Which plot would be linear for a first-order reaction in A → products?

Which plot would be linear for a first-order reaction in A → products?

• A. Plot ln[A] vs t
• B. Plot [A] vs t
• C. Plot 1/[A] vs t
• D. Plot log[A] vs t

Answer: (A)

For a first-order reaction, the rate law is -d[A]/dt = k[A]. Integrating gives ln[A] = -kt + ln[A]0, so the natural logarithm of the concentration versus time is a straight line with slope -k and intercept ln[A]0. This linear relationship is why plotting ln[A] against time yields a line for a first-order process. Plotting [A] versus time shows exponential decay, not a straight line. Plotting 1/[A] versus time would be linear for a second-order reaction, not first-order. Plotting log[A] versus time would be linear if log means the natural logarithm (they differ only by a constant factor), but the standard, unambiguous way to identify a first-order process is the ln[A] versus t plot.

For a first-order reaction, the rate law is -d[A]/dt = k[A]. Integrating gives ln[A] = -kt + ln[A]0, so the natural logarithm of the concentration versus time is a straight line with slope -k and intercept ln[A]0. This linear relationship is why plotting ln[A] against time yields a line for a first-order process.

Plotting [A] versus time shows exponential decay, not a straight line. Plotting 1/[A] versus time would be linear for a second-order reaction, not first-order. Plotting log[A] versus time would be linear if log means the natural logarithm (they differ only by a constant factor), but the standard, unambiguous way to identify a first-order process is the ln[A] versus t plot.