Concentration-Time Graphs

In a zero-order reaction, the rate is independent of the reactant concentration. This results in a constant rate of consumption, which manifests as a straight line with a negative gradient on a concentration-time graph.

The mathematical relationship is expressed as $[A]_t = -kt + [A]_0$, where the gradient of the line is equal to the negative of the rate constant ($-k$). This is the only order where the rate constant can be read directly from the slope of the concentration-time plot.

Because the rate never changes, the reactant is consumed at a steady pace until it is entirely exhausted. This linear behavior is common in reactions limited by surface area, such as catalysis on a solid metal surface.

A first-order reaction shows a rate that is directly proportional to the concentration of the reactant. On a graph, this produces a characteristic exponential decay curve that levels off as it approaches the x-axis.

The most defining feature of a first-order concentration-time graph is a constant half-life ($t_{1/2}$). This means the time taken for the concentration to drop from $1.0$ to $0.5$ is identical to the time taken to drop from $0.5$ to $0.25$.

To calculate the rate constant $k$ from this graph, one must either use the half-life formula $k = \frac{\ln(2)}{t_{1/2}}$ or measure the gradient of a tangent at a specific concentration and divide that rate by the concentration at that point.

Tangent Precision: When asked to find the rate at a specific time, always use a long ruler to draw a tangent that extends as far as possible across the grid. This reduces the percentage error when calculating $\frac{\Delta y}{\Delta x}$.

Half-life Verification: To prove a reaction is first-order in an exam, you must show at least two successive half-lives on the graph. Clearly mark the points where the concentration halves and show that the time intervals are equal.

Units of k: Remember that the units for the rate constant $k$ change depending on the order. For zero-order, $k$ has the same units as rate ($mol\,dm^{-3}\,s^{-1}$), but for first-order, it is simply $s^{-1}$.

Check the Axes: Always verify if the graph is plotting Concentration vs. Time or Rate vs. Concentration. Students often confuse the straight line of a zero-order concentration-time graph with the straight line of a first-order rate-concentration graph.