓 𝑟𝑒𝑎𝑐𝑡𝑖𝑜𝑛 = − = 𝑘1 𝐴 𝑑𝑡  Rate determining steps When k1 ˃˃ k2, A is rapidly converted to B B is consumed in the slow (rate determining) step The conversion of B to C dominates the kinetics of the process 𝑑𝐵 𝑅𝑎𝑡𝑒 𝑜𝑓 𝑟𝑒𝑎𝑐𝑡𝑖𝑜𝑛 = − = 𝑘2 𝐵 𝑑𝑡 However, [B] is not directly observable Steady state approximation k1 ˃˃ k2, k2[B] determines reaction rate, but [B] not directly observable Under certain circumstances, it is reasonable to assume that the concentration of B is low and relatively constant. I.e., B is being consumed at approximately the same rate that it is being produced. If so, can apply the steady state approximation, i.e. 𝑑𝐵 𝑑𝐵 =− =0 𝑑𝑡 𝑑𝑡 Steady state approximation B produced by the first step, k1[A] B consumed by the second step, k2[B] i.e: 𝑑𝐵 = 𝑘1 𝐴 − 𝑘2 𝐵 = 0 𝑑𝑡 𝑘1 𝐴 = 𝑘2 𝐵 𝑘1 𝐴 𝐵 = 𝑘2 Steady state approximation Now have expression for [B] in terms of observable parameter [A] 𝑘1 𝐴 𝐵 = 𝑘2 Applying the steady state approximation to the case of k1 ˃˃ k2 𝑘1 𝐴 𝑅𝑎𝑡𝑒 𝑜𝑓 𝑟𝑒𝑎𝑐𝑡𝑖𝑜𝑛 = 𝑘2 𝐵 = 𝑘2 = 𝑘1 𝐴 𝑘2 Rate laws for zero order, first order and second order kinetics Differential form Integrated form Zero order [A] = −kt + [A]o First order ln [A] = −kt + ln [A]0 Second order Half-life formulas for zero order, first order and second order kinetics t1/2 = [A]0 ln 2/k = t1/2 2k First order Zero order Second order  Tutorial Sucrose (C12H22O11) reacts to form glucose and fructose; C12H22O11 + H2O → 2C6H12O6. At 23C and in 0.5M HCl, the following data were obtained for the disappearance of sucrose: Time (mins) [C12H22O11] M 0 0.316 39 0.274 80 0.238 140 0.190 210 0.146 (a) Fit the data to both first order and second order kinetics (i.e. construct plots for both). Does the data better fit first order or second order kinetics? (b) Determine the rate coefficient for the better fitting data?  Tutorial: half-life A solution of a drug contained 0.50 mmol per millilitre when prepared. It was analysed after 40 days and was found to contain 0.30 mmol per millilitre. Assuming the decomposition is first-order, at what time will the drug have decomposed to one half its original concentration? First need to calculate k and as have been told it is a first order reaction we use the first order equation: ln[A] = -kt + ln[A]0  Tutorial: Arrhenius equation The following data for the decomposition of penicillin was obtained: Temperature / °C 37 54 Rate coefficient (k) / hr−1 0.0216 0.119 Calculate the activation energy Ea (R = 8.314 J K−1 mol−1); need T values in K (+273) Need Or 𝑘1 𝐸𝑎 𝑇1 − 𝑇2 ln = 𝑘2 𝑅 𝑇1 𝑇2