Reaction rate experiments are an important topic frequently covered in physical chemistry laboratories. Changes in reactant concentration or absorbance over time are measured to determine reaction order, rate constants, half-life, and other quantities. In particular, for a first-order reaction, plotting the logarithm of concentration against time gives a linear relationship, allowing the rate constant to be determined by graphical analysis.
In a discussion of reaction rate experiments, it is not sufficient simply to write that “the concentration decreased with time,” “the reaction was first order,” or “the rate constant was determined.” It is necessary to explain why an ln plot becomes linear for a first-order reaction, what the slope of the fitted line means, how the rate constant and half-life are determined, and how changes in temperature or deviations in the reaction start time affect the results.
This article clearly explains reaction rate experiments focusing on first-order reactions, including how to determine the rate constant and half-life, how to interpret graphs, sources of error, points for improvement, and discussion examples that can be used in reports.
Note: This article is a reference intended to assist with discussions of results obtained in physical chemistry experiments at universities and similar institutions. For the actual reagents, measuring instruments, temperature conditions, absorbance measurements, titration procedures, data processing, and specified report format, always follow the instructions in your university’s laboratory manual and those given by your instructor or TA.
- What Is a Reaction Rate Experiment?
- Main Items to Include in the Results
- Reference Experimental Values for a First-Order Reaction and Examples of Rate Constant and Half-Life Analysis
- Reference Experimental Conditions
- Rate Law and Integrated Rate Equation for a First-Order Reaction
- Example Measurement of Concentration Change Over Time
- Confirming a First-Order Reaction Using an ln[A] Plot
- Example Calculation of the Rate Constant k
- Example Calculation of the Rate Constant Using Two Points
- Example Calculation of Half-Life
- Example Confirming That Half-Life Is Independent of Initial Concentration
- Example Measurements With Different Initial Concentrations
- Example of Determining Concentration From Absorbance
- Comparison of Concentration and ln Concentration Plots
- Changes in the Rate Constant and Half-Life With Temperature
- Example of Confirming Half-Life From Measured Values
- Treatment of Low-Concentration Data in the Later Stage of the Reaction
- Example When an Outlier Is Present
- Reference Example of a Pseudo-First-Order Reaction
- Main Sources of Measurement Error
- Example of How to Write the Results
- Points for Connecting the Results to the Discussion
- Example Discussion
- Summary
- What Is a First-Order Reaction?
- Integrated Rate Equation for a First-Order Reaction
- How to Read a First-Order Reaction Graph
- How to Determine the Rate Constant k
- How to Determine Half-Life
- Why the Half-Life Remains Constant
- Discussion When Absorbance Is Used
- Discussion of Blank Correction in Absorbance Measurements
- Error in Time Measurement
- Effect of Temperature on the Rate Constant
- When Initial Reaction Data Deviate
- When Data in the Later Stage of the Reaction Deviate
- Discussion of R2 for the Fitted Line
- Discussion When the Reaction Is Not First Order
- Discussion of a Pseudo-First-Order Reaction
- Discussion of the Unit of the Rate Constant
- Reasons the Rate Constant May Differ From a Literature Value
- When the Half-Life Does Not Agree With the Measured Value
- Error in Concentration Conversion
- Error Caused by Insufficient Mixing
- How to Handle Outliers
- When the Results Can Be Considered Good
- Example Discussion When the Experiment Did Not Go Well
- How to Write Points for Improvement
- Difference Between a Superficial Discussion and a Good Discussion
- Examples of Expressions That Can Be Used in Reports
- Points to Check When Discussing a Reaction Rate Experiment
- Summary
What Is a Reaction Rate Experiment?
A reaction rate experiment is an experiment used to investigate how quickly a chemical reaction proceeds. Changes in reactant concentration over time, or how the amount of product increases, are measured to determine the rate law and rate constant.
In physical chemistry experiments, changes in absorbance, electrical conductivity, pH, pressure, titration volume, and other quantities may be used instead of directly measuring concentration. If the measured value is proportional to the reactant or product concentration, it can be used for rate analysis.
Example Discussion:
In this experiment, the change in the measured value over time as the reaction proceeded was followed, and the reaction rate was analyzed. If the measured value changes in correspondence with the reactant concentration, the rate law can be examined from the concentration change with time. Therefore, in a reaction rate experiment, it is important to convert the time-dependent measured values into an appropriate form and graph them.
Main Items to Include in the Results
In the results of a reaction rate experiment, organize the table of time and measured values, concentration conversion, graphs, fitted lines, rate constant, half-life, coefficient of determination, and other items. Particularly for a first-order reaction, ln concentration or the ln of the measured value is plotted against time, and the linearity is used to determine whether the reaction can be treated as first order.
Main Items to Include in the Results
- Reaction temperature
- Measurement start time
- Measurement time
- Concentration or measured value at each time
- Values obtained by converting absorbance or other measurements to concentration
- ln concentration or ln measured value
- Graph of time versus ln value
- Equation of the fitted line
- Coefficient of determination R2
- Rate constant k
- Half-life t1/2
- Comparison with theoretical or literature values
- Presence or absence of outliers or scatter
Example of How to Write the Results:
The reactant concentration at each time was determined, and ln[reactant] was plotted against time, producing an approximately linear relationship. The slope of the fitted line was negative, and the rate constant k was determined from the absolute value of this slope. The half-life of the first-order reaction was also calculated using the obtained k.
Reference Experimental Values for a First-Order Reaction and Examples of Rate Constant and Half-Life Analysis
Here, concentration changes, ln concentration plots, rate constants, and half-lives obtained in a first-order reaction experiment are organized as reference experimental values that are easy to discuss in a report. Changes in concentration with time, conversion from absorbance to concentration, how to determine the rate constant k, half-life, temperature dependence, and sources of error are covered.
In a first-order reaction, the reaction rate is proportional to the reactant concentration. Therefore, the reactant concentration [A] decreases exponentially with time, and plotting ln[A] against time gives a straight line. The rate constant k can be determined from the slope of this line.
Reference Experimental Conditions
| Item | Details |
|---|---|
| Measurement targets | Decomposition of dyes, ester hydrolysis, decomposition of hydrogen peroxide, decomposition reactions of chemicals, etc. |
| Measurement methods | Absorbance measurement, titration, conductivity measurement, measurement of evolved gas volume |
| Measured quantities | Time, reactant concentration, absorbance, ln[A] |
| Analysis items | Determination of first-order behavior, rate constant k, half-life, linearity, temperature dependence, sources of error |
| Temperature conditions | 25°C as the reference. Comparative experiments may also use 15°C, 35°C, 45°C, etc. |
Rate Law and Integrated Rate Equation for a First-Order Reaction
In a first-order reaction, the reaction rate v is proportional to the reactant concentration [A].
v = −d[A]/dt = k[A]
Integrating this equation gives the following expression.
ln[A] = −kt + ln[A]0
Therefore, if ln[A] is plotted on the vertical axis and time t on the horizontal axis, a first-order reaction gives a straight line whose slope is −k.
| Item | Meaning in a First-Order Reaction | Point to Check in a Report |
|---|---|---|
| [A] | Reactant concentration | Decreases exponentially with time |
| ln[A] | Natural logarithm of concentration | Becomes linear with respect to time |
| Slope | −k | The absolute value of the slope is the rate constant |
| Intercept | ln[A]0 | Corresponds to the logarithm of the initial concentration |
| Half-life | ln2/k | Independent of the initial concentration |
Example Measurement of Concentration Change Over Time
The following is reference data in which the concentration of reactant A was measured at different times. In this example, the data are arranged so that ln[A] is approximately linear with time and can be readily analyzed as a first-order reaction.
| Time | [A] | ln[A] | [A]/[A]0 | Observation |
|---|---|---|---|---|
| 0 min | 0.100 mol/L | −2.303 | 1.00 | Initial concentration |
| 5 min | 0.078 mol/L | −2.551 | 0.78 | Decrease |
| 10 min | 0.061 mol/L | −2.797 | 0.61 | Further decrease |
| 15 min | 0.047 mol/L | −3.058 | 0.47 | Close to approximately half |
| 20 min | 0.037 mol/L | −3.297 | 0.37 | Reaction proceeds |
| 25 min | 0.029 mol/L | −3.540 | 0.29 | Low-concentration region |
| 30 min | 0.022 mol/L | −3.817 | 0.22 | More susceptible to error |
The concentration [A] decreases with time, but the concentration itself does not decrease linearly. For a first-order reaction, whether the relationship between ln[A] and time is linear is checked.
Confirming a First-Order Reaction Using an ln[A] Plot
The following is an example of the fitted equation obtained by plotting ln[A] against time using the data above.
ln[A] = −0.0500t − 2.303
| Analysis Item | Value | Meaning |
|---|---|---|
| Fitted equation | ln[A] = −0.0500t − 2.303 | Corresponds to the integrated equation for a first-order reaction |
| Slope | −0.0500 | Corresponds to −k |
| Rate constant k | 0.0500 min−1 | Constant representing the reaction rate |
| Intercept | −2.303 | Corresponds to ln(0.100) |
| Coefficient of determination R2 | 0.999 | High linearity |
Because the relationship between ln[A] and time shows high linearity, this reaction is considered to be treatable as a first-order reaction.
Example Calculation of the Rate Constant k
The integrated rate equation for a first-order reaction is as follows.
ln[A] = −kt + ln[A]0
If the fitted line is ln[A] = −0.0500t − 2.303, the slope is −0.0500.
−k = −0.0500
k = 0.0500 min−1
Therefore, the rate constant of this reaction is 0.0500 min−1. The unit of the rate constant for a first-order reaction is the reciprocal of time.
Example Calculation of the Rate Constant Using Two Points
The rate constant can also be roughly estimated from the data at 0 min and 20 min.
k = {ln[A]0 − ln[A]t} / t
When [A]0 = 0.100 mol/L and [A]20 = 0.037 mol/L,
k = {ln(0.100) − ln(0.037)} / 20
k = {−2.303 − (−3.297)} / 20 = 0.994 / 20 = 0.0497 min−1
The rate constant determined from two points is close to 0.0500 min−1, which was obtained from the fitted line. However, because a calculation using only two points is more susceptible to measurement error, linear fitting using multiple points is more reliable.
Example Calculation of Half-Life
The half-life of a first-order reaction is determined using the following equation.
t1/2 = ln2 / k
When k = 0.0500 min−1,
t1/2 = 0.693 / 0.0500 = 13.9 min
Therefore, in this reaction, it takes approximately 13.9 minutes for the reactant concentration to decrease by half.
Example Confirming That Half-Life Is Independent of Initial Concentration
In a first-order reaction, the half-life does not change even if the initial concentration changes. This is an important characteristic of first-order reactions.
| Initial Concentration [A]0 | Half Concentration | Time to Reach It | Rate Constant k | Judgment |
|---|---|---|---|---|
| 0.200 mol/L | 0.100 mol/L | 13.9 min | 0.0500 min−1 | Same half-life |
| 0.100 mol/L | 0.050 mol/L | 13.9 min | 0.0500 min−1 | Same half-life |
| 0.050 mol/L | 0.025 mol/L | 13.9 min | 0.0500 min−1 | Same half-life |
Even when the initial concentration differs, if the time required for the concentration to decrease by half is the same, the result agrees with the characteristics of a first-order reaction.
Example Measurements With Different Initial Concentrations
Even when the initial concentration is changed, the rate constant k should be almost the same if the temperature and other reaction conditions are identical.
| Sample | Initial Concentration [A]0 | Slope of ln[A] vs t | Rate Constant k | Half-Life | Judgment |
|---|---|---|---|---|---|
| Sample A | 0.050 mol/L | −0.0495 | 0.0495 min−1 | 14.0 min | Typical |
| Sample B | 0.100 mol/L | −0.0500 | 0.0500 min−1 | 13.9 min | Reference |
| Sample C | 0.200 mol/L | −0.0508 | 0.0508 min−1 | 13.6 min | Nearly identical |
| Sample D | 0.100 mol/L | −0.0430 | 0.0430 min−1 | 16.1 min | Possible outlier |
Samples A–C have almost the same k values, which is consistent with a first-order reaction. If k is smaller, as in Sample D, possible causes include a decrease in temperature, deviation in the reaction start time, insufficient mixing, and concentration-measurement error.
Example of Determining Concentration From Absorbance
If the reactant or product is colored, its concentration can be determined from absorbance. According to the Beer-Lambert law, absorbance is proportional to concentration.
A = εcl
If the calibration curve is expressed as A = 8.00 × [A], the concentration corresponding to an absorbance of 0.488 is as follows.
[A] = 0.488 ÷ 8.00 = 0.061 mol/L
| Time | Absorbance | Converted Concentration [A] | ln[A] | Use in Analysis |
|---|---|---|---|---|
| 0 min | 0.800 | 0.100 mol/L | −2.303 | Initial concentration |
| 5 min | 0.624 | 0.078 mol/L | −2.551 | Rate analysis |
| 10 min | 0.488 | 0.061 mol/L | −2.797 | Rate analysis |
| 15 min | 0.376 | 0.047 mol/L | −3.058 | Rate analysis |
| 20 min | 0.296 | 0.037 mol/L | −3.297 | Rate analysis |
| 25 min | 0.232 | 0.029 mol/L | −3.540 | Low-concentration region |
After converting absorbance to concentration, ln[A] is calculated and the linearity with respect to time is checked.
Comparison of Concentration and ln Concentration Plots
| Plot | Fitted Equation | Coefficient of Determination R2 | Evaluation |
|---|---|---|---|
| [A] vs t | [A] = −0.00255t + 0.093 | 0.945 | Low linearity |
| ln[A] vs t | ln[A] = −0.0500t − 2.303 | 0.999 | Good for a first-order reaction |
| 1/[A] vs t | 1/[A] = 1.15t + 8.9 | 0.970 | Insufficient for a second-order reaction |
In a first-order reaction, not concentration itself but ln concentration becomes linear with respect to time. Therefore, it is important not to determine the reaction order from the [A] vs t plot alone.
Changes in the Rate Constant and Half-Life With Temperature
As temperature increases, the reaction rate generally increases and the rate constant k also increases. As a result, the half-life becomes shorter.
| Temperature | Slope of ln[A] vs t | Rate Constant k | Half-Life | How to Interpret the Result |
|---|---|---|---|---|
| 15°C | −0.025 | 0.025 min−1 | 27.7 min | Reaction is slow |
| 25°C | −0.050 | 0.050 min−1 | 13.9 min | Standard condition |
| 35°C | −0.095 | 0.095 min−1 | 7.3 min | Reaction is fast |
| 45°C | −0.170 | 0.170 min−1 | 4.1 min | Reaction is considerably fast |
As temperature increases, the proportion of molecules with energies exceeding the activation energy increases, so the reaction rate becomes larger. Data obtained under different temperature conditions should not be compared as though they have the same rate constant.
Example of Confirming Half-Life From Measured Values
In a first-order reaction, the time required for the concentration to decrease by half is approximately constant.
| Concentration Change | Start Time | Time Reached | Time Required for Halving | Judgment |
|---|---|---|---|---|
| 0.100 → 0.050 mol/L | 0 min | Approximately 14 min | Approximately 14 min | Half-life |
| 0.050 → 0.025 mol/L | Approximately 14 min | Approximately 28 min | Approximately 14 min | Similar |
| 0.025 → 0.0125 mol/L | Approximately 28 min | Approximately 42 min | Approximately 14 min | Similar |
The fact that the time required for each halving is approximately the same is consistent with the characteristics of a first-order reaction.
Treatment of Low-Concentration Data in the Later Stage of the Reaction
As the reaction proceeds and the concentration becomes low, the relative error in measured values becomes larger. Particularly in absorbance measurements, the smaller the absorbance, the more noticeable the effects of noise and blank correction become.
| Time | Concentration [A] | Example Concentration Error | Effect on ln[A] | Treatment |
|---|---|---|---|---|
| 5 min | 0.078 | ±0.002 | Small | Easy to use |
| 20 min | 0.037 | ±0.002 | Moderate | Use with caution |
| 30 min | 0.022 | ±0.002 | Large | Check linearity |
| 40 min | 0.014 | ±0.002 | Very large | Consider exclusion |
If points in the low-concentration region deviate greatly from the straight line, measurement error and the detection limit must be checked rather than simply concluding that the reaction mechanism changed.
Example When an Outlier Is Present
| Time | Measured [A] | Expected [A] | Judgment | Possible Cause |
|---|---|---|---|---|
| 0 min | 0.100 | 0.100 | Good | – |
| 5 min | 0.078 | 0.078 | Good | – |
| 10 min | 0.061 | 0.061 | Good | – |
| 15 min | 0.060 | 0.047 | Possible outlier | Time-recording error, insufficient mixing, absorbance-reading error |
| 20 min | 0.037 | 0.037 | Good | – |
If only the value at 15 min is too high, check the sampling time, measurement cell, absorbance, and whether the wrong sample was used before concluding that the reaction actually became slower.
Reference Example of a Pseudo-First-Order Reaction
Even in a reaction involving two types of reactants, if one reactant is present in large excess, its concentration can be treated as approximately constant, and the reaction may be analyzed as though it were first order. This is called a pseudo-first-order reaction.
| Condition | [A]0 | [B]0 | Apparent Rate Constant kobs | Interpretation |
|---|---|---|---|---|
| Small amount of B | 0.010 mol/L | 0.010 mol/L | Low first-order linearity | Changes in both concentrations are important |
| B in 10-fold excess | 0.010 mol/L | 0.100 mol/L | 0.040 min−1 | Easy to treat as pseudo-first-order |
| B in 20-fold excess | 0.010 mol/L | 0.200 mol/L | 0.080 min−1 | kobs increases according to B concentration |
In a pseudo-first-order reaction, after determining the apparent rate constant kobs, the original rate law can be discussed by considering its relationship with the concentration of the reactant present in excess.
Main Sources of Measurement Error
| Source of Error | Effect on Measured Values | Trend in Results | Improvement |
|---|---|---|---|
| Deviation in reaction start time | Time axis shifts | k becomes inaccurate | Standardize immediately after mixing as 0 min |
| Insufficient mixing | Initial concentration is not uniform | Initial data become irregular | Mix quickly and consistently |
| Temperature change | Rate constant changes | k becomes larger or smaller | Measure under constant-temperature conditions |
| Absorbance-reading error | Concentration conversion shifts | ln[A] plot becomes scattered | Perform blank correction and multiple measurements |
| Calibration-curve error | All concentrations shift | Affects evaluation of k and half-life | Prepare standard solutions accurately |
| Measurement limit in the low-concentration region | Relative error becomes large | Linearity in the later stage of the reaction worsens | Use a reliable concentration range |
Example of How to Write the Results
When the concentration of reactant A was measured at each time, [A] decreased from 0.100 mol/L at 0 min to 0.022 mol/L at 30 min. To determine whether the reaction was first order, ln[A] was plotted against time, producing an approximately linear relationship. The fitted line was ln[A] = −0.0500t − 2.303, and the coefficient of determination R2 was 0.999.
Comparing this with the integrated rate equation for a first-order reaction, ln[A] = −kt + ln[A]0, the slope of the line corresponds to −k. Therefore, the rate constant was determined as k = 0.0500 min−1. In addition, the half-life was 13.9 min from t1/2 = 0.693/k. This value approximately agreed with the time required for the concentration to decrease from 0.100 mol/L to about 0.050 mol/L in the measured data.
Even in experiments with different initial concentrations, the obtained rate constants fell within the range of 0.0495–0.0508 min−1 and were almost constant. In a first-order reaction, the rate constant depends on temperature and reaction conditions rather than on the initial concentration, so this result agrees with the characteristics of a first-order reaction. The fact that the half-life was independent of initial concentration also supports the characteristics of a first-order reaction.
Points for Connecting the Results to the Discussion
In a discussion of a first-order reaction, it is important to explain not only that the concentration decreased but also the linearity of the ln[A] plot, the relationship between the slope and the rate constant, the meaning of half-life, and the effects of temperature and errors in relation to one another.
- Can you explain that the rate is proportional to the reactant concentration in a first-order reaction?
- Can you explain why ln[A] vs t becomes linear in relation to the integrated rate equation?
- Can you determine the rate constant k from the slope of the fitted line?
- Have you confirmed that the unit of the first-order rate constant is the reciprocal of time?
- Can you calculate the half-life using t1/2 = ln2/k?
- Can you explain that the half-life of a first-order reaction is independent of the initial concentration?
- If concentration is determined from absorbance, can you explain the calibration curve and Beer-Lambert law?
- Can you discuss that increasing temperature increases the rate constant and shortens the half-life?
- Can you explain reaction start time, insufficient mixing, temperature changes, and measurement error at low concentrations as sources of error?
- For a pseudo-first-order reaction, can you explain the idea of treating the concentration of the excess component as constant?
Example Discussion
In this experiment, the change in the concentration of reactant A was measured at each time to determine whether the reaction could be analyzed as a first-order reaction. [A] decreased with time, but when [A] itself was plotted against time, the relationship was not perfectly linear. In contrast, plotting ln[A] against time gave high linearity, so this reaction is considered to be first order with respect to A.
The integrated rate equation for a first-order reaction is ln[A] = −kt + ln[A]0. The fitted line obtained from the experimental data was ln[A] = −0.0500t − 2.303. Because this slope corresponds to −k, the rate constant is k = 0.0500 min−1. In addition, the intercept −2.303 corresponds to ln(0.100), confirming consistency with the initial concentration.
The half-life determined from the rate constant was 13.9 min. In a first-order reaction, the half-life is independent of the initial concentration and is determined only by the rate constant. In fact, even when the initial concentration was changed to 0.050, 0.100, and 0.200 mol/L, the obtained k values were almost the same. This agrees with the characteristics of a first-order reaction.
As the temperature increased, the rate constant became larger and the half-life became shorter. This is considered to occur because increasing the temperature increases the kinetic energy of the molecules and increases the proportion of molecules that can react by exceeding the activation energy. Therefore, when comparing rate constants, the temperature must be kept constant.
Possible sources of error include deviation in the reaction start time, insufficient mixing, errors in converting absorbance to concentration, errors in the calibration curve, and temperature changes. Particularly in the later stage of the reaction, the concentration and absorbance become small, so the relative error becomes larger and the linearity of the ln[A] plot may decrease. When determining the rate constant, it is important to check the reliability of outliers and low-concentration data.
Summary
In a first-order reaction, the relationship between ln[A] and time is linear, and the rate constant k can be determined from the absolute value of its slope. The half-life is determined using t1/2 = ln2/k and is characterized by being independent of the initial concentration.
This reference example covered concentration changes over time, ln[A] plots, rate constants, two-point calculations, half-life, comparisons using different initial concentrations, concentration conversion from absorbance, temperature dependence, errors in the low-concentration region, outliers, and pseudo-first-order reactions. In a report, it is useful to relate linearity, slope, rate constant, and half-life and clearly show the basis for judging the reaction to be first order.
What Is a First-Order Reaction?
A first-order reaction is a reaction in which the reaction rate is proportional to the first power of the reactant concentration. Considering a reaction in which reactant A decreases, the reaction rate is proportional to [A], so the reaction is faster when the concentration is high and slower as the concentration becomes lower.
Rate = k[A]
In a first-order reaction, the reactant concentration decreases exponentially with time. Therefore, plotting concentration itself against time gives a curve, whereas taking the natural logarithm of concentration gives a straight line.
Example Discussion:
In a first-order reaction, the reaction rate is proportional to the reactant concentration. As the reaction proceeds, the reactant concentration decreases, so the reaction rate also becomes smaller with time. Therefore, the concentration does not change linearly with time but decreases exponentially, and a linear relationship can be confirmed by plotting ln[reactant] against time.
Integrated Rate Equation for a First-Order Reaction
In a first-order reaction, the change in reactant concentration [A] with time can be expressed by the integrated rate equation. If the initial concentration is [A]0 and the concentration at time t is [A], the following relationship applies.
ln[A] = ln[A]0 − kt
This equation corresponds to the straight-line equation y = ax + b. If ln[A] is placed on the vertical axis and t on the horizontal axis, the slope is −k and the intercept is ln[A]0. Therefore, the rate constant k can be determined from the slope of the graph.
Example Discussion:
The integrated rate equation for a first-order reaction is expressed as ln[A] = ln[A]0 − kt. Therefore, plotting ln[A] against time t gives a straight line with slope −k and intercept ln[A]0. Because the graph obtained in this experiment was approximately linear, the reaction is considered to be treatable as a first-order reaction within the measured range.
How to Read a First-Order Reaction Graph
To confirm a first-order reaction, a graph is created with time t on the horizontal axis and ln[A] on the vertical axis. If the measurement points lie on a straight line, the reaction is more likely to be treatable as a first-order reaction. The slope of the fitted line becomes negative, and its absolute value is the rate constant k.
| Graph Element | Meaning | Interpretation for a First-Order Reaction |
|---|---|---|
| Horizontal axis t | Reaction time | Represents the passage of time |
| Vertical axis ln[A] | Natural logarithm of reactant concentration | Becomes linear for a first-order reaction |
| Slope | Related to the rate of concentration decrease | Corresponds to −k |
| Intercept | Value at t = 0 | Corresponds to ln[A]0 |
Example Discussion:
When ln[A] was plotted against time, the measurement points lay approximately on a straight line. From this result, the reaction in this experiment is considered to follow the integrated rate equation for a first-order reaction. Because the slope of the fitted line corresponds to −k, the rate constant can be determined from the absolute value of the slope.
How to Determine the Rate Constant k
The rate constant k for a first-order reaction is determined from the slope of the graph of ln[A] versus time t. If the fitted line is expressed as y = ax + b, the slope a corresponds to −k. Therefore, the rate constant k is the value obtained by reversing the sign of the slope.
k = − slope
The unit of the rate constant for a first-order reaction is the reciprocal of time. If time is treated in seconds, the unit is s−1, and if time is treated in minutes, the unit is min−1. In a report, always write the unit of k according to the unit of time.
Example Discussion:
The slope of the fitted line was negative and corresponds to −k in the integrated rate equation for a first-order reaction. Therefore, the absolute value of the slope was taken as the rate constant k. Because time was treated in seconds, the unit of the rate constant is s−1.
How to Determine Half-Life
Half-life is the time required for the reactant concentration to decrease to half of its initial concentration. In a first-order reaction, the half-life is independent of the initial concentration and depends only on the rate constant k. The half-life of a first-order reaction is determined using the following equation.
t1/2 = ln2 / k ≒ 0.693 / k
The larger the rate constant k, the faster the reaction proceeds and the shorter the half-life becomes. Conversely, the smaller k is, the slower the reaction and the longer the half-life.
Example Discussion:
In a first-order reaction, the half-life is expressed as t1/2 = 0.693/k. Using the rate constant k determined in this experiment, the time required for the reactant concentration to decrease by half was calculated. Because a larger k means a greater reaction rate, the half-life becomes shorter.
Why the Half-Life Remains Constant
In a first-order reaction, the half-life is independent of the initial concentration. This is because the fractional change required for the concentration to decrease by half follows the same rate law regardless of the starting concentration. Therefore, in a first-order reaction, the time required for [A]0 to become [A]0/2 is the same as the time required for [A]0/2 to become [A]0/4.
Example Discussion:
In a first-order reaction, the half-life is independent of the initial concentration and is determined only by the rate constant k. This is because the reaction rate is proportional to the reactant concentration at that moment and the concentration decreases by a constant fraction. Therefore, the time required for the concentration to decrease by half is considered to be the same in any concentration range.
Discussion When Absorbance Is Used
In reaction rate experiments, changes in reactant or product concentration may be followed using absorbance. If absorbance is proportional to concentration, rate analysis may sometimes be performed using absorbance instead of concentration. However, it is necessary to confirm which component’s absorbance is being measured and whether the absorbance lies within the range proportional to concentration.
If the absorbance of a reactant decreases with time, a first-order analysis may be performed by plotting ln absorbance against time. If the absorbance of a product increases, it may be necessary to convert the value to a quantity corresponding to the remaining reactant concentration before analysis.
Example Discussion:
In this experiment, the reaction rate was analyzed using absorbance proportional to the reactant concentration. Within the range where absorbance is proportional to concentration, an ln plot can be prepared using absorbance instead of concentration. Because plotting ln(absorbance) against time gave a linear relationship, the reaction is considered to be treatable as a first-order reaction.
Discussion of Blank Correction in Absorbance Measurements
Blank correction is important when absorbance is used. If background absorption from the solvent, cell, or reagents remains, the measured absorbance includes absorption from components other than the target substance. Taking the ln of this value without correction may cause error in the rate analysis.
Particularly in the later stage of the reaction, when absorbance becomes small, deviation in the blank has a relatively large effect. Therefore, insufficient blank correction can be discussed as a cause of later points deviating from the fitted line.
Example Discussion:
In absorbance measurements, if blank correction is insufficient, absorption from components other than the target substance is included in the measured value. Particularly in the later stage of the reaction, absorbance becomes small, so deviation in the blank has a large effect on the ln value and may cause deviation from the fitted line. Therefore, accurate blank correction before measurement is important for determining an accurate rate constant.
Error in Time Measurement
Time measurement is extremely important in reaction rate experiments. If the reaction start time is unclear or there is a delay before measurement begins, large errors occur in the initial measured values. Particularly for a fast reaction, even a deviation of several seconds may greatly affect the rate constant.
Example Discussion:
A possible reason the initial measurement point deviated from the fitted line is a deviation in the reaction start time. If there is a time lag between mixing the reaction solution and actually starting measurement, the concentration at t = 0 cannot be measured accurately. Particularly when the reaction rate is high, a deviation of only a few seconds at the beginning may greatly affect calculation of the rate constant.
Effect of Temperature on the Rate Constant
Reaction rate depends strongly on temperature. In general, as temperature increases, the reaction rate increases and the rate constant k also becomes larger. If the temperature is not kept constant during a reaction rate experiment, the rate constant may change during measurement and the linearity expected for a first-order reaction may become poor.
When comparing a rate constant with a literature value, it is also necessary to confirm whether the measurement temperature of the literature value matches the experimental temperature.
Example Discussion:
A difference in measurement temperature may explain why the rate constant differed from the literature value. The reaction rate depends strongly on temperature, and k tends to become larger at higher temperatures. Therefore, if the temperature was not constant during measurement, the individual measurement points may have deviated from the linear relationship expected for a first-order reaction.
When Initial Reaction Data Deviate
Initial reaction data are susceptible to insufficient mixing, delayed measurement start, and unstable temperature. If measurement is performed before the reaction solution becomes uniform, the measured value may not correctly reflect the actual concentration. The time required to place the sample in the instrument can also cause error in the initial data.
Example Discussion:
A possible reason the initial measurement point deviated from the fitted line is insufficient mixing of the reaction solution. Immediately after the reaction starts, the concentration and temperature may not be uniform, making it difficult for the measured value to reflect the actual average concentration. A time lag before starting the measurement may also have contributed to the deviation of the initial data.
When Data in the Later Stage of the Reaction Deviate
In the later stage of the reaction, the reactant concentration and absorbance become small, so the effect of measurement error becomes relatively larger. Even slight reading errors or deviations in blank correction may become large when converted to ln values. Side reactions or the effect of equilibrium may also cause deviation from a simple first-order reaction.
Example Discussion:
A possible reason the measurement points in the later stage of the reaction deviated from the fitted line is that the measured values became small and the relative error increased. In regions of low absorbance or concentration, even slight reading errors or deviations in blank correction have a large effect on ln values. Therefore, data in the later stage of the reaction may increase the error in calculating the rate constant.
Discussion of R2 for the Fitted Line
If the coefficient of determination R2 for a graph of ln[A] versus time is close to 1, the measurement points are considered to fit the fitted line well. This supports the possibility that the reaction can be analyzed as a first-order reaction. However, a high R2 alone does not prove that the reaction is definitely first order.
If the measurement range is narrow or the number of data points is small, the data may appear linear by chance. Therefore, in addition to R2, the theoretical equation, reaction mechanism, residuals, and comparison with plots for other reaction orders should also be considered.
Example Discussion:
In the graph of ln[A] versus time, the coefficient of determination R2 was close to 1. This indicates that the measured values followed the integrated rate equation for a first-order reaction well. However, reaction order cannot be determined from a high R2 alone, so it is necessary to make a judgment together with plots for other reaction orders and the reaction mechanism.
Discussion When the Reaction Is Not First Order
If the graph of ln[A] versus time is not linear, the reaction may not be first order. Alternatively, possible causes include nonconstant measurement conditions, a multistep reaction, side reactions, approaching equilibrium, or a measurement range that is too broad.
If the reaction may not be first order, comparing plots of [A] versus time and 1/[A] versus time, which correspond to other reaction orders, makes the discussion easier.
Example Discussion:
Because plotting ln[A] against time did not produce a linear relationship, this reaction may not be treatable as a simple first-order reaction. Possible reasons include the reaction proceeding through multiple steps, the presence of side reactions, and changes in the reaction conditions during measurement. In addition, to determine the reaction order, comparison with plots based on other rate laws is also necessary.
Discussion of a Pseudo-First-Order Reaction
Even when multiple reactants are involved in a reaction, if one reactant is present in large excess, its concentration change may be negligible. In this case, the reaction can appear to behave as a first-order reaction and is called a pseudo-first-order reaction. Pseudo-first-order conditions may be used experimentally to simplify analysis.
In a pseudo-first-order reaction, the obtained rate constant may not be the true rate constant but an apparent rate constant that includes the concentration of the excess component. It is important to distinguish between these in a report.
Example Discussion:
In this experiment, one reactant was used in large excess, so its concentration change could be regarded as approximately constant during the reaction. As a result, the rate law took an apparent first-order form, and the rate constant could be determined from an ln plot. However, this rate constant is an apparent rate constant containing the concentration of the excess component and must be distinguished from the true rate constant.
Discussion of the Unit of the Rate Constant
The unit of the rate constant differs depending on the reaction order. For a first-order reaction, the unit of the rate constant k is the reciprocal of time. For example, if time is expressed in seconds, the unit is s−1, and if time is expressed in minutes, the unit is min−1.
An incorrect unit also leads to an incorrect half-life calculation. For example, if k is determined in min−1, the half-life is obtained in minutes. Conversion is necessary to express it in seconds.
Example Discussion:
The first-order rate constant k has units of reciprocal time. Because time was treated in minutes in this experiment, the unit of k obtained from the slope of the fitted line is min−1. The same time unit must also be used when calculating the half-life, and an error in unit conversion causes a large deviation in the result.
Reasons the Rate Constant May Differ From a Literature Value
Possible reasons an experimentally determined rate constant differs from a literature value include temperature differences, concentration errors, deviation in the reaction start time, instrument error, insufficient blank correction, and differences in reaction conditions. Because the rate constant depends on reaction conditions, check whether the conditions match when comparing with a literature value.
Example Discussion:
One possible reason the obtained rate constant k differed from the literature value is that the measurement temperature did not completely match the literature conditions. Reaction rates are strongly temperature-dependent, and even a small temperature difference may change the value of k. In addition, deviation in the reaction start time and insufficient blank correction in absorbance measurements may also affect the slope of the fitted line and lead to error in the rate constant.
When the Half-Life Does Not Agree With the Measured Value
The half-life calculated from the rate constant may not agree with the half-life read from the graph or measured values. Possible causes include the reaction not being perfectly first order, an incorrect initial concentration, too few measurement points, errors in concentration conversion, and increased measurement error in the later stage of the reaction.
Example Discussion:
A difference occurred between the half-life calculated from the rate constant and the half-life read from the measured values. One possible reason is that the measurement points were widely spaced and the exact time at which the concentration became half could not be read accurately. In addition, deviation from a simple first-order reaction or error in the initial concentration may also cause a difference in half-life.
Error in Concentration Conversion
When concentration is determined from absorbance, titration volume, or another measured quantity, errors in the conversion equation or calibration curve affect the rate analysis. If an error occurs during conversion to concentration, the ln[A] value also shifts and the slope of the fitted line changes. As a result, errors also occur in the rate constant and half-life.
Example Discussion:
One possible cause of error in the rate constant is error in converting absorbance to concentration. If the slope or intercept of the calibration curve contains error, the determined concentration deviates from the actual concentration. Because this concentration is logarithmically transformed and used in rate analysis, concentration-conversion error affects the slope of the fitted line, that is, the rate constant.
Error Caused by Insufficient Mixing
In reaction rate experiments, the reaction begins at the moment the reaction solutions are mixed. If mixing is insufficient, the concentration may not be uniform during measurement and the measured value may not reflect the actual average concentration. The effect of insufficient mixing is particularly large in the early stage of the reaction.
Example Discussion:
Insufficient mixing of the reaction solution may explain why the initial measured values deviated from the fitted line. When mixing is insufficient, the concentration differs depending on the measurement location and the measured value does not correctly represent the concentration of the entire reaction solution. Therefore, large scatter may have occurred in the data immediately after the reaction started.
How to Handle Outliers
In reaction rate experiments, some measurement points may deviate greatly from the fitted line. Possible causes of outliers include errors in recording the measurement time, absorbance-reading errors, bubbles in the sample, contamination of the cell, temperature changes, and insufficient mixing.
If an outlier is excluded, a clear experimental reason rather than simple inconvenience must be given. A more careful discussion can examine how the rate constant changes when the outlier is included and when it is excluded.
Example Discussion:
Some measurement points deviated greatly from the fitted line. Possible causes include errors in recording the measurement time, bubbles inside the cell, and absorbance-reading errors. Because outliers affect the slope of the fitted line and change the rate constant, an experimental basis must be clearly stated when excluding them.
When the Results Can Be Considered Good
Reaction rate experiment results can be considered good when the graph of ln[A] versus time is close to linear, the coefficient of determination is high, and the rate constant and half-life do not greatly contradict theoretical or literature values. It is also important that there be few outliers and that temperature and measurement conditions be kept constant.
Example Discussion:
When ln[A] was plotted against time, the measurement points agreed well with the fitted line and the coefficient of determination was also high. The rate constant determined from the slope of the fitted line did not greatly contradict the literature value. From these results, the reaction in this experiment is considered to be analyzable as a first-order reaction within the measured range, and the rate constant and half-life are considered reasonable.
Example Discussion When the Experiment Did Not Go Well
If a reaction rate experiment does not go well, possible causes are considered from results such as an ln plot that is not linear, deviations in the initial or later points, a low R2, a rate constant that differs greatly from the literature value, or a half-life that does not agree. Organizing the causes separately into time measurement, temperature, mixing, concentration conversion, blank correction, and measurement range makes the discussion easier to write.
Example Discussion:
In this experiment, sufficient linearity was not observed in the graph of ln[A] versus time. Possible causes include deviation in the reaction start time, insufficient mixing of the reaction solution, nonconstant temperature, and errors in concentration conversion. In addition, in the later stage of the reaction the measured values became small and the relative error increased, which may have caused deviation from the fitted line. Therefore, to improve the reliability of the rate constant, it is necessary to keep the measurement conditions constant and measure accurately from the initial to the later stages.
How to Write Points for Improvement
In a discussion of a reaction rate experiment, including points for improvement as well as sources of error makes the report easier to organize. Improvements are easier to organize when divided into time measurement, temperature control, sample preparation, measurement operations, and data analysis.
Improvements to Time Measurement
- Clearly define the reaction start time
- Start time measurement simultaneously with mixing
- Keep the measurement interval constant
- If the reaction is fast, collect initial data at shorter intervals
- Record the measurement times accurately
Improvements to Temperature and Sample Conditions
- Keep the temperature constant using a thermostatic bath or similar equipment
- Bring the reaction solutions to the same temperature before measurement
- Mix the sample sufficiently
- Prepare concentrations accurately
- Avoid contamination and bubbles in the reaction solution
Improvements to Measurement and Analysis
- Perform blank correction appropriately
- Ensure that absorbance lies within the linear range
- Check the causes of outliers
- Check the linearity of the ln plot
- Compare with plots for other reaction orders
- Use consistent units for the rate constant and half-life
Example of How to Write Points for Improvement:
To improve the accuracy of the rate constant, the reaction start time must be clearly defined and the measurement interval kept constant. In addition, because the reaction rate depends on temperature, it is important to keep the temperature constant during measurement. When absorbance measurements are used, accurate blank correction and avoidance of bubbles in the reaction solution and contamination of the cell can reduce scatter in the ln plot.
Difference Between a Superficial Discussion and a Good Discussion
In a discussion of a reaction rate experiment, simply writing that “it was a first-order reaction” or “the rate constant was determined” results in a superficial discussion. Relating the rate law, ln plot, slope, rate constant, half-life, and sources of error produces a more persuasive discussion.
| Superficial Discussion | Good Discussion |
|---|---|
| It was a first-order reaction. | When ln[A] was plotted against time, an approximately linear relationship was obtained. Therefore, within the measured range, the reaction is considered to follow the integrated rate equation for a first-order reaction, ln[A] = ln[A]0 − kt. |
| The rate constant was determined. | In an ln plot for a first-order reaction, the slope of the fitted line corresponds to −k, so the rate constant k was determined from the absolute value of the slope. Because time was treated in minutes, the unit of k is min−1. |
| There was an error. | Possible reasons the rate constant differed from the literature value include inadequate temperature control, deviation in the reaction start time, insufficient blank correction in absorbance measurements, and errors in concentration conversion. |
Examples of Expressions That Can Be Used in Reports
The following expressions can be used when writing the results and discussion of reaction rate experiments. Adjust the necessary parts according to your own experimental results.
- In a first-order reaction, the reaction rate is proportional to the reactant concentration.
- The integrated rate equation for a first-order reaction is expressed as ln[A] = ln[A]0 − kt.
- When ln[A] was plotted against time, an approximately linear relationship was obtained.
- Because the slope of the fitted line corresponds to −k, the rate constant was determined from the absolute value of the slope.
- The half-life of a first-order reaction is determined using t1/2 = 0.693/k.
- The larger the rate constant, the faster the reaction proceeds and the shorter the half-life becomes.
- Deviation in the reaction start time affects the rate constant as an error in the initial data.
- Because the reaction rate depends on temperature, inadequate temperature control leads to deviation in k.
- In the later stage of the reaction, where absorbance is small, the effects of blank correction and reading errors become larger.
- If the ln plot is not linear, the reaction may not be a simple first-order reaction.
Points to Check When Discussing a Reaction Rate Experiment
Checking the following points before writing the report makes the discussion easier to write.
- Have you written the rate law for a first-order reaction?
- Have you explained the relationship between the integrated rate equation and the graph?
- Have you created a graph of ln[A] versus time?
- Have you determined k from the slope of the fitted line?
- Have you written the unit of k correctly?
- Have you determined the half-life using t1/2 = 0.693/k?
- Have you checked R2 and linearity?
- Have you discussed deviations in the initial and later data?
- Have you considered the effect of temperature control?
- Have you considered errors in the reaction start time and measurement interval?
- Have you considered errors in absorbance and concentration conversion?
- Do the points for improvement correspond to the sources of error?
Summary
In reaction rate experiments, changes in reactant concentration, absorbance, and similar quantities over time are measured to determine the rate law, rate constant, and half-life. In a first-order reaction, the integrated rate equation ln[A] = ln[A]0 − kt applies, so plotting ln[A] against time gives a linear relationship. The slope of this fitted line corresponds to −k, and the rate constant can be determined from the absolute value of the slope.
The half-life of a first-order reaction is determined using t1/2 = 0.693/k, and the larger the rate constant, the shorter the half-life becomes. Because the half-life is independent of the initial concentration, it is an important value for explaining the characteristics of a first-order reaction.
In a report, do not simply write that “it was a first-order reaction.” Explain the linearity of the ln plot, the slope of the fitted line, the rate constant, the half-life, and the sources of error in relation to one another. A persuasive reaction rate experiment report can be produced by discussing temperature control, the reaction start time, insufficient mixing, blank correction, concentration conversion, and measurement errors in the later stage of the reaction.
