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Chemical KineticsPart 2

Chapter 13

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The Change of Concentration with Time

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Zero-Order Reactions (or zeroth order)• Goal: convert rate law into a convenient

equation to give concentrations as a function of time.

• For a zero order rxn, the rate is unchanged or is independent of the concentration of a reactant.

• However, you must have some of the reactant for the rxn to occur!

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Zero-Order Reactions (or zeroth order)• One example of a rxn which is 0 order is:

2HI(g) H2(g) + I2(g)

• The rate law for this rxn has been determined experimentally and is:

rate = k[HI]0 = k or rate = k• What are the k units?

Rate = M/s so k units are M/s or M•s-1

Au⏐ →⏐

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Zero-Order Reactions (or zeroth order)• But the rate is also equal to the change in

[reactant] over the change in time:

• But if rate = k, this means that this is true:

rate = -Δ HI[ ]Δt

k = -Δ HI[ ]Δt

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Zero-Order Reactions (or zeroth order)• We rearrange this equation:

• We then integrate:

Δ HI[ ] = -kΔt

Δ HI[ ]HI[ ]0

HI[ ]t

∫ = −kΔtt=0

t

∫

Which becomes:

HI[ ]t - HI[ ]0 = -kt

If we rearrange the above, we get the eq. of a line:

HI[ ]t = HI[ ]0 - kt

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Zero-Order Reactions (or zeroth order)

• This eq. means that a graph of [HI] vs. time is a straight line with a slope of -k and a y-intercept of [HI]0.

• Here are typical 0-order graphs:

HI[ ]t = HI[ ]0 - kt

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Zero-Order Reactions (or zeroth order)• We can find the half-life, t1/2, for a 0-order rxn.

• The t1/2 is defined as the time it takes for half of the

reactant to disappear.• But this is the time required for [A] to reach

0.5[A]0

• Mathematically, this is:0.5 A[ ]0 = A[ ]0 - kt1/2

So: t1/2 = 0.5k

A[ ]0

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First-Order Reactions• For a first order rxn, the rate doubles as the

concentration of a reactant doubles.• We can show that

• A plot of ln[A]t versus t is a straight line with slope -

k and y-intercept ln[A]0.

• In the above we use the natural logarithm, ln, which is log to the base e.

-kt = lnA[ ]tA[ ]0

⎛

⎝⎜

⎞

⎠⎟

This becomes: ln A[ ]t =−kt+ ln A[ ]0

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First-Order Reactions

[ ] [ ]0AlnAln +−= ktt

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Half-Life for 1st-Order Rxns• Half-life is the time taken for the concentration of a

reactant to drop to half its original value.• That is, half life, t1/2 is the time taken for [A]0 to reach

½[A]0.

• Mathematically,

• Note the half-life is independent of the [reactant]0.

t 12=−

ln0.5k

=0.693k

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The Change of Concentration with TimeHalf-Life

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Second-Order Reactions• For a second order reaction with just one reactant

• A plot of 1/[A]t versus t is a straight line with slope k and intercept 1/[A]0

• For a second order reaction, a plot of ln[A]t vs. t is not linear.

[ ] [ ]0A1

A1

+= ktt

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Second-Order Reactions

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Second-Order Reactions• We can show that the half life is:

• The half-life of a 2nd-order rxn changes as the rxn progresses.

• Each half-life is twice as long as the one before!• This makes these problems harder (and less common).

[ ]0A1

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kt =

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Second-Order Reactions• A reaction can also have a rate constant expression of

the form:

rate = k[A][B]• This is second order overall, but has first order

dependence on A and B.• This is more complicated and you won’t have to solve

for half-lives of these rxns.

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Temperature and Rate

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Temperature and Rate• Most reactions speed up as temperature increases.

(E.g. food spoils when not refrigerated.)• When two light sticks are placed in water: one at

room temperature and one in ice, the one at room temperature is brighter than the one in ice.

• The chemical reaction responsible for chemiluminescence is dependent on temperature: the higher the temperature, the faster the reaction and the brighter the light.

• As temperature increases, the rate increases.

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Temperature and Rate• As temperature increases, the rate increases.• Since the rate law has no temperature term in it, the

rate constant must depend on temperature.• Consider the first order reaction CH3NC → CH3CN.

– As temperature increases from 190°C to 250°C the rate constant increases from 2.52 x 10-5 s-1 to 3.16 x 10-3 s-1.

• A rule of thumb is that for every 10°C increase in temperature, the rate doubles!

• The temperature effect is quite dramatic. Why?

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Temperature and Rate

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The Collision Model• Observations: rates of reactions are affected by

concentration and temperature.• Goal: develop a model that explains why rates of

reactions increase as concentration and temperature increases.

• The collision model: in order for molecules to react they must collide.

• The greater the number of collisions the faster the rate.

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The Collision Model• The more molecules present, the greater the

probability of collision and the faster the rate.• The higher the temperature, the more energy

available to the molecules and the faster the rate.• However, not all collisions lead to products. In fact,

only a small fraction of collisions lead to product.• In order for reaction to occur the reactant molecules

must collide in the correct orientation and with enough energy to form products.

• These are called effective collisions.

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The Collision Model

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The Collision Model

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Orientation Factor in Effective Collisions• The orientation of a molecule during collisions is

critical in whether a rxn takes place.• Consider the reaction between Cl and NOCl:

Cl + NOCl →NO + Cl2

• If the Cl collides with the Cl of NOCl then the products are Cl2 and NO.

• If the Cl collided with the O of NOCl then no products are formed.

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Orientation Factor in Effective Collisions

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Activation Energy• Arrhenius: molecules must possess a minimum

amount of energy to react. Why?– In order to form products, bonds must be broken in the

reactants.– Breaking bonds always requires energy.

• Activation energy, Ea, is the minimum energy required to initiate a chemical reaction.

• It is also called the Energy of Activation.

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Activation Energy

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Activation Energy• Consider the rearrangement of methyl isonitrile to

form acetonitrile:

– In H3C-N≡C, the C-N≡C bond bends until the C-N bond breaks and the N≡C portion is perpendicular to the H3C portion. This structure is called the activated complex or transition state.

– The energy required for the above twist and break is the activation energy, Ea.

– Once the C-N bond is broken, the N≡C portion can continue to rotate forming a C-C≡N bond.

H3C N CC

NH3C H3C C N

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Activation Energy

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Activation Energy• The change in energy for the reaction is the difference

in energy between CH3NC and CH3CN.

• The activation energy is the difference in energy between reactants, CH3NC and transition state.

• The rate depends on Ea.

• The higher the Ea, the slower the rate!

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Activation Energy• Notice that if a forward reaction is exothermic

(CH3NC → CH3CN), then the reverse reaction is endothermic (CH3CN →CH3NC).

• What is the ΔH and the Ea for the reverse rxn?

• Is Ea rev just -Ea?

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Activation Energy• How does the Ea relate to temperature?

• At any particular temperature, the molecules (or atoms) have an average kinetic energy.

• However, some molecules have less energy while others have more energy than the average value.

• This gives us an energy distribution curve where we plot the fraction of molecules with a given energy.

• We can graph this for different temperatures as well.

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Activation Energy• We can see on the graph that some molecules do have

enough kinetic energy to react.• This is called f, the fraction of molecules with an

energy ≥ Ea.

• The equation for f is:

f = e-Ea

RT

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Activation Energy• Molecules with an energy ≥ Ea have sufficient energy

to react.• What happens to the kinetic energy as we increase the

temperature?• It increases!• So, as we increase the temperature, more molecules

have an energy ≥ Ea.

• So more molecules react per unit time, and the rate increases.

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Arrhenius Equation• Arrhenius discovered that most rxn-rate data obeyed

an equation based on 3 factors:• The number of collisions per unit time.• The fraction of collisions that occur with the correct

orientation.• f, the fraction of colliding molecules with an energy ≥

Ea.

• From this, he developed the Arrhenius Equation.

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Arrhenius Equation

• In the above, k is the rate constant: it depends on temperature!

• R is the Ideal Gas Constant, 8.314J/mol•K• Ea is the Energy of Activation in J

• T is the temperature in Kelvin• A is the frequency factor

k = Ae-Ea

RT

or ln k = -Ea

RT + lnA

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Arrhenius Equation

• A is related to the frequency of collisions & the probability that a collision occurs with the correct orientation.

• This is related to the molecular size, mass, and shape.

• Usually the larger or more complicated the shape, the lower A is.

• Important: Both Ea and A are rxn-specific!

k = Ae-Ea

RT

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Arrhenius Equation• How do we find Ea and A? By experiments!

• You will do this in the lab!• If we have data from 2 different temperatures, we

can find Ea mathematically:

lnk2

k1

⎛

⎝⎜⎞

⎠⎟ =

Ea

R1T1 -

1T2

⎛

⎝⎜⎞

⎠⎟

or lnk1k2

⎛

⎝⎜⎞

⎠⎟ =

Ea

R1T2

- 1T1

⎛

⎝⎜⎞

⎠⎟

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Arrhenius Equation• But we can’t find A with only 2 temperatures. • If we have data from 3 or more different

temperatures, we can find Ea and A graphically.

• According to the Arrhenius Equation:

• If we graph lnk vs. 1/T, we get a straight line with a slope of -Ea/R and a y-intercept of lnA.

ln k = -Ea

RT + lnA

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Arrhenius Equation• Here’s a typical graph of the Arrhenius Equation.