Reaction Rates and Rate Laws

Reaction Rates

Definition of Rate

Reactions don't have built-in speedometers.
Reaction rate is determined by measuring changes in concentrations of reactants and products over time.
For a generic reaction: 2A + B \rightarrow C, where 1 mole of C is produced from 2 moles of A and 1 mole of B.
Rate can be described by:
- Disappearance of reactants over time.
- Appearance of products over time.
Reactants are consumed, so rate expressions have a negative sign.
- Rate with respect to A: \frac{-\Delta[A]}{\Delta t}
- Rate with respect to B: \frac{-\Delta[B]}{\Delta t}
- Rate with respect to C: \frac{\Delta[C]}{\Delta t}
Stoichiometric coefficients are not equal, meaning rates of concentration changes are not equal.
- 2 moles of A consumed for every 1 mole of B consumed.
- Rate of consumption of A is twice the rate of consumption of B.
- Rate of consumption of A is twice the rate of production of C.
- Rate of consumption of B is equal to the rate of production of C.

Standard Rate of Reaction

To show a standard rate where rates are equal with respect to all species, divide the rate of concentration change by the stoichiometric coefficient.
For the general reaction: aA + bB \rightarrow cC + dD
- Rate = \frac{-\Delta[A]}{a\Delta t} = \frac{-\Delta[B]}{b\Delta t} = \frac{\Delta[C]}{c\Delta t} = \frac{\Delta[D]}{d\Delta t}
Rate is expressed in units of moles per liter per second: mol/(L \cdot s) or M/s (Molarity per second).

Determination of Rate Law

MCAT will likely provide experimental data to determine the rate law.
Rate is proportional to the concentrations of reactants raised to experimentally determined exponents for forward irreversible reactions.
For the general reaction: aA + bB \rightarrow cC + dD
- Rate is proportional to [A]^x [B]^y
Rate law equation:
- Rate = k[A]^x[B]^y
- k = rate constant or rate coefficient.
- x and y are the orders of the reaction.
Rate is always in units of concentration over time (Molarity/second).
Exponents (x, y, z) indicate the order of the reaction with respect to each reactant.
- x = order with respect to reactant A.
- y = order with respect to reactant B.
- Overall order = sum of x and y.
Exponents can be integers or fractions and must be determined experimentally.
MCAT focuses on 0, 1st, and 2nd order reactions, with integer exponents.

Common Traps in Chemical Kinetics

Assuming reaction orders are the same as stoichiometric coefficients.
- Orders of a reaction must be determined experimentally.
Two cases where stoichiometric coefficients match reaction orders:
- Reaction mechanism is a single step.
- Complete reaction mechanism is given, and the rate-determining step is indicated (reactant side coefficients equal reaction orders).
Complication: Rate-determining step involves an intermediate reactant.
- Derive intermediate molecule's concentration using the law of mass action (equilibrium constant expression).
Mistaking equilibrium constant expression (law of mass action) for the rate law.
- Equilibrium includes concentrations of all species (reactants and products).
- Rate law includes only reactants.
- K_{eq} indicates equilibrium position; rate indicates how quickly equilibrium is reached.
Rate constant k is technically not a constant because it depends on activation energy and temperature.
- However, for a specific reaction at a specific temperature, k is constant.
- For reversible reaction: K{eq} = \frac{k}{k{-1}} (k is forward reaction rate constant, k_{-1} is reverse reaction rate constant).
Equilibrium principles apply only at the end of the reaction (after equilibrium is reached).
Reaction rate is usually measured at or near the beginning of the reaction to minimize the effects of the reverse reaction.

Experimental Determination of Rate Law

Values of k, x, and y in Rate = k[A]^x[B]^y must be determined experimentally at a given temperature.
MCAT uses straightforward reaction mechanisms, experimental data, and rate laws.
Experimental data is usually a chart with initial reactant concentrations and initial rates of product formation.
Identify trials where only one reactant concentration changes while others remain constant.
Change in rate is attributable to the change in concentration of that one reactant.
Example: Reactants A and B, forming product C.
- [A] is constant; [B] doubles; rate quadruples.
- Rate = k[A]^x[B]^y
- Doubling [B] quadruples the rate, so 2^y = 4, thus y = 2.
Repeat for other reactants.
Once orders are determined, write the complete rate law.
To find k, plug in values from any trial.

Example

Reaction: A + B \rightarrow C + D at 300 K
Data:
- Trial 1: [A] = 1.00 M, [B] = 1.00 M, Rate = 2.0 M/s
- Trial 2: [A] = 1.00 M, [B] = 2.00 M, Rate = 8.1 M/s
- Trial 3: [A] = 2.00 M, [B] = 2.00 M, Rate = 15.9 M/s
Solution:
- Trials 1 & 2: [A] is constant, [B] doubles, rate increases by ~4.
- Rate ∝ [B]^y
- \frac{\Delta Rate}{\Delta [B]^y}
- 4 = 2^y, so y = 2
- Rate = k[A]^x[B]^2
- Trials 2 & 3: [B] is constant, [A] doubles, rate increases by ~2.
- Rate ∝ [A]^x
  - \frac{\Delta Rate}{\Delta [A]^x}
- 2 = 2^x, so x = 1
- Rate = k[A][B]^2
- Order is 1 with respect to A, 2 with respect to B. Overall order is 3.
- Calculate k using Trial 1: 2.0 M/s = k(1.00 M)(1.00 M)^2
- k = 2.0 M^{-2}s^{-1}
- Final rate law: Rate = 2.0 M^{-2}s^{-1}[A][B]^2

Reaction Orders

Classify reactions based on kinetics: zero order, first order, second order, higher order, or mixed order.
Generic reaction: aA + bB \rightarrow cC + dD

Zero Order Reaction

Rate is independent of reactant concentrations.
Constant reaction rate equals the rate constant k.
Rate law: Rate = k[A]^0[B]^0 = k, where k has units of M/s.
Rate constant depends on temperature; change the temperature to change the rate.
Catalyst addition lowers activation energy, increasing k and changing the rate.
Concentration vs. time plot is linear; slope is -k.

First Order Reaction

Rate is directly proportional to one reactant.
Doubling reactant concentration doubles the rate.
Rate law: Rate = k[A]^1 or Rate = k[B]^1, where k has units of s^{-1}.
Example: radioactive decay.
- Rate = \frac{-\Delta[A]}{\Delta t} = k[A]
- Concentration at time t: [A]t = [A]0e^{-kt}
  - [A]_t = concentration of A at time t.
  - [A]_0 = initial concentration of A.
  - k = rate constant.
  - t = time.
A first-order rate law with a single reactant suggests the molecule undergoes a chemical change by itself, without interaction with other molecules.
Concentration vs. time plot is nonlinear.
ln[A] vs. time plot is linear; slope is -k.

Second Order Reaction

Rate is proportional to the concentrations of two reactants or the square of one reactant.
Rate laws:
- Rate = k[A]^1[B]^1
- Rate = k[A]^2
- Rate = k[B]^2, where k has units of M^{-1}s^{-1}.
Often suggests a physical collision between two reactant molecules.
Concentration vs. time plot is nonlinear.
\frac{1}{[A]} vs. time plot is linear; slope is k.

Higher Order Reactions

Reactions involving a termolecular process are rare (few reactions with 3rd order rates).
Simultaneous collision of three particles with correct orientation and sufficient energy is rare.

Mixed Order Reactions

Sometimes refers to non-integer orders (fractions) or reactions with rate orders that vary.
Fractions are more specifically described as broken order.
Now refers solely to reactions that change order over time.
Example: Rate = \frac{k1[C][A]^2}{k2 + k_3[A]}, where A is a reactant and C is a catalyst.
If [A] is large: k3[A] >> k2, reaction appears 1st order with respect to A.
If [A] is low: k2 >> k3[A], reaction appears 2nd order with respect to A.
Must recognize how the rate order changes as the reacting concentration changes; deriving the rate expression not required.

Conclusion

Reaction rate laws are derived through analysis of experimental data.
Factors can affect chemical reaction rates.
Chemical principles in the human body depend on chemical kinetics.
- Body maintains a certain temperature to stabilize enzymes for metabolic reactions.
- Body maintains a pH buffer.
  - Altering [H^+] affects enzyme structure and reactant collisions.
Clinical perspective throughout medical career.
Next topic: chemical equilibria (related to kinetics but distinct).