che441 analysis of rate data-1

Ch E 441 - Chemical Kinetics and Reaction Engineering

Analysis of Rate Data

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Extracting Kinetic Constants• To design reactors, rate equation must be known.• Determining the rate equation and its associated

constants requires analysis of experimental data.

Experimental reactor types:

• batch• differential

Experimental methods used to collect data:

• half-lives• initial rates

• differential• Integral• linear regression• non-linear

regression

Analysis & interpretation techniques:

Batch Reactor Data• Differential Method of Analysis• Consider the following reaction that occurs in a

constant volume batch reactor productsA

AA r

dtdN

V1

AA kCr

Mole Balance Rate Law Stoichiometry

oVV

Combine A

A kCdt

dC

take the natural log of the combined equation

Batch Reactor Data• Differential Method of Analysis• Consider the following reaction that occurs in a

constant volume batch reactor

xm b y

Clnklndt

dCln A

A

fit by least squares regression

productsA

AA kC

dtdC

Differential method example• When arterial blood enters a tissue capillary, it exchanges oxygen and

carbon dioxide with its environment. The kinetics of this deoxygenation of hemoglobin in blood was studied with the aid of a tubular reactor by Nakamura and Staub [J. Physiol., 173, 161 (1967)].

• Although this is a reversible reaction, measurements were made in the initial phases of the decomposition so that the reverse reaction could be neglected. Consider a system similar to the one used by Nakamura and Staub: the solution enters a tubular reactor (0.158 cm diameter) that has oxygen electrodes placed at 5-cm intervals down the tube. The solution flow rate into the reactor is 19.6 cm3/s.

Electrode position 1 2 3 4 5 6 7% decomposition of HbO2 0.00 1.93 3.82 5.68 7.48 9.25 11.0

22 OHbHbO

Differential method example• For a tubular reactor assuming rate equation is

nth order in CA: nn

AoAo X1kCdVdX

F

n

Ao

cnAo X1F

AkCdzdX

X1lnnF

AkCln

dzdX

lnAo

cnAo

Batch Reactor Data• Experimental Technique: Method of Excess• Consider the following reaction that occurs in a

constant volume batch reactor

BAAA CCkr

Rate Law

component A in excess such that [A] constant

BA C"kr

AA C'kr component B in excess such that [B] constant

kA can now be determined

AA Cln vs. rln

BA Cln vs. rln

BAAA CCrk

productsBA

Batch Reactor Data• Experimental Technique: Method of Initial Rates• used when differential method is ineffective: e.g.,

– Competing reactions,– Reverse reactions, and/or– Pressure changes due to reaction.

• series of reactions at different Ci,o performed• initial rate: differentiate data, extrapolate to t = 0• relate –rAo to CAo, to find appropriate rate law.

Batch Reactor Data• Protocol for analyzing Rate Data:

1. Postulate a rate law.2. Process data in terms of the measured variable.3. Look for simplifications.4. Calculate –rA as a function of reactant concentration

to determine reaction order.• Batch: -dCA/dt• Differential PBR: -rA = FAoX/W

5. Determine specific reaction rate, k.

Batch Reactor Data• Integral Method of Analysis

– Assume order– integrate rate law/design equation– Linearize result

• Consider reaction in constant V batch reactor

productsA

AA r

dtdC

kdt

dCA

Zero Order Reaction

ktCC AoA

Thus, a plot of CA vs. t will be linear with a slope of k and an intercept of CAo if the rate law is correct




productsA

AA r

dtdC

AA kC

dtdC

1st Order Reaction

ktCC

lnA

Ao

Thus, a plot of ln(CAo/ CA) vs. t will be linear with a slope of k if the rate law is correct




productsA

AA r

dtdC

2A

A kCdt

dC kt

C1

C1

AoA

Thus, a plot of 1/ CA vs. t will be linear with a slope of k if the rate law is correct

2nd Order Reaction

Integral method of analysis• Penicillin G reacts with hydroxylamine (NO2OH) to form

hydroxamic acid, producing a colored complex with iron(III).• To determine overall reaction order, equal concentrations of

penicillin and NH2OH were mixed together in a 250-mL flask [J. Chem. Educ., 49, 539 (1972)]. – Samples were withdrawn every 10 minutes and added to a solution

containing iron(III) chloride. – Using a colorimeter, concentration of the colored complex (hence

concentration of hydroxamic acid) was obtained as function of time (absorbance is directly to hydroxamic acid concentration).

Time(min) 0 10 20 30 40 50 Absorbance 0.00 0.337 0.433 0.495 0.539 0.561 0.665

Integral method of analysisHAOHP

1X when AA

constant analyticalKabsorbanceA

acid hydroxamicHAinehydroxylamOH

penicillinP

Mole Balance Rate Law Stoichiometry

PP r

dtdC

XCC

X1CC

PoHA

PoP

n

PP kCr

define relationship tomeasured variable

AKCHA

Combine

nnPoPo X1kC

dtdX

C

nnPo

Po

A

A1kC

dtdA

AC

nAAKdtdA

1nnPo

AC

kK

Integral method of analysis

IntegralMethod nAAK

dtdA

1nnPo

AC

kK

0AAKdtdA

Zero

Order KtdAA

0

1AAKdtdA

First

Order

t

0

A

0dtK

AAdA

KtA

AAln

2AAKdtdA

SecondOrder

t

0

A

0 2 dtKAA

dA Kt

A1

AA1

KtA

HAOHP

Express in terms of measured variable• Do not convert measured data into conversion.

Express design equation in terms of measured variable.• Consider constant V & T, gas-phase, batch reaction for

which total pressure vs time is measured

X1CC AoA

o

o

o T

TX1

PP

VV

oAo

ooAo

oo

PPP1

PPPy

1PP

P1

X

RT

PPPC oo

A

substituting

• Do not convert measured data into conversion. Express design equation in terms of measured variable.

• Consider constant V & T, gas-phase, batch reaction

Express in terms of measured variable

RT2

PP3RT

2PPPRT

PPPC ooooo

A

C2BA .,g.e 2121

A

AA kC

tC

r

PP3'ktP

o PP3ln'kln

tP

ln o




C2BA

AA r

dtdC

PP3'ktP

o

assume = 1

t'kPP3

P2ln

o

o

Batch Reactor Data• Experimental technique: Method of Half Lives• Half life is the time required for concentration of

the reactant to fall to half of its initial value.• By determining half life as a function of initial

concentration, order and specific rate can be determined.

• Experimental technique: Method of Half Lives

Batch Reactor Data

AA kCr

productsA

AA kC

dtdC

1

Ao1

A C1

C1

1k1

t

1CC

1kC1

t1

A

Ao1

Ao

• Experimental technique: Method of Half Lives

Batch Reactor Data

AoA

½

C½Ctt

1Ao

1

½ C1k12

t

Ao

1

½ Cln11k12

lntln

obtain order from slopeobtain k from intercept after obtaining

AA kCr

productsA

1CC

1kC1

t1

A

Ao1

Ao

Method of Half Lives Example• Nitrogen oxide is a pollutant in automobile exhaust that

can react with oxygen to form nitrogen dioxide:

• At 298 K the specific reaction rate is:

Find t½ of 3000 ppm NO (typical pre-control exhaust level) in air?

What is t½ of 1 ppm NO (a typical polluted atmosphere value)?

2k

2 NO2ONO2 2minppm104.1k 29

2

since oxygen is highly in excess,

NO2ONONO CkCkCr






2k


2

3NO2NO2ONONO CkCkCkCr

units on specific reaction rate imply = 3






2k


2

2Ao

913Ao

13

1Ao

1

½ C108.23

C13k12

C1k12

t

substituting = 3 into half-life expression:






2k


2

substituting CNO = 3000 ppm:

min 1.119

3000108.23

t 29½

substituting CNO = 1 ppm:

min 10071.1

1108.23

t 929½

Differential Reactors• Used to determine rate for heterogeneous catalytic

reactions as a function of Ci or Pi.• Uses a method similar to initial rates.• Reactor is a tube with a small (differential) wafer of

catalyst, causing a small conversion of reactant.

FAo FAe

L

inert packingcatalyst

Differential Reactors• Known, controlled, or measured:

– Volumetric flow rate ()– Concentrations (Cao, CA)– Catalyst weight (W)

FAo, FA

0W'rFF AAeAo

WF

WXF

WFF

'r PAoAeAoA

FAo FAe

L

WC

WCC

'r PoAeAooA

for constant o

Differential Reactors• By using very little catalyst and operating at high

volumetric flow rates, (CAo - CAe) can be made small.• Catalyst bed is approximated as being gradientless.

AbAA C'r'r 2

CCC AeAo

Ab

AoAb CC

FAo FAe

L

WC

WCC

'r PoAeAooA

for constant o

che441 analysis of rate data-1

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