Integrated Solution to the Rate Law Linear Form

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Integrated Solution to the Rate LawLinear Form

• Edward A. Mottel
• Integrated, First-Year Curriculum
• in Science, Engineering and Mathematics

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Please Sit with Your Group

• Please be sure each member of your team has a
  copy of
• Integrated Solution to the Rate Law Linear Form
  Lecture Notes
• Rate Laws Problem Set
• Todays reporter is the tallest member of the
  team.

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Determination of theRate Constant and Order of a
Reaction

• The speed of a chemical reaction can be described
  by a differential rate equation.
• Integration of this rate equation provides a
  method to determine the order and rate constant
  for the reaction.
• The integrated solution of the rate equation can
  be analyzed in either a linear or a non-linear
  form.

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Methods To Determine theOrder of a Reaction

• Initial rate method
• Differential rate method
• Integrated solution to the rate law
• Linear form
• Non-linear form

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Initial Rate Method

• Hold all but one of the concentrations of
  reactants constant, and measure the initial rate
  of reaction as a function of the reactant
  concentration being varied.

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Differential Rate Method

• Measure the rate of reaction at several times
  during the course of the reaction and relate this
  to the concentration at these times.

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Integrated Solution to the Rate Equation Linear
Form Method

• Assume a reasonable order for the reaction.
• Integrate (anti-differentiate) the differential
  rate equation.
• Rewrite the equation in linear form.
• Plot the corresponding function of the reactant
  concentration versus time.
• A linear relationship indicates that the data is
  consistent with that order of reaction.

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Order of Reaction What is a reasonable order of
reaction to assume?
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Order of Reaction
What is a reasonable order of reaction to assume?
Because the rate of a chemical reaction is
based on one molecule breaking apart or colliding
with another molecule, the order of a chemical
reaction generally involves a small integer
order or a ratio of small integers.
Examples
0, 1, 2, -1, 1/2
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Decomposition of NO2Test for a zero-order
reaction
2 NO2(g) 2 NO(g) O2(g)
What is the differential equation for a
zero-order reaction?
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Decomposition of NO2Test for a zero-order
reaction
What is the integrated solution to the
differential equation for a zero-order reaction?
Rewrite the equation so that the differentials
are on opposite sides of the equation and similar
terms are on the same side.
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Integrationis the reverse process of
differentiation.
Power series integration, increase the exponent
by one
There is no expressed t term, but t0 is implied.
C
and divide by the new exponent.
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Integrationis the reverse process of
differentiation.
C -(PNO2) - k t
-(PNO2) k t - (PNO2)o - k to
At t 0, t to and (PNO2) (PNO2)o
(PNO2) -k (t - to) (PNO2)o
Integrated solution to the zero-order rate law.
C -(PNO2)o - k to
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Decomposition of NO2Test for a zero-order
reaction
2 NO2(g) 2 NO(g) O2(g)
(PNO2) -k (t - to) (PNO2)o
y m x b
Explain how this equation is like a linear
equation.
Which terms should be plotted on the x and y axes?
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Test for a Zero-Order Reaction(PNO2) -k (t -
to) (PNO2)o
If this reaction is zero-order in NO2, the data
will be linear.
The line is not linear, the order (n) is not zero.
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Decomposition of NO2Test for a first-order
reaction
2 NO2(g) 2 NO(g) O2(g)
What is the differential equation for a
first-order reaction?
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Decomposition of NO2Test for a zero-order
reaction
What is the integrated solution to the
differential equation for a first-order reaction?
Rewrite the equation so that the differentials
are on opposite sides of the equation and similar
terms are on the same side.
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Integrationis the reverse process of
differentiation.
Special form needed for n1 because power series
integration fails!
- (PNO2)-11
C
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Integrationis the reverse process of
differentiation.
Special form needed
C
Constant of integration
What is the special form?
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Integrationis the reverse process of
differentiation.
C -ln(PNO2) - k t
-ln(PNO2) k t - ln(PNO2)o - k to
At t 0, t to and ln(PNO2) ln(PNO2)o
ln(PNO2) -k (t - to) ln(PNO2)o
Integrated solution to the first-order rate law.
C -ln(PNO2)o - k to
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Decomposition of NO2Test for a first-order
reaction
2 NO2(g) 2 NO(g) O2(g)
ln(PNO2) -k (t - to) ln(PNO2)o
y m x b
Explain how this equation is like a linear
equation.
Which terms should be plotted on the x and y axes?
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Test for a First-Order Reactionln(PNO2) -k (t
- to) ln(PNO2)o
If this reaction is first-order in NO2, the data
will be linear.
The line is not linear, the order (n) is not one.
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Decomposition of NO2Test for a second-order
reaction
2 NO2(g) 2 NO(g) O2(g)
What is the differential equation for a
second-order reaction?
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Decomposition of NO2Test for a second-order
reaction
What is the integrated solution to the
differential equation for a second-order reaction?
Rewrite the equation so that the differentials
are on opposite sides of the equation and similar
terms are on the same side.
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Integrationis the reverse process of
differentiation.
Power series integration, increase the exponent
by one
C
and divide by the new exponent.
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Integrationis the reverse process of
differentiation.
C (PNO2)-1 - k t
(PNO2)-1 k t (PNO2)o-1 - k to
At t 0, t to and (PNO2)-1 (PNO2)o-1
(PNO2)-1 k (t - to) (PNO2)o-1
Integrated solution to the second-order rate law.
C (PNO2)o-1 - k to
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Decomposition of NO2Test for a second-order
reaction
2 NO2(g) 2 NO(g) O2(g)
(PNO2)-1 k (t - to) (PNO2)o-1
y m x b
Explain how this equation is like a linear
equation.
Which terms should be plotted on the x and y axes?
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Test for a Second-Order Reaction(PNO2)-1 k (t
- to) (PNO2)o-1
If this reaction is second-order in NO2, the data
will be linear.
The line is linear, the order (n) is two!
What is the rate constant for the reaction? What
are the units?
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Test for a Second-Order Reaction(PNO2)-1 k (t
- to) (PNO2)o-1
Write the rate law for this reaction.
How is the rate of reaction determined at any
time?
How is the time determined for any concentration?
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SummaryLinear Form of the Integrated Solution to
the Rate Equation Method

• Assume a reasonable order for the reaction.
• Integrate (anti-differentiate) the differential
  rate equation.
• Rewrite the equation in linear form.
• Plot the corresponding function of the reactant
  concentration versus time.
• A linear relationship indicates that the data is
  consistent with that order of reaction.

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Integrated Solution to the Rate EquationGeneral
Form
The same result is obtained using the integral
sign.
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Integrated Solution to the Rate Equation General
Form
Substitute in the limits of integration
final - initial

k t
- k to
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Integrated Solution to the Rate Equation General
Form
Rewrite this equation in a straight-line form.
y m x b
Which terms can vary?
(PNO2)t-n1

(t - to)
(n-1) k
(PNO2)o-n1
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Integrated Solution to the Rate Equation
(PNO2)t-n1 (n-1) k (t - to) (PNO2)o-n1
What is the rate constant for this reaction?
What does the y-intercept represent?
(PNO2)o 474 torr
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Integrated Solution to the Rate Equation First
Order
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Integrated Solution to the Rate Equation First
Order
ln (PNO2)t - ln (PNO2)o - k (t - to)
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