KINETICS OF DEGRADATION REACTIONS

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KINETICS OF DEGRADATION REACTIONS
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Kinetics

• rate of reaction
• relation between concentration and time

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Kinetics of a chemical reaction

• kf
• aA bB cC dD
• kb
• 6 unknowns, need to simplify

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• Ass
• B gtgt A
• B not limiting
• kb ltlt kf

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• Apply in practice not theoretically true.
• Assumptions
• Backward reaction is negligible
• Other reaction species are not limiting
• Reaction conditions are constant pH, T, aw,
  redox potential, concentration of other species
• Therefore, k is a pseudo rate constant particular
  for a given food system.

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Reaction order

• n 0 A Ao kt
• B Bo kt
• n 1 ln A/Ao - kt
• A Ao e-kt B Bo
  ekt

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Units are different, cannot compare.
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• Accurate data ? accurate k ? need to measure over
  50 change in reactant species.
• but in foods 20-30 change is enough for quality
  degradation, can use simple zero order kinetics
• minimize error in measurements
• s std. dev.
• mean value.

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Half-life time for decrease in quality by
50 If n 1
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Order determination

• Method of differentiation

ln dA dt
slopen
lnA
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• 2. Method of integration
• Assume order n 0 or 1 or 2
• Integrate rate equation, plot equation
• Evaluate the fit of linear line

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A
t
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lnA
t
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1/A
t
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• 3. When n ? 1

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Microbial growth
First order reaction N number of
microorganisms kG growth rate constant
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G time for one doubling DG time for 1 log cycle
increase in number of microorganisms G is
determined from the log phase of growth
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• Reaction order for some common reactions in foods
• n 0 enzymatic degradation, lipid oxidation, NEB
  
• n 1 microbial growth, rancidity
• n 2 vitamin C loss
• For shelf life study
• Need to determine
• Quality criteria to be measured
• Environmental conditions affect the reaction
  rates
• T, RH

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Temperature effect

• Activation energy
• Arrhenius relation
• ko Arrhenius equation constant
• EaActivation energy (cal/mol) excess energy
  barrier to over come
• R universal gas constant (1.9872 cal/mol K)
• T absolute temperature, K

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• Important points
• Mode of deterioration changes with temperature
• Need to have more than two temperatures to obtain
  reliable results
•  
• Activation energies for some degradation
  reactions (kcal/mol)
• Hydrolysis 10-20
• Lipid oxidation 15-25
• NEB 20-40
• Enzymatic or microbial degradation 50-150

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Temperature effect

• Q10 value
• measure of sensitivity to temperature
• change in rate by 10C change in temperature

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• Q10
• 1.5-2 sensory quality loss in canned foods
• 1.5-3 rancidity
• 4-10 browning
• 20-40 quality loss for frozen fruits and
  vegetables

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• Deviations from Arrhenius relation
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• change in moisture
• change in physical state, phase change, ice or
  glass formation
• change in mode of deterioration with T
  increase
• partitioning of reactants between two phases,
  such as concentration of reactants upon
  freezing
• temperature history effects

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Temperature effect

• Tg considerations
• T lt Tg glassy, Arrhenius model
• Tg - (Tg 100C) rubbery, WLF model
• T gt Tg 100C solution, Arrhenius model
• slope of Arrhenius plot changes,
  William-Landel-Ferry (WLF) equation applies which
  empirically models T dependence of mechanical and
  dielectric relaxations within the rubbery state.

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ln k
Arrhenius
rubbery Ea 45.1 kcal/mol
glassy Ea 20.8 kcal/mol
Tg
1/T
NEB Model System (Nelson and Labuza, 1994) Break
in the Arrhenius plot due to changes in the
properties of the rubbery and glassy systems
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• In diffusioncontrolled systems
• where diffusion is free volume dependent, WLF
  equation is needed to express reaction rate
  constants as a function of T.

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Calculate kg from known Tg and the average
coefficients C1 and C2. To find real C1 and C2
use 2nd equation
kref rate constant at Tref gt Tg C1, C2
system-dependent coefficients. C1 -17.44 C2
51.6 for Tref Tg for various polymers
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• 10-100?C above Tg
• diffusion-dependent changes in food quality
• viscosity, crystallization, textural changes fit
  WLF model.
• In model systems,
• deterioration reactions seen to occur at T lt Tg
• oxidation, NEB, ascorbic acid degradation
• therefore, porosity of the system, physical
  changes such as collapse and crystallization
  should be considered

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• But chemical reactions may be kinetically and/or
  diffusion limited. How to decide which equation
  to use?
• Effective reaction rate constant k
  k/(1k/?D)
• D Diffusion coefficient
• ? constant, independent of T
• k Arrhenius-type T dependence constant
• D follows Arrhenius equation with a change in
  slope at Tg, or follow WLF equation in the
  rubbery state and especially in the range
  10-100?C above Tg,
• k/?D defines relative influence of k and D.
• If k/?D lt 0.1 Arrhenius equation can be used for
  modeling T dependency.
• Slope changes in Arrhenius plot at Tg with either
  constant slope above Tg or with a gradually
  changing slope.
• For complex food systems involving multiple
  reaction steps and phases, either model can be
  used for controlled-temperature functions like
  sine, square wave T fluctuations to verify the
  shelf life model.

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Temperature effects

• Fluctuating temperatures
• sine wave
• square wave

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aw and temperature

• Clasius-Clapeyron equation describes temperature
  dependence of aw

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BET equation

• Brunauer-Emmett-Teller equation
• Sorption isotherm moisture content vs aw
• can determine monolayer value (m1)
• valid for aw 0 - 0.5

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GAB equation

• valid for aw 0 - 0.9
• can determine monolayer value
• There are other sorption isotherm models
• Need to find which fits for particular food using
  experimental data

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Moisture gain or loss

• Can estimate changes in moisture in packaged
  foods
• k/x permeability of the package

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Kinetics of enzymatic reactions

• k1 k2
• E S ES E P
• k-1

P
Vo
t
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Michaelis Menten equation
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Kinetics of enzymatic reactions

• Vmax Maximum velocity when enzyme is saturated
  with substrate
• Km Substrate concentration at half maximum
  velocity, (k-1 k2) / k1

Vmax/2
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• Use linear form of the equation to find Km and
  Vmax
• 6-10 substrate concentration
• So 0.1Km-10Km
• Lineweaver-Burk method

1/Vo
?Km/Vmax
1/Vmax
-1/Km
1/So
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Linear regression

• To estimate changes in a quality index with time
• Need to fit experimental data to equations and
  calculate equation parameters
• Can convert equations to linear form
• Use statistics for fitting data to equations
• accurate estimates of the parameters

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Linear regression

• Relation of a dependent variable y with an
  independent variable x
• y bo b1x
• bo , b1 parameters to estimate
• y a quality index
• x time
• Assumptions
• 1. Data are normally distributed
• 2. Constant variance
• 3. Independence of error
• 4. Linear relation

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Linear regression
Minimize sum of squares of error
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Linear regression

• More data more accurate prediction
• df degree of freedom, depend on number of data,
  more data more df
• lose 1 df to estimate 1 parameter
• confidence level (1-?) 90, 95, 99

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Linear regression

• t?/2 a test statistic, student t value at a
  given confidence level
• If n 3 df 1 t?/2 at 95 12.71
• If min n 8 df 6 t?/2 2.45

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Linear regression

• Confidence interval for a parameter estimate
• change in the parameter estimate

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Linear regression

• Correlation Coefficient (R2)
• Proportion of variability in y explained by the
  linear relation
• Total variability in y
• variability explained by linear relation
  residuals
• R2 lt 1

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Linear regression

• Strength of a linear relation
• 1. Small confidence intervals for parameter
  estimates
• 2. High Correlation Coefficient (R2) 1
• If R2 is small
• 1. no relation between x and y
• 2. relation not linear, use nonlinear equations