Which of the following curves represent a first order reaction?
Understanding First Order Reaction Kinetics
A first order reaction is a reaction whose rate depends on the concentration of only one reactant raised to the power of one. The rate law for a first order reaction involving a reactant A is given by:
\begin{equation*} \text{Rate} = k[A] \end{equation*}
where $[A]$ is the concentration of reactant A and $k$ is the rate constant.
Integrated Rate Law for First Order Reactions
To understand how concentration changes with time in a first order reaction, we look at the integrated rate law. The differential rate law is $\frac{d[A]}{dt} = -k[A]$. Integrating this gives:
\begin{equation*} \ln[A]_t - \ln[A]_0 = -kt \end{equation*}
Rearranging this equation, we get:
\begin{equation*} \ln[A]_t = -kt + \ln[A]_0 \end{equation*}
This equation is in the form of a straight line, $y = mx + c$, where:
- $y = \ln[A]_t$ (natural logarithm of the concentration of A at time $t$)
- $m = -k$ (the slope, which is a negative constant)
- $x = t$ (time)
- $c = \ln[A]_0$ (the y-intercept, natural logarithm of the initial concentration of A)
Therefore, plotting $\ln[A]_t$ against time $(t)$ for a first order reaction yields a straight line with a negative slope.
Analyzing the Given Reaction Curves
Let's examine the types of curves typically associated with different reaction orders when plotted in specific ways:
- Zero Order Reaction: $[A]_t = -kt + [A]_0$. A plot of $[A]_t$ versus $t$ is a straight line with a negative slope.
- First Order Reaction: $\ln[A]_t = -kt + \ln[A]_0$. A plot of $\ln[A]_t$ versus $t$ is a straight line with a negative slope.
- Second Order Reaction (Type 1: Rate = k[A]2): $\frac{1}{[A]_t} = kt + \frac{1}{[A]_0}$. A plot of $\frac{1}{[A]_t}$ versus $t$ is a straight line with a positive slope.
We are looking for a curve that represents a first order reaction. Based on the integrated rate law, the characteristic linear plot for a first order reaction is $\ln[A]_t$ vs. $t$. Let's consider the shapes of the curves provided as options:
Option 1: Shows a linear relationship, possibly Rate vs. Concentration, which could be first order (Rate = k[A]). However, the question asks for curves representing the reaction itself, usually concentration or a function of concentration vs. time.
Option 2: Shows a curve increasing with time. This could represent the concentration of a product over time, but the shape is not typically linear or a simple exponential for product formation in a standard plot type.
Option 3: Shows a curve decreasing with time in a non-linear fashion. This could represent the concentration of a reactant decaying over time for a first or second-order reaction, but neither gives a straight line when plotting concentration vs. time.
Option 4: Shows a straight line with a negative slope. This is characteristic of a plot where the y-axis represents $\ln[Reactant]$ and the x-axis represents time for a first order reaction, or potentially concentration vs time for a zero-order reaction. Given the context of reaction orders and common plots, a straight line often implies a specific manipulation of the concentration term (like $\ln[A]$ or $1/[A]$) plotted against time.
Comparing the options with the expected plots for different reaction orders, the straight line with a negative slope (Option 4) is the definitive representation of a first order reaction when the y-axis is $\ln[Reactant]$ and the x-axis is time.
Conclusion on First Order Reaction Curve
The curve that represents a first order reaction is the one showing a linear relationship with a negative slope when the natural logarithm of the reactant concentration is plotted against time. Among the given options, only Option 4 depicts a straight line with a negative slope, which fits this description for a first order reaction.
| Reaction Order | Integrated Rate Law Plot | Expected Graph Shape |
|---|---|---|
| Zero Order | $[A]_t$ vs. Time ($t$) | Straight line with negative slope |
| First Order | $\ln[A]_t$ vs. Time ($t$) | Straight line with negative slope |
| Second Order (Rate=k[A]2) | $\frac{1}{[A]_t}$ vs. Time ($t$) | Straight line with positive slope |
Revision Table: Key Kinetic Plots
| Plot (y vs. x) | Slope | Intercept | Reaction Order Indicated |
|---|---|---|---|
| $[A]$ vs. $t$ | $-k$ | $[A]_0$ | Zero Order |
| $\ln[A]$ vs. $t$ | $-k$ | $\ln[A]_0$ | First Order |
| $\frac{1}{[A]}$ vs. $t$ | $k$ | $\frac{1}{[A]_0}$ | Second Order |
Additional Information on First Order Reaction Kinetics
The half-life ($t_{1/2}$) of a first order reaction is independent of the initial concentration. It is given by the formula:
\begin{equation*} t_{1/2} = \frac{\ln(2)}{k} = \frac{0.693}{k} \end{equation*}
This is a unique characteristic of first order reactions and is often used to identify them. Radioactive decay is a common example of a process that follows first order kinetics.
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