Sequential Reactions and Intermediates (25.7)• Sequential reactions (elementary) involve multiple reactions in which one

or more intermediates are formed– The differential form of the rate is written with respect to product formation– Rate law only involves intermediate since it is the only species that generates products

• Intermediate concentration is extremely hard to measure, so we need to relate it to quantities we can measure– We should be able to measure reactant concentration, so we rely on the fact that A only

decays in one way

• Since the intermediate only forms from the reactant, we can express its concentration in terms of A– The first reaction creates I, the second reaction depletes it (hence the negative sign)

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A kA ⏐ → ⏐ I k I ⏐ → ⏐ P

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Rate =d P[ ]dt

= kI I[ ]

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−d A[ ]dt

= kA A[ ]

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A[ ] = A[ ]0e−kA t

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d I[ ]dt

= kA A[ ] − kI I[ ]

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I[ ] =kA

kI − kAe−kA t − e−k I t( ) A[ ]0

Sequential Reactions and Rate Determining Steps (25.7)• Concentration of product can be determined using a simple mass balance

principle– The sum of concentrations of all substances at any time must be equal to the initial

concentration of reactant

• The size of the rate constants is an indication of how quickly each part of the reaction proceeds– If kI is much larger than kA, the intermediate does not last very long and thus does not

build up in the reaction– If kA is much larger than kI, then the reactant decomposes quickly and a significant

amount of intermediate will form

• The rate-determining (or rate-limiting) step is the slowest step in the mechanism and it dictates how quickly products will form– If first step is rate-limiting, the reaction looks like it is one step (i.e., decay of reactant

and formation of product follow first order kinetics)– If second step is rate-limiting, reaction follows first-order for intermediate

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A[ ]0= A[ ] + I[ ] + P[ ]

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P[ ] =kAe

−k I t − kIe−kA t

kI − kA+1

⎛

⎝ ⎜

⎞

⎠ ⎟ A[ ]0

Parallel Reactions (25.8)• Parallel reactions involve the reactant decaying into more than one

product– The rate of decay of the reactant is related to the rate constants of both processes

• The rates of formation of each product has a simple form due to the simplicity of the differential rate equation– The product concentration (P = B or C) differs based on the rate constant

• The ratio of concentrations is related to the ratio of the rate constants for a parallel set of reactions– The overall yield (ϕ) of a product in the reaction is the ratio of the concentration of the

product of interest over the sum of all product concentrations

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−d A[ ]dt

= kB B[ ] + kC C[ ]€

B kB← ⏐ ⏐ A kC ⏐ → ⏐ C

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A[ ] = A[ ]0e− kB +kC( )t

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d P[ ]dt

= kP A[ ] = kP A[ ]0e− kB +kC( )t

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P[ ] =kP

kB + kCA[ ]0

1− e− kB +kC( )t( )

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B[ ]C[ ]

=kBkC

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φ=k1

k1 + k2 + k3 + ...

Reversible Reactions and Equilibrium (25.10)• Reversible reactions are ones in which the reactants can be generated

from products– Each direction of the reaction has a rate constant associated with it

• If the reaction starts with only A, after a certain length of time the concentrations of A and B stabilize (i.e., equilibrium is obtained)

• At equilibrium, rate of change of A and B are zero– Equilibrium constant (K) can be expressed as a ratio of rate constants

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AkA ⏐ → ⏐kB

← ⏐ ⏐ B

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Rate =d A[ ]dt

= −kA A[ ] + kB B[ ] = −d B[ ]dt

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A[ ]eq = A[ ]0

kBkA + kB

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B[ ]eq = A[ ]01−

kBkA + kB

⎛

⎝ ⎜

⎞

⎠ ⎟

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d A[ ]eqdt

= −d B[ ]eqdt

= −kA A[ ]eq + kB B[ ]eq = 0

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kAkB

=B[ ]eqA[ ]eq

=K

Concentration Profile for Sequential Reactions

Rate-Limiting Behavior in Sequential Reactions

Concentration Profile for Parallel Reactions

Concentration Profile for Reversible Reaction