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Time

Figure 1 Plot of concentration remaining versus time for a first-order process (e.g., radioactive decay).

Time

Figure 1 Plot of concentration remaining versus time for a first-order process (e.g., radioactive decay).

curved, which means that the rate is constantly changing and therefore cannot be expressed in terms of a finite time interval. By resorting to differential calculus, it is possible to express the rate of decay in terms of an infinitesimally small change in concentration (dC) over an infinitesimally small time interval (dt). The resulting function, dC/dt, is the slope of the line, and it is this function that is proportional to concentration in a first-order process. Thus,

The negative sign is introduced because the concentration is falling as time progresses.

Equation (8) is the differential rate expression for a first-order reaction. The value of the rate constant, k, could be calculated by determining the slope of the concentration versus time curve at any point and dividing by the concentration at that point. However, the slope of a curved line is difficult to measure accurately, and k can be determined much more easily using an integrated rate expression.

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