Calculation Session 1 Lecture 2: Rate equations from experimental data; concentration vs. time method of initial rates isolation method integrated rate equations method of half lives Defining Question: How to best assess graphically an empirical fit to experimental data?   Discussion Quizzes or No Quizzes? Exercises 1, 2, 4, and 3. Work in groups Solutions will be presented in class Solutions will be posted: www.cbe.cornell.edu/courses/cheme390

Rate Equations from Experimental Data A  P irreversible rate equation? assume a = ? usually a = 0, 1/2, 1, 3/2, or 2

Rate Equations from Experimental Data 0.06 0.05 0.04 [A] (mol/L) . 0.03 0.02 0.01 100 200 300 400 500 600 700 800 time (sec)

half order: rate = k[A]1/2 Reaction Order? How to Find Guess “a” and fit a line to the data? 2nd order: rate = k[A]2 half order: rate = k[A]1/2 1st order: rate = k[A] zero order: rate = k

Reaction Order? Better to plot the data as a straight line. equation for a straight line on a log-log plot x y slope intercept

How to convert concentration vs. time to rate vs. concentration? Measure slopes! 0.06 (0, 0.044) 0.05 0.0440 0.04 [A] (mol/L) . 0.03 0.0252 0.02 0.0110 0.01 100 200 300 400 500 600 700 800 time (sec) (500, 0)

Method of Initial Rates 0.001 (0.3, 0.001) Reasonable? dt [A]/ 0.0001 d - (0.01, 0.000033) 0.00001 0.01 0.1 1 [A]

Isolation Method More than one reactant? Isolate the effects of [A]. A + B  P B is in excess. [B]  [B]0 [B] is ~constant

Integrated Rate Equations: First-Order Reactions 1 important: add integration limits separate … and integrate. equation for a straight line

Integrated Rate Equations: First-Order Reactions, A  P (0, 1) straight line confirms 1st-order reaction (800, 0.052)

Integrated Rate Equations: Second-Order Reactions separate and integrate. equation for a straight line

Integrated Rate Equations: Second-Order Reactions, 2A  P Example: Dimerization of Butadiene 2C4H6  C8H12 slope = k = 0.86 L/(mol·min) straight line confirms 2nd-order reaction

Integrated Rate Equations: Half Lives t½  the time for a reactant concentration to decrease to ½ its initial value First Order Reaction: substitute [A] = ½[A]0 at t = t½ If t½ is independent of [A], reaction is first order and irreversible.

Integrated Rate Equations: Half Lives t½  the time for a reactant concentration to decrease to ½ its initial value Second Order Reaction: substitute [A] = ½[A]0 at t = t½ Plot t½ vs. 1/[A]0. If line is straight, reaction is second order and irreversible.

Integrated Rate Equations: Half Lives nth Order Reaction: need an integrated equation. substitute [A] = ½[A]0 at t = t½ okay for n = 1? L’Hôpital! straight line! y x slope intercept

Integrated Rate Equations: Half Lives Dimerization of Butadiene: 2C4H6  C8H12 18 17 1 16 2 14 3 12 11 mol/L 4 3 10 ] x 10 5 8 H 6 4 [C 6 4 2 20 40 60 80 100 120 140 160 180 200 35 min time (min)

Integrated Rate Equations: Half Lives Dimerization of Butadiene: 2C4H6  C8H12 (0.0012, 1000) slope = -1/2 (0.1, 12) reasonable? slope = -2

7 Logarithmic Scales log102 = 0.30 log1020 = 1.30 log10x x: x 10x: 4 5 6 8 9 20 11 12 30 40 60 80 100 = 1 100.4 = 2.51 101 = 10 101.4 = 25.1 102 = 100

Logarithmic Scales log(ab) = log(a) + log(b) +0.6 1.2 + 0.6 = 1.8 5 8  5 = 40 3 4  3 = 12 How to plot 12? 3 50  3 = 16.7 How to plot 17?

Logarithmic Scales Another method to plot numbers. distance from 1 to 10 = 105 mm distance from 1 to 17 = x mm 129 mm A method to read numbers. 77 mm distance from 1 to 10 = 105 mm distance from 1 to x = 77 mm