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: $K<em>{eq} = \frac{k}{k</em>{-1}}$ (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: $[A]<em>t = [A]</em>0e^{-kt}$
    • $[A]_t$ = 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 = $\frac{k<em>1[C][A]^2}{k</em>2 + k_3[A]}$, where A is a reactant and C is a catalyst.
• If [A] is large: $k<em>3[A] >> k</em>2$, reaction appears 1st order with respect to A.
• If [A] is low: $k<em>2 >> k</em>3[A]$, 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).