You wish to determine the activation energy for the following first-order reaction: A → B + C. How would you use the Arrhenius equation to determine the activation energy?

• A. Plot the natural logarithm of the concentration of A against time and determine the slope.
• B. Plot the concentration of A against time and determine the slope.
• C. Plot the rate constant (k) against the temperature (T) and determine the slope.
• D. Plot the natural logarithm of the rate constant (ln k) against the inverse of the temperature (1/T) and determine the slope.

Answer: (D)

Verified step by step guidance

1. Understand that the Arrhenius equation is given by: \( k = A e^{-\frac{E_a}{RT}} \), where \( k \) is the rate constant, \( A \) is the pre-exponential factor, \( E_a \) is the activation energy, \( R \) is the gas constant, and \( T \) is the temperature in Kelvin.
2. Take the natural logarithm of both sides of the Arrhenius equation to linearize it: \( \ln k = \ln A - \frac{E_a}{R} \cdot \frac{1}{T} \). This equation is in the form of \( y = mx + c \), where \( y = \ln k \), \( m = -\frac{E_a}{R} \), \( x = \frac{1}{T} \), and \( c = \ln A \).
3. To determine the activation energy \( E_a \), plot \( \ln k \) (the natural logarithm of the rate constant) on the y-axis against \( \frac{1}{T} \) (the inverse of the temperature in Kelvin) on the x-axis.
4. Determine the slope of the resulting straight line from the plot. The slope \( m \) of this line is equal to \( -\frac{E_a}{R} \).
5. Calculate the activation energy \( E_a \) by rearranging the slope equation: \( E_a = -m \cdot R \), where \( R \) is the gas constant (8.314 J/mol·K).