Question

Sketch a graph for the generic first-order reaction \(\mathrm{A} \longrightarrow \mathrm{B}\) that has concentration of \(\mathrm{A}\) on the vertical axis and time on the horizontal axis. (a) Is this graph linear? Explain. (b) Indicate on your graph the half-life for the reaction.

Short answer

The graph of the generic first-order reaction is plotted with concentration of A on the vertical axis and time on the horizontal axis. It follows an exponential decay function, which means the graph is not linear, but a nonlinear downward-sloping curve. The half-life of the reaction can be found by locating the point where the concentration of A decreases to half its initial value, \([\frac{[A]_0}{2}]\). The time value where this point intersects the horizontal axis represents the half-life of the reaction.

Step-by-step solution

Write the expression for concentration as a function of time

For a first-order reaction, the concentration of A as a function of time is given by the following formula: \[ [A] = [A]_0 e^{-kt} \] Where \([A]\) is the concentration of A at time \(t\), \([A]_0\) is the initial concentration of A, \(k\) is the rate constant, and \(t\) is time.

Sketch the graph

Plot the concentration of A on the vertical axis and time on the horizontal axis. Now, remember that the equation for a first-order reaction has an exponential decay function. This means that the concentration of A decreases over time, but it never completely reaches zero. Start your curve at the point \((0,[A]_0)\), which represents the initial concentration. As time progresses, the concentration of A decreases exponentially, making the curve gradually flatten but never touch the horizontal axis.

Determine if the graph is linear

Examine the shape of the curve. Since it is an exponential decay function, the graph is not linear. Rather, it is a nonlinear, downward-sloping curve that tapers off as time progresses.

Indicate the half-life on the graph

The half-life of a first-order reaction is defined as the amount of time it takes for the concentration of A to decrease by half. Using the formula for a first-order reaction, the half-life (\(t_{1/2}\)) can be expressed as: \[ t_{1/2} = \frac{\ln{2}}{k} \] On the graph, find the point where the concentration of A has reduced by half. To do this, locate point with \(y\)-coordinate equal to \([\frac{[A]_0}{2}]\). Draw a horizontal line from this point on the vertical axis until it intersects the curve, and then draw a vertical line from this intersection point to the horizontal (time) axis. The point where the vertical line intersects the time axis is the half-life of the reaction.

Key concepts

Concentration-Time Relationship

Understanding the concentration-time relationship is essential when studying chemical kinetics, especially in first-order reactions. Imagine a common scenario where a substance A is transforming into substance B over time. In a graph representing a first-order reaction, where the concentration of A is plotted on the vertical axis and time is on the horizontal axis, the curve we get is not a straight line.

This curve displays a characteristic pattern. It starts high when time is zero (think of it as the moment we start observing the reaction), representing the initial concentration, \( [A]_0 \). As time increases, the curve descends, showing that the concentration of A is decreasing. However, unlike dropping off a cliff, the decline in concentration slows down as time passes; the graph approaches, but never quite touches, the horizontal axis, indicating the concentration of A is decreasing progressively slower.

This pattern tells us something fundamental: in a first-order reaction, the rate at which the concentration of A decreases is proportional to its current concentration. When the concentration is high, A transforms into B rapidly. But as less of A remains, the transformation slows down. This integral relationship between concentration and time is what characterizes first-order reactions and is described mathematically by an exponential decay function.

Exponential Decay Function

The graph we discussed earlier best illustrates an exponential decay function, a concept crucial to first-order reactions. Unlike linear relationships where changes occur at a constant rate, exponential decay describes a process that decreases at a rate proportional to its current value.

Specifically, for a first-order reaction, the concentration of A at any given time, t, is given by the formula \( [A] = [A]_0 e^{-kt} \), where \( e \) is the base of the natural logarithm, and \( k \) is the rate constant. The negative sign in the exponent of \( e \) ensures that as time increases, the value of \( e^{-kt} \) gets smaller, indicating that the concentration of A is decaying over time.

To visualize this, think of the exponential decay function as a swift drop in concentration followed by an ever-increasing reluctance to drop further, resulting in a curve that 'flattens out' as time progresses. This shape is key to understanding how the concentration of a substance in a first-order reaction declines over time.

Reaction Half-Life

Reaction half-life (\(t_{1/2}\)) provides a simple way to understand the kinetics of a reaction without getting lost in complex calculations. It refers to the time required for the concentration of a reactant to reach half of its initial value.

For a first-order reaction, half-life is particularly noteworthy because it is constant. This means no matter how much reactant A we start with, it will always take the same amount of time for its concentration to reduce by half. To express half-life mathematically, we use the formula \( t_{1/2} = \frac{\ln{2}}{k} \), where \( \ln{2} \) (approximately 0.693) is the natural logarithm of 2, and \( k \) is the rate constant.

On the concentration-time graph, the half-life is visually represented as the point in time at which the curve has fallen to half of the initial concentration of A, \( [A]_0 \). This fixed time interval underscores the predictability and uniform nature of the half-life in first-order reactions.

Rate Constant

The rate constant (\( k \) is a pivotal element in the kinetics of a reaction. For first-order reactions, \( k \) not only dictates the rate at which reactant A will convert to product B but also relates directly to the half-life and the overall speed of the reaction.

The rate constant can be derived from experimental data and once determined, it remains a fixed value for a given reaction at a specific temperature. Its units are reciprocal seconds (\( s^{-1} \)), underscoring that it represents the change of concentration per unit time. The larger the rate constant, the faster the reaction progresses.

Because \( k \) impacts the steepness of the exponential decay function, it plays a definitive role in shaping the graph's contour on a concentration-time plot. A high value of \( k \) would result in a steeper descent in concentration, implying a quick-reacting scenario. Conversely, a lower \( k \) signifies a more gradual decline in reaction rate, translating to a flatter curve on the graph. The relationship between the rate constant and reaction progress is therefore fundamental in understanding chemical kinetics.