Arrhenius Equation Forms and Applications

Introduction

• Arrhenius equation relates rate constant ( k ) to temperature ( T ) and activation energy ( Ea ).
• Basic form: [ k = A \times e^{-Ea/RT} ]
  • ( A ) = frequency factor
  • ( R ) = gas constant

Derivation of Alternate Forms

Natural Log Form

• Take natural log on both sides: [ \ln k = \ln A + \ln (e^{-Ea/RT}) ]
• Simplifies to: [ \ln k = \ln A - \frac{Ea}{RT} ]
• Rewritten as: [ \ln k = -\frac{Ea}{R} \frac{1}{T} + \ln A ]
• This form resembles ( y = mx + b ):
  • ( y = \ln k ), ( x = \frac{1}{T} )
  • Slope ( m = -\frac{Ea}{R} )
  • Y-intercept ( b = \ln A )
• Useful for graphing:
  • Plot ( \ln k ) vs. ( \frac{1}{T} ) to find:
    • Slope: reveals ( Ea )
    • Y-intercept: reveals ( A )

Temperature-Specific Form

• At temperature ( T_1 ): [ \ln k_1 = -\frac{Ea}{R} \frac{1}{T_1} + \ln A ]
• At temperature ( T_2 ): [ \ln k_2 = -\frac{Ea}{R} \frac{1}{T_2} + \ln A ]
• Subtracting these equations: [ \ln \left(\frac{k_2}{k_1}\right) = -\frac{Ea}{R} \left(\frac{1}{T_2} - \frac{1}{T_1}\right) ]
• Advantage: No ( A ) involved
• Application: Knowing rate constants for two temperatures allows calculation of ( Ea ).

Conclusion

• Alternate forms of Arrhenius equation provide different insights and uses.
• Use form best suited to the problem or available data.
• Textbook variations may exist in presentation of forms.
• Future discussions/videos will address specific use-cases for each form.