Using the integrated rate law for a first-order reaction, ln[A]t = -kt + ln[A]0, how can you determine the rate constant (k) if you have concentration data over time?

• A. Plot [A]t versus time and find the slope of the line.
• B. Plot ln[A]0 versus time and find the slope of the line.
• C. Plot ln[A]t versus time and find the slope of the line.
• D. Plot 1/[A]t versus time and find the slope of the line.

Answer: (C)

Verified step by step guidance

1. Understand the integrated rate law for a first-order reaction: \( \ln[A]_t = -kt + \ln[A]_0 \). This equation describes how the natural logarithm of the concentration of reactant \( [A] \) at time \( t \) changes over time.
2. Recognize that the equation \( \ln[A]_t = -kt + \ln[A]_0 \) is in the form of a linear equation \( y = mx + b \), where \( y \) is \( \ln[A]_t \), \( m \) is the slope \( -k \), \( x \) is time \( t \), and \( b \) is the y-intercept \( \ln[A]_0 \).
3. To determine the rate constant \( k \), plot \( \ln[A]_t \) on the y-axis and time \( t \) on the x-axis. This will create a straight line if the reaction is first-order.
4. Find the slope of the line from the plot. The slope \( m \) of the line is equal to \( -k \). Therefore, the rate constant \( k \) can be determined by taking the negative of the slope.
5. Ensure that the data used for plotting is accurate and covers a sufficient range of time to confirm the linear relationship, which validates the first-order nature of the reaction.