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Chemistry andDifferentialEquationsWhat happens whenequations in the real worldare not linear?
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Importance
Chemistsoccasionally runtests on chemical
kinetics. Currently, many
chemists arelooking into the
effectiveness ofcatalysts, toincrease the yieldof ethanol
production or
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Current Applications
Supposedly 90% ofcommerciallyproduced
chemical productsinvolve catalystsduringmanufacturing.
Petroleum refining,biodiesel, and fuelcells all rely onsome sort of
catalytic reaction.
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Simple Chemical Kinetics
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Michaelis-Menten Kinetics
Givens:
K1 K2
E + S ESE+P
K-1
From the law of
mass action: d[S]/dt=-k1[E][S]+k-1[ES]
d[E]/dt=-k1[E][S]+k-1[ES]
+k2[ES]
d[ES]/dt=k1[E][S]-k-1[ES]-
Difficulties System of
equations is notlinear.
Law ofSuperpositiondoes not apply.
Ramifications Much more difficult
to solve.
Variation ofparameters and
undetermined
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Michaelis-Menten Kinetics
Assumptions Reaction is in Equilibrium
Assume a Steady-State
Assume that the maximum reaction rateoccurs when the enzyme is completely used
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Equilibrium Reaction
Assumptions:The ratio of
substrate (S) toenzyme (E) isvery large.
All enzymes bindas muchsubstrate as
possible. No enzymes are
damaged andlose their ability
to react.
Implications: Reaction rate at
time t=0 ispractically zeroso V0 = K2[ES],
where V0 is the
production rate attime t=0.
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Steady-State Assumption
Assumptions:The concentration
of the enzyme-substrateintermediate (ES)is constant.
Implications:This is only true
when theformation anddeformation ratesfor ES are equalor:
K1 [E][S] = (K-1 +
K2) [ES]
Also, [E] = [ET]
[ES] where [ET] is
the total enzymeconcentration.
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Maximum Rate Assumption
Assumptions: Maximum rate is
obtained whenthe enzyme issaturated withsubstrate.
Implications Maximum rate is
achieved if [ES] =[E
T].
[ES] = enzymesubstrateconcentration
[ET] = total
enzymeconcentration
[E] = freeenzymeconcentration
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Putting It All Together
Givens: K1 K2 E + S ESE+P K-1
From the law of
mass action: d[S]/dt=-k1[E][S]+k-1[ES]
(1)
d[E]/dt=-k1[E][S]+k-1[ES]+k2[ES] (2)
d[ES]/dt=k1[E][S]-k-1[ES]-
Implications:
V0 = K2[ES] (5)
K1 [E][S] = (K-1 +
K2) [ES] (6) Maximum rate is
achieved if [ES] =[ET] (7)
[E] = [ET] [ES] (8)
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Putting It All Together
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Practicality of Michaelis-Menten
It is much easier to regulate and monitor thesubstrate concentration than it is to monitor therate of product formation.
The theory allows chemists to find optimum
substrate concentrations quickly and allowsthem to avoid solving nonlinear differentialequations.
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Conclusion
Differential equations are applied to kineticsmodels in chemical reactions.
Carefully placed assumptions and conditions allowfor the simplification of complex systems.
More complicated and real-world chemicalreactions may require one to solve nonlineardifferential equations.
Linear differential equations are the tip of the
iceberg when it comes to the world ofdifferential equations.
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Sources:
Lehninger Principles of Biochemistry
http://www.engr.ucr.edu/news/2010news.html
http://en.wikipedia.org/wiki/Catalyst#cite_note-9
http://www.lawrencepumps.com/industries/petroleum.htm
http://www.uwsp.edu/chemistry/tzamis/chem10609home.
Notes of Jeremy Lai
http://www.engr.ucr.edu/news/2010news.htmlhttp://en.wikipedia.org/wiki/Catalyst#cite_note-9http://www.lawrencepumps.com/industries/petroleum.htmhttp://www.lawrencepumps.com/industries/petroleum.htmhttp://www.uwsp.edu/chemistry/tzamis/chem10609home.htmlhttp://www.uwsp.edu/chemistry/tzamis/chem10609home.htmlhttp://www.uwsp.edu/chemistry/tzamis/chem10609home.htmlhttp://www.uwsp.edu/chemistry/tzamis/chem10609home.htmlhttp://www.lawrencepumps.com/industries/petroleum.htmhttp://www.lawrencepumps.com/industries/petroleum.htmhttp://en.wikipedia.org/wiki/Catalyst#cite_note-9http://www.engr.ucr.edu/news/2010news.htmlhttp://www.engr.ucr.edu/news/2010news.htmlhttp://www.engr.ucr.edu/news/2010news.htmlTop Related