Table of Contents
What Does “Time Course” of a Reaction Mean?
When chemists talk about the time course of a chemical reaction, they mean how the amounts (or concentrations) of reactants and products change as time passes. Instead of only asking “What are the starting materials and final products?”, kinetics asks:
- How fast are reactants consumed?
- How fast are products formed?
- How does this speed change over time?
To describe and measure this systematically, we use:
- Concentration–time curves (graphs)
- Reaction rate as a function of time
- Rate laws that link rate with concentration
In this chapter, we focus on how these quantities behave over time, without yet going deeply into temperature effects or catalysis (covered in later chapters).
Reaction Rate as a Function of Time
For a general reaction
$$
\text{a A} + \text{b B} \rightarrow \text{c C} + \text{d D}
$$
we define the instantaneous rate at a certain time $t$ via the change in concentration with time:
$$
v(t) = -\frac{1}{a} \frac{d[\text{A}]}{dt}
= -\frac{1}{b} \frac{d[\text{B}]}{dt}
= \frac{1}{c} \frac{d[\text{C}]}{dt}
= \frac{1}{d} \frac{d[\text{D}]}{dt}
$$
Key points for the time course:
- Reactant concentrations decrease with time: $d[\text{A}]/dt < 0$, $d[\text{B}]/dt < 0$.
- Product concentrations increase with time: $d[\text{C}]/dt > 0$, $d[\text{D}]/dt > 0$.
- The rate itself usually changes with time, because concentrations change.
On a concentration–time graph:
- Reactant curves slope downward.
- Product curves slope upward.
- The slope at any point gives the rate at that moment.
Experimental Observation of the Time Course
To follow the time course, chemists monitor some observable quantity that depends on concentration:
- Color (absorbance) for colored reactants/products
- Conductivity for ionic species
- pH for acid–base reactions
- Pressure for gas-phase reactions
- Mass or volume changes in some setups
Experimental procedure (conceptual):
- Mix reactants quickly to define $t = 0$.
- Measure an observable quantity at defined times.
- Convert observables into concentrations (using calibration or known relations).
- Plot concentration vs. time and analyze the curve.
The resulting curves show the time course directly.
Rate Laws and Their Consequences for Time Course
For many reactions, the rate law has a simple power-law form. For a reaction involving one reactant A,
$$
v(t) = -\frac{d[\text{A}]}{dt} = k \,[\text{A}]^{n}
$$
- $k$ is the rate constant (depends on temperature, etc., but not on concentration).
- $n$ is the order of the reaction with respect to A.
Different orders produce characteristic concentration–time curves. Understanding these shapes is central to describing the time course.
We will look qualitatively at zero-, first-, and second-order behavior, which are commonly encountered ideal cases.
Zero-Order Reactions
Rate law:
$$
-\frac{d[\text{A}]}{dt} = k
$$
- The rate is constant, independent of $[\text{A}]$.
- A decreases linearly with time.
Integration gives:
$$
\text{A} = [\text{A}]_0 - kt
$$
where $[\text{A}]_0$ is the initial concentration at $t = 0$.
Time course characteristics:
- Straight-line decline of $[\text{A}]$ vs. $t$.
- Eventually, $[\text{A}]$ would reach zero at $t = [\text{A}]_0/k$ (in practice, reaction effectively stops earlier as approximations break down).
- The rate is the same at the beginning and near the end, as long as the zero-order approximation holds.
Zero-order behavior can appear when a surface or catalyst is saturated, so that the limiting factor is not how much A is left in solution.
First-Order Reactions
Rate law:
$$
-\frac{d[\text{A}]}{dt} = k\,[\text{A}]
$$
- Rate is proportional to the concentration.
- As A is consumed, the rate automatically slows.
Integration gives:
$$
\text{A} = [\text{A}]_0\, e^{-kt}
$$
This is an exponential decay.
Half-Life for First-Order Reactions
The half-life $t_{1/2}$ is the time required for the concentration to drop to half its initial value:
$$
\text{A} = \frac{1}{2} [\text{A}]_0
$$
Insert into the integrated law:
$$
\frac{1}{2} [\text{A}]_0 = [\text{A}]_0 e^{-kt_{1/2}}
\Rightarrow e^{-kt_{1/2}} = \frac{1}{2}
\Rightarrow t_{1/2} = \frac{\ln 2}{k}
$$
Important property:
- For a first-order reaction, $t_{1/2}$ is independent of $[\text{A}]_0$.
- The time course therefore shows successive equal time intervals each reducing the concentration by a constant factor (e.g., halves).
Time course characteristics:
- Concentration decreases quickly at first, then more slowly.
- The curve approaches zero but never truly reaches it in finite time.
- A plot of $\ln [\text{A}]$ vs. $t$ is a straight line (often used to identify first-order behavior experimentally).
Many processes such as radioactive decay and some simple decompositions follow first-order kinetics.
Second-Order Reactions (Simple Cases)
Consider a simple second-order case with only one reactant A:
$$
-\frac{d[\text{A}]}{dt} = k\,[\text{A}]^2
$$
Integration gives:
$$
\frac{1}{\text{A}} = \frac{1}{[\text{A}]_0} + kt
$$
Time course characteristics:
- At high initial concentration, the rate at early times is very high (because it depends on $[\text{A}]^2$).
- As A is consumed, the rate drops off more rapidly than in a first-order reaction.
- A plot of $1/[\text{A}]$ vs. $t$ is a straight line for a simple second-order reaction.
The half-life now depends on the initial concentration:
$$
t_{1/2} = \frac{1}{k\,[\text{A}]_0}
$$
- If $[\text{A}]_0$ is larger, the half-life is shorter.
- This contrasts with first-order kinetics.
Another common second-order situation: reaction of two different species with equal initial concentrations:
$$
\text{A} + \text{B} \rightarrow \text{Products}, \quad [\text{A}]_0 = [\text{B}]_0
$$
This often shows a similar integrated form and similar time-course behavior (fast initial decay that slows markedly with time).
Initial Rate and Early-Time Behavior
The initial rate $v_0$ is the rate very near $t = 0$. It is often easier to measure accurately because:
- Conditions are well defined (known initial concentrations).
- Reverse reaction and side reactions often have not yet become significant.
For many rate laws, the early-time behavior can be approximated by:
$$
v_0 = v(t \approx 0) = k \, f([\text{A}]_0, [\text{B}]_0, \dots)
$$
where $f$ expresses the dependence on initial concentrations (e.g., $f = [\text{A}]_0^n$).
The slope at the very beginning of a concentration–time curve equals the initial rate. The shape just after $t = 0$ gives information on the order and mechanism.
Typical Shapes of Concentration–Time Curves
For a single reactant A converting to products:
- Zero-order: Straight-line decrease; constant slope until A is nearly exhausted.
- First-order: Smooth exponential decay; slope (rate) is steep at first, then becomes gradually flatter.
- Second-order (simple): Initially very steep decline for high $[\text{A}]_0$, then curve flattens more strongly than for first-order as $[\text{A}]$ drops.
For products, the curves usually show the complementary, increasing trends, often approaching a plateau when reactants are depleted (or when equilibrium is reached in reversible systems).
Time Course in Reversible Reactions (Qualitative)
For a reversible reaction
$$
\text{A} \rightleftharpoons \text{B}
$$
there are forward and reverse rates:
- Forward: A → B, with rate depending on $[\text{A}]$.
- Reverse: B → A, with rate depending on $[\text{B}]$.
Typical time course:
- At $t = 0$, only A present: forward rate is high, reverse rate is zero.
- As B forms, reverse rate increases; forward rate decreases as A is consumed.
- Eventually, forward and reverse rates become equal: the system reaches dynamic equilibrium.
- Concentrations then remain constant over time (though both directions continue at equal rates).
On a concentration–time plot:
- $[\text{A}]$ decreases from its initial value toward an equilibrium value.
- $[\text{B}]$ increases from zero toward its equilibrium value.
- Both approach their equilibrium values asymptotically (flattening curves as time goes on).
The detailed connection between time course and equilibrium constants is treated in the chapters on chemical equilibrium and the law of mass action; here we only note that reversible reactions often show a characteristic approach to constant concentrations.
Pseudo-Order and Its Effect on Time Course
In many practical cases, several reactants are involved, but one is in large excess. For example:
$$
\text{A} + \text{B} \rightarrow \text{Products}, \quad [\text{B}]_0 \gg [\text{A}]_0
$$
The rate law might be:
$$
v = k\,[\text{A}][\text{B}]
$$
If $[\text{B}]$ remains approximately constant over the time of interest (because B is in huge excess), we can define an effective rate constant:
$$
k' = k [\text{B}] \approx \text{const.}
$$
and write:
$$
v = k' [\text{A}]
$$
So although the true reaction is second order, the observed time course of $[\text{A}]$ behaves like a first-order reaction (this is called pseudo-first-order behavior).
Time-course consequence:
- $[\text{A}]$ will show exponential decay similar to a first-order reaction.
- The apparent half-life depends on $[\text{B}]$ via $k' = k[\text{B}]$.
This trick is widely used to simplify kinetic analysis: by using large excess of one reactant, the time course becomes easier to interpret and model.
Relating Time Course and Mechanism (Qualitative)
The way concentrations change with time often gives clues about the reaction mechanism (the sequence of elementary steps).
Some qualitative examples:
- Single-step (elementary) reaction: Often shows simple order behavior (e.g., first order in A, or second order in A and B).
- Consecutive reactions: A → B → C
The time course of B typically shows a rise to a maximum, then a decline, as B is first formed from A and then consumed to form C. - Parallel reactions: A can react to form different products (A → B and A → C). The time courses of B and C reflect relative speeds of the competing pathways.
- Autocatalytic reactions: A product speeds up its own formation; rates can show slow start then rapid acceleration followed by slowing as reactant is used up.
Even without doing full mathematical treatments, recognizing such shapes in concentration–time plots helps infer possible mechanisms.
Summary: Key Ideas for the Time Course of Reactions
- The time course describes how concentrations of reactants and products change with time.
- The rate is the slope of the concentration–time curve; it usually changes over time because concentrations change.
- Rate laws (with given orders) determine the mathematical form of the concentration–time relationship:
- Zero-order: linear decrease $[\text{A}] = [\text{A}]_0 - kt$
- First-order: exponential decay $[\text{A}] = [\text{A}]_0 e^{-kt}$, constant half-life
- Second-order (simple): $1/[\text{A}] = 1/[\text{A}]_0 + kt$, half-life depends on $[\text{A}]_0$
- In reversible reactions, concentrations typically move from initial values toward constant equilibrium values, where forward and reverse rates are equal.
- Pseudo-order approaches (e.g., pseudo-first-order) simplify the observed time course by holding some concentrations effectively constant.
- The shapes of concentration–time curves contain information about reaction order and can give hints about the mechanism of the reaction.