Which plot yields a straight line for first-order elimination?

Which plot yields a straight line for first-order elimination?

• A. Plot of C versus time
• B. Semi-log plot of log C versus time
• C. Plot of time versus C
• D. Double-log plot of C versus time

Answer: (B)

First-order elimination produces exponential decay, meaning the rate of loss is proportional to the current concentration. Mathematically, dC/dt = -kC, and integrating gives ln C = -k t + ln C0. Because of this, plotting the logarithm of concentration against time yields a straight line with slope -k (or -k/2.303 if using base-10 log). This makes the semi-log plot of log C versus time the standard way to visualize first-order elimination and to determine the elimination rate constant from the slope. Plotting concentration versus time gives a curved, exponential decline rather than a straight line. Plotting time versus concentration isn’t linear either, since t = (ln C0 - ln C)/k, which isn’t a straight line when time is graphed against concentration. A double-log plot of C versus time likewise won’t be linear for first-order decay, because the relationship is exponential in time, not a power-law.

First-order elimination produces exponential decay, meaning the rate of loss is proportional to the current concentration. Mathematically, dC/dt = -kC, and integrating gives ln C = -k t + ln C0. Because of this, plotting the logarithm of concentration against time yields a straight line with slope -k (or -k/2.303 if using base-10 log). This makes the semi-log plot of log C versus time the standard way to visualize first-order elimination and to determine the elimination rate constant from the slope.

Plotting concentration versus time gives a curved, exponential decline rather than a straight line. Plotting time versus concentration isn’t linear either, since t = (ln C0 - ln C)/k, which isn’t a straight line when time is graphed against concentration. A double-log plot of C versus time likewise won’t be linear for first-order decay, because the relationship is exponential in time, not a power-law.