Rate constants are crucial in understanding chemical reactions. They tell us how fast a reaction happens at a specific temperature. These constants link reaction speed to the amount of reactants present, helping us predict how quickly chemicals will transform.

Knowing rate constants lets us compare reaction speeds and figure out how temperature affects them. We can calculate these constants from experiments, watching how reactant amounts change over time. This knowledge is key for controlling reactions in labs and industries.

Rate Constants

Definition of rate constant

• Proportionality constant $k$ relates reaction rate to reactant concentrations
• Specific to a particular reaction at a given temperature
• Units depend on overall reaction order
  • First-order reaction: $s^{-1}$ (reciprocal seconds)
  • Second-order reaction with rate law $rate = k[A][B]$: $M^{-1}s^{-1}$ (reciprocal molar-seconds)
  • Second-order reaction with rate law $rate = k[A]^2$: $M^{-1}s^{-1}$ (reciprocal molar-seconds)
  • Zero-order reaction: $Ms^{-1}$ (molar per second)

Significance in reaction rates

• Determines reaction speed at given temperature and reactant concentrations
  • Larger $k$ indicates faster reaction, smaller $k$ indicates slower reaction
• Intrinsic property of reaction depends on factors like reactant nature, catalysts, and temperature
• Used in rate law equation to calculate reaction rate
  • First-order reaction example: $rate = k[A]$, $[A]$ is reactant A concentration

Temperature dependence via Arrhenius equation

• Arrhenius equation relates rate constant to temperature: $k = Ae^{-E_a/RT}$
  • $A$: pre-exponential factor related to reactant molecule collision frequency
  • $E_a$: activation energy, minimum energy for reaction to occur
  • $R$: universal gas constant (8.314 J/mol·K)
  • $T$: absolute temperature (K)
• Rate constant increases exponentially with increasing temperature
  • Higher temperature leads to more molecules with energy to overcome activation energy barrier
• Activation energy and pre-exponential factor determined from Arrhenius plot slope and y-intercept (ln($k$) vs. $1/T$)

Calculation from experimental data

• Rate constant calculated by measuring reactant or product concentrations over time
• First-order reaction uses integrated rate law: $ln[A]_t = -kt + ln[A]_0$
  • $[A]_t$: reactant A concentration at time $t$
  • $[A]_0$: initial reactant A concentration
  • Plot $ln[A]_t$ vs. $t$, slope is $-k$
• Second-order reaction with rate law $rate = k[A][B]$ uses integrated rate law: $1/[A]_t = kt + 1/[A]_0$ (assuming $[A]_0 = [B]_0$)
  • Plot $1/[A]_t$ vs. $t$, slope is $k$
• Zero-order reaction uses integrated rate law: $[A]_t = -kt + [A]_0$
  • Plot $[A]_t$ vs. $t$, slope is $-k$