Kinetics Part III: Integrated Rate Laws

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Kinetics Part IIIIntegrated Rate Laws

• C. Yau
• Spring 2009

Based on Brady Senese 5/eChap. 13 Sec 4
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Integrated Rate Laws
How are integrated rate laws different from
rate laws? A rate law gives the speed of
reaction at a given point in time, and how it is
affected by the concentration of the
reactants. An integrated rate law gives the
change in concentration from time zero up to a
given point in time.
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Integrated Rate Law for First-Order Reactions

• We already know the rate law for a 1st order
  reaction is
• Rate kA
• Using calculus, the integrated rate law is
• Where Ao molarity of A at time zero
• At molarity of A at time t
• and k rate constant for the reaction

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Integrated Rate Law of First-Order Reactions
It is often more useful to rewrite the equation
by finding the anti natural log of both sides of
the equation
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Treatment of Sig. Fig. in Log

• Log 3.4 x 10-1 -0.468521083
• -0.47
• 2 sig.fig. 2 decimal places
• antilog -3.213 10-3.213 6.123503x10-4
• 6.12x10-4
• 3 decimal places 3 sig. fig.

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Treatment of Sig. Fig. in Ln

• Ln 3.4 x 10-1 -1.078809
• - 1.08
• antiln -4.273 e-4.273 1.393990x10-2
• 1.39x10-2

2 sig. fig.
3 decimal places
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Use of the Integrated Rate of 1st Order Rxns

• Example 13.6 p. 535 Dinitrogen pentoxide is not
  very stable. In the gas phase or dissolved in a
  nonaqueous solvent, like CCl4, it decomposes by a
  1st order rxn into dinitrogen tetroxide and
  molecular oxygen.
• 2 N2O5 2 N2O4 O2
• The rate law is Rate kN2O5
• At 45oC, the rate constant for the rxn is 6.22 x
  10-4s-1. If the initial concentration of N2O5 at
  45oC is 0.500 M, what will its concentration be
  after exactly one hour?
• Ans. 0.055 M

Do Practice Exercises 14 15 p.536
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Determination of the Rate Constantfor the
Integrated Rate Law

• The rate constant can be determined graphically.
  Again we manipulate the rate law to allow us to
  obtain a linear graph Variables are At and t

How does this allow us to determine the rate
constant, k, graphically?
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Determination of the Rate Constantfor the
Integrated Rate Law
What is the slope equal to? slope - k
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Half-life (t½ ) of a Reactant

• "Half-life" of a reactant is the time it takes
  for ½ of the reactant to disappear.
• It is NOT half the time it takes for all of the
  reactant to disappear.
• For example, t½ of the radioisotope I-131 is 8.0
  days. Starting with 20 g of I-131, after 8.0
  days, there would be 10 g left.
• After a total of 16.0 days (two half-lives),
  there would be
• 5 g left

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• Half-life is a measure of the speed of reaction
  The shorter the half-life, the faster is the
  reaction.
• By definition, after one half-life,
• At ½ Ao
• For a 1st order rxn,
• Note that half-life isnot affected
  byconcentration.

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• Application of Half-life

Fig.13.7 p.538
I-131 has a half-life of 8 days. After 4
half-lives (32 days), it is down to 1/16 of its
original concentration.
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• Example 13.7 A patient is given a certain amt of
  I-131 as part of a diagnostic procedure for a
  thyroid disorder. What fraction of the initial
  iodine-131 would be present in a patient after 24
  days if none of it were eliminated through
  natural body processes?
• Can we solve this problem using the integrated
  rate law for 1st order rxn?

Ans 1/8 of the initial dosage.
Given Half-life is 8.0 days.
Do Practice Exercises 16, 17 p. 539
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• The half-life of I-132 is 2.295h. What
  percentage remains after 24 hours?

Ans. 0.0711
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• C-14 dating Determination of the age of organic
  substances.
• When object is still living, it is ingesting C
  with a constant ratio of
• C-14/C-12 1.2x10-12
• Once the object dies, the amount of C becomes
  constant and no more C-14 is incorporated. As a
  result the C-14/C-12 ratio begins to decrease due
  to the decay of C-12.
• By measuring the C-14/C-12 ratio of the object,
  we can estimate how long it has been dead.

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• C-14 has a half-life of 5730 years and
  radioactive decay is a first-order process.
• This means
• and

What is the rate constant, knowing that the
half-life is 5730 yrs?
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• Example 13.8 p.540
• An ancient wooden object was found to have ratio
  of 14C to 12C equal to 3.3x10-13. What is the age
  of the object?

Ans. 1.1x104 years
Do Pract Exer 18, 19 p.540
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Integrated Rate Law of 2nd Order Reactions

• They are of several types RatekA2,
  RatekA1B1 and RatekA2B0, etc
• The integrated equation is of the form

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Example 13.9 p.540 Nitrosyl chloride, NOCl,
decomposes slowly to NO and Cl2. 2NOCl
2NO Cl2 The rate law shows that the
rate is second order in NOCl. Rate
kNOCl2 The rate constant k equals 0.020 L mol-1
s-1 at a certain temp. If the initial conc of
NOCl in a closed reaction vessel is 0.050 M, what
will the concentration be after 30 minutes?
Do Pract Exer 20 21 p.541
Ans. 0.018 M
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Integrated Rate Law of 2nd Order Reactions

• How can we determine rate constant graphically?
  Which are the variables?

Rearrange
y mx b
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Half-life of 2nd Order Reactions
Note that unlike 1st order reactions, half-life
of 2nd order rxns depend on the initial conc of
the reactant. Ex 13.10 p.542 2HI(g)
H2(g) I2 (g) has the rate law, Rate kHI2
with k 0.079 L mol-1s-1 at 508oC. what is the
half-life of this rxn at this temp when the
initial HI concentration is 0.10 M?
Ans. 1.3x102s
Do Prac Ex 22 23 p.543
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Summary of Integrated Rate Laws
1st Order Rxn
2nd Order Rxn