Exponential Decay CHEM 1231 Fall 2009 Claudia Neuhauser
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1 Exponential Decay Propylene glycol monometyl eter (PGME) is a solvent used in surface coatings or cleaners. A study by Domoradzki et al. 003 examined ow quickly PGEM was eliminated from blood in vitro. Doses of 10 or 100 mg/kg were injected into te blood of rats and te concentration of PGME in te blood was monitored. Te following table lists te concentration of PGME in μg per g blood at successive time points for te two initial concentrations (low and ig). Time [min] Concentration PGME [μg/g blood] low ig In class Activity 1 Te data is available in te accompanying Excel spreadseet under tab PGME data. Plot te data in Excel as a Scatter Plot. Transform te axes until te data for eac experiment lie along a straigt line. Use Trendline to fit an appropriate function. Display te equation of te function and te R value for eac of te two graps. Exponential Decay In In class Activity 1, you found tat te best fit was an exponential decay function of te form (1) yt () = y(0) e kt We will investigate tis function in te following. Funding: Tis work was partially supported by a HHMI Professors grant from te Howard Huges Medical Institute. Page 1
2 Successive Ratios Suppose we measured y every units of time, tat is, we would obtain te data y(0), y( ), y( ), y(3 ),... If we calculated successive ratios y() y() y(3),,, y(0) y( ) y( ) We would find tat all ratios are identical, namely In class Activity y () = e y(0) In te spreadseet under tab Ratio, values of te function yt () = y(0) e kt at different times are calculated (Column A and B). In Column C, we list successive ratios. (a) Confirm tat te value of te ratio does not depend on time t and te value of y (0), but depends on te parameter k and te time step. (b) For y (0) = 10, determine te value of te ratio for te following combinations of and k: k (c) Define te quantity p as k yt ( + ) = 1 p yt () Interpret te quantity p. Average Velocity Te average velocity of a quantity is defined as te ratio of te amount of cange during a time interval and te lengt of te time interval yt ( + ) yt ( ) average velocity = Sometimes, it is useful to scale tis ratio relative to te current amount of te quantity. We call tis new quantity ten relative average velocity Funding: Tis work was partially supported by a HHMI Professors grant from te Howard Huges Medical Institute. Page
3 () 1 yt ( + ) yt ( ) relative average velocity = yt () In class Activity 3 In te spreadseet under tab RAV, calculate te relative average velocity for = 100,10,1,0.1,0.01,0.001, wen (a) y(0) = 10, k=, (b) y(0) = 10, k= 3, (c) y(0) = 100, k=, and (d) y(0) = 100, k= 3. Enter te results in te table provided and grap te relative average velocity as a function of for eac of te four cases (a) (d) above. As approaces 0, wat value does te relative average velocity approac? To answer te question, logaritmically transform te orizontal axis were is displayed. A Differential Equation Describing Exponential Decay In In class Activity 3, you found tat te relative average velocity (3) 1 yt ( + ) yt ( ) k yt () as approaces 0 Te cange in y values, yt ( + ) yt ( ), is often denoted by Δ y, were te Greek letter Delta, Δ, refers to difference. Te quantity is te time difference of te times wen te two data points yt () and yt ( + ) were collected and is often denoted by Δ t. We can ten write for te relative average velocity 1 yt ( + ) yt ( ) 1 Δy relative average velocity = = yt () yδt As, or Δ t, approaces 0, te relative average velocity becomes a relative instantaneous velocity, or a relative rate of cange, wic we denote formally as 1 dy. Wit te result in (3), we tus find ydt (4) 1 dy k ydt = If we multiply bot sides by y, we find for te instantaneous velocity, or rate of cange, Funding: Tis work was partially supported by a HHMI Professors grant from te Howard Huges Medical Institute. Page 3
4 (5) dy ky dt = Equations (4) or (5) are examples of a first order differential equation. Tese kinds of equations relate an instantaneous velocity of a quantity to te quantity itself (or to a function of te quantity). Tey are studied in detail in calculus. In calculus, you will learn ways to figure out wat functions satisfy a given differential equation. We will not concern ourselves wit learning tese metods at tis point. Tere is computer software tat can do tis task. For instance, if we wanted to find a function y tat satisfied Equation (5), we could use te WolframAlpa searc engine. For tis, we need to know first tat instead of dy, te alternative dt notation y is used. We would enter into te WolframAlpa searc engine ten y () t + k* y() t = 0, y(0) = 10 and find tat yt () = 10e kt. Here is a screensot of te result Figure 1: Screensot of WolframAlpa solving te differential equation dy = ky wit (0) 10 dt y =. We ave now related exponential decay of te form yt () = y(0) e kt to te rate of cange of yt (), namely, dy ky dt =. Funding: Tis work was partially supported by a HHMI Professors grant from te Howard Huges Medical Institute. Page 4
5 Half Life An important quantity describing exponential decay is te alf life of te quantity, denoted by T, wic is defined as te amount of time it takes to reduce te quantity to alf its amount. Matematically, tis is expressed as Wit yt () = y(0) e kt, we find 1 yt ( + T) = yt ( ) Tis simplifies to Solving for T, In class Activity 4 kt ( + T) 1 kt y(0) e = y(0) e kt 1 e = ln T = k For te data in In class Activity 1, find te alf life for eac of te two experiments. In class Activity 5 (Callenge) Te quantity p, defined above, is related to te parameters k and as follows p = 1 If we interpret tis as te probability tat a molecule will decay during te time interval of lengt, we can build a stocastic model in Excel tat reflects tis and compare it to deterministic decay, as defined by Equation (1). Develop te simulation in an Excel spreadseet and grap bot te stocastic decay curve and its corresponding deterministic decay curve in te same grap. Since we know tat te functional relationsip is described by exponential decay, use a semi log grap. Funding: Tis work was partially supported by a HHMI Professors grant from te Howard Huges Medical Institute. Page 5 k e
6 Additional Application Hydrolysis Consider a cemical reaction were a cemical compound is ydrolyzed k R X+ H O R OH+ H + In tis reaction, te compound R X reacts wit water. If water is abundant and te amount of water available remains essentially uncanged during te reaction, te reaction can be considered as a decay of a compound. If we denote te amount of te compound R X by [RX], ten te rate of cange of disappearance of te compound is described by d[rx] = k[rx] dt Tis is of te same form as Equation (5). We tus expect tat were [RX] 0 is te amount available at time 0. [RX] = [RX] 0 e kt References J.Y. Domoradzki, K.A. Brzak, and C.M. Tornton Hydrolysis kinetics of propylene glycol monometyl eter acetate in rats in vivo and in rat and uman tissue in vitro. Toxicological Sciences 75: Funding: Tis work was partially supported by a HHMI Professors grant from te Howard Huges Medical Institute. Page 6
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