Reaction Rate Graphs | ln Concentration Plots and Discussion of the Rate Constant

What Is a Reaction Rate Graph?

A reaction rate graph is a graph in which concentration, absorbance, titration volume, electrical conductivity, or another quantity that changes as a reaction proceeds is plotted against time.
In experiments that follow the decrease of a reactant, concentration or absorbance may decrease with time.
In experiments that follow a product, the measured value may increase with time.

The purpose of creating a reaction rate graph is to determine what rate law the reaction follows and to obtain the rate constant.
For a first-order reaction, a graph of concentration itself is curved, but a plot of ln concentration against time becomes linear.
This linearity is used to discuss the reaction order and rate constant.

Main Items to Include in the Results

In a report using reaction rate graphs, it is important to show not only the table of measured values but also how each value was converted and plotted.
In particular, concentration, ln concentration, time, the equation of the fitted line, and the unit of the rate constant should be clearly organized.

Main Items to Include in the Results

• Reaction temperature
• Measurement time
• Concentration or absorbance at each time
• Values converted to concentration
• ln concentration or ln absorbance
• Time-concentration graph
• Time-ln concentration graph
• Equation of the fitted line
• Slope of the fitted line
• Intercept
• Coefficient of determination R²
• Rate constant k
• Presence or absence of outliers
• Sources of error in the rate constant

Reference Experimental Values for Reaction Rate Graphs and Examples of Rate Constant Analysis

Here, reference experimental values are organized for measuring changes in reactant concentration over time and discussing reaction order and the rate constant from concentration-time, ln concentration-time, and reciprocal concentration-time graphs.
Methods for distinguishing zero-order, first-order, and second-order reactions, the rate constant k, half-life, and measurement errors are summarized in a form that is easy to use in reports.

In reaction rate experiments, the decrease in reactant concentration with time is examined.
If the reaction is first order, plotting ln[A] against time gives a straight line, and the rate constant k can be determined from its slope.
By comparing which plot is closest to linear, the reaction order can be estimated.

Reference Experimental Conditions

Relationship Between Rate Laws and Their Integrated Forms

When determining the reaction order, multiple graphs are prepared and the plot with the best linearity is identified.

Example Measurement of Concentration Change Over Time

The following is reference data for a reaction in which reactant A decreases with time.

The concentration [A] decreases with time.
However, it is necessary to compare whether the concentration itself decreases linearly, whether ln[A] is linear, or whether 1/[A] is linear.

Example of Analysis as a First-Order Reaction

For a first-order reaction, the integrated rate equation is expressed as follows.

ln[A] = −kt + ln[A]₀

Therefore, when ln[A] is plotted on the vertical axis and time t on the horizontal axis, a straight line with slope −k is obtained.

If the relationship between ln[A] and time is close to linear, the reaction is highly likely to be treated as a first-order reaction.

Example Calculation of the Rate Constant k

Suppose the fitted line for ln[A] vs t is obtained as follows.

ln[A] = −0.0500t − 2.303

Comparing this with the first-order equation ln[A] = −kt + ln[A]₀, the slope is −k.

−k = −0.0500

k = 0.0500 min⁻¹

Therefore, the rate constant of this reaction is determined to be 0.0500 min⁻¹.

Example of Determining the Rate Constant From Two Points

The first-order rate constant is estimated using the data at 0 min and 20 min.

ln[A] = −kt + ln[A]₀

Rearranging,

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

This is almost identical to 0.0500 min⁻¹, which was obtained from the fitted line.
However, using only two points makes the result more susceptible to measurement error, so linear fitting using multiple points is preferable.

Example Calculation of Half-Life

The half-life of a first-order reaction is determined using the following equation.

t_1/2 = ln2 / k

When k = 0.0500 min⁻¹,

t_1/2 = 0.693 / 0.0500 = 13.9 min

In this reaction, it is considered to take approximately 14 minutes for the reactant concentration to decrease by half.

Comparison of Linearity of Zero-Order, First-Order, and Second-Order Plots

The following example compares the fitted linearity of [A] vs t, ln[A] vs t, and 1/[A] vs t for the same data.

In this example, the R² value for ln[A] vs t is the highest, so analysis as a first-order reaction is considered appropriate.

Reference Data for a Zero-Order Reaction

In a zero-order reaction, the reaction rate does not depend on concentration and [A] decreases linearly with time.

A zero-order reaction is characterized by the same decrease in concentration over equal time intervals.

Reference Data for a Second-Order Reaction

In a second-order reaction, 1/[A] increases linearly with time.

In a second-order reaction, the reaction becomes slower as the concentration decreases, and the half-life depends on the initial concentration.

Example of Determining Concentration From Absorbance

When concentration cannot be measured directly, it may be determined from absorbance.
According to the Beer-Lambert law, absorbance A is proportional to concentration c.

A = εcl

If the calibration curve is expressed as A = 8.00 × c, the concentration corresponding to an absorbance of 0.488 is

c = 0.488 ÷ 8.00 = 0.061 mol/L

In this way, absorbance at each time is converted to concentration and used for reaction-rate analysis.

Change in the Rate Constant With Temperature

As temperature increases, reaction rates generally increase and the rate constant k also becomes larger.

As temperature increases, the proportion of molecules with energies exceeding the energy required for reaction increases, so the reaction rate becomes larger.

Effect of Error at Low Concentrations

As the reaction proceeds and the concentration becomes low, absorbance and titration values become smaller, and relative error becomes larger.

Including data from the low-concentration region may shift the slope of the fitted line.
When using data from the later stage of the reaction, the reliability of the measured values must be checked.

Example Comparison With the Initial-Rate Method

In addition to analyzing concentration changes over time, the initial-rate method can also be used to determine the rate law.

In this example, doubling the concentration also doubles the initial rate, so the reaction is considered first order with respect to A.

Main Sources of Measurement Error

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.
Three plots, [A] vs t, ln[A] vs t, and 1/[A] vs t, were prepared for these data.
As a result, the ln[A] vs t plot showed the highest linearity, with a coefficient of determination of 0.999.
Therefore, this reaction is considered to be treatable as a first-order reaction with respect to A.

The integrated rate equation for a first-order reaction is ln[A] = −kt + ln[A]₀.
Because the fitted line for ln[A] vs t was ln[A] = −0.0500t − 2.303, the rate constant k was determined as 0.0500 min⁻¹ from the slope −0.0500.
In addition, the half-life was 13.9 min from t_1/2 = 0.693/k.

In the later stage of the reaction, the concentration becomes small and the influence of measurement error becomes large.
Particularly in a region of low absorbance, even a slight reading error may cause a large deviation in ln[A].
Therefore, when determining the rate constant, it is necessary to check whether low-concentration data disturb the linearity.

Points for Connecting the Results to the Discussion

In a discussion of reaction rate graphs, it is important not merely to state that the concentration decreased, but to explain which plot becomes linear, what the slope means, and how the rate constant and half-life are determined.

• Can you compare [A] vs t, ln[A] vs t, and 1/[A] vs t and determine the reaction order?
• Can you explain that for a first-order reaction, ln[A] vs t becomes linear and the slope is −k?
• Have you determined the rate constant from a fitted line using multiple points rather than only from a two-point calculation?
• Have you confirmed that the unit of the rate constant differs depending on the reaction order?
• Can you calculate the half-life from the rate constant and relate it to how the reaction proceeds?
• If concentration is determined from absorbance, do you understand the calibration curve and Beer-Lambert law?
• Can you discuss why increasing temperature increases the rate constant in relation to molecular collisions and activation energy?
• Can you explain that the relative error becomes larger for low-concentration data in the later stage of the reaction?
• Can you discuss the reaction start time, insufficient mixing, temperature changes, and absorbance-measurement errors as sources of error?

Example Discussion

In this experiment, the change in concentration of reactant A was measured over time, and the reaction order and rate constant were determined.
[A] decreased with time, but the [A] vs t plot was not perfectly linear.
In contrast, when ln[A] was plotted against time, the relationship was almost linear and the coefficient of determination was also the highest.
From this result, the reaction is considered to be first order with respect to A.

For a first-order reaction, the integrated rate equation ln[A] = −kt + ln[A]₀ applies.
Because the slope of the fitted line was −0.0500, the rate constant was determined as k = 0.0500 min⁻¹.
Using this value, the half-life was 0.693/k = 13.9 min.
Because the half-life of a first-order reaction is independent of the initial concentration, the time required for the concentration to decrease by half remains approximately constant at each stage.

When the temperature was increased, the rate constant became larger and the half-life became shorter.
This is because increasing the temperature increases the kinetic energy of the molecules and increases the proportion of molecules that can react by overcoming the activation energy.
Therefore, it is important to keep the temperature constant when comparing reaction rates.

Possible sources of error include deviation in the reaction start time, insufficient mixing, temperature changes, and errors in converting absorbance to concentration.
The period immediately after the reaction starts is particularly susceptible to the effects of mixing operations, while in the later stage of the reaction, concentration and absorbance become small and the relative error increases.
These errors may reduce the linearity of the plot and cause deviations in the estimated rate constant.

Summary

In reaction rate analysis, the reaction order can be determined not only by directly examining concentration changes over time but also by converting the data to ln[A] or 1/[A] and plotting them.
For a first-order reaction, ln[A] vs t becomes linear, and the rate constant k can be determined from the absolute value of the slope.

This reference example covered concentration-time data, ln concentration plots, the rate constant, half-life, comparison of zero-order, first-order, and second-order reactions, concentration conversion from absorbance, temperature dependence, errors at low concentrations, and the initial-rate method.
In a report, it is useful to determine the reaction order based on which graph is closest to linear and discuss the slope, rate constant, and half-life in relation to one another.

Why Create a Time-Concentration Graph First?

In reaction rate analysis, first graphing the relationship between time and concentration makes it easier to understand the overall trend of the reaction.
It is possible to check whether the reactant concentration decreases with time, where rapid changes occur, and whether any measured values appear unnatural.

However, for a first-order reaction, a time-concentration graph is generally not linear.
Because the change is larger at the beginning when the reactant concentration is high and becomes smaller in the later stage as the concentration decreases, the graph shows a curved decrease.
Therefore, an ln concentration plot is prepared to determine the rate constant.

What Is an ln Concentration Plot?

An ln concentration plot is a graph in which the natural logarithm ln[A] of the reactant concentration [A] is plotted on the vertical axis and time t on the horizontal axis.
For a first-order reaction, a linear relationship exists between ln[A] and time t.
Therefore, this plot is used to check whether the reaction is first order and to determine the rate constant.

ln[A] = ln[A]₀ − kt

This equation corresponds to the linear equation y = ax + b.
Here, y is ln[A], x is t, the slope a is −k, and the intercept b is ln[A]₀.
Therefore, the rate constant can be determined from the fitted line of the ln concentration plot.

How to Create an ln Concentration Plot

To create an ln concentration plot, first determine the reactant concentration at each time.
Next, calculate the natural logarithm of each concentration and plot time on the horizontal axis and ln concentration on the vertical axis.
Finally, add a fitted straight line to the measurement points and display the fitted equation and R².

Basic Procedure

1. Prepare a table of time t and concentration [A].
2. Calculate ln[A] for each concentration.
3. Set time t on the horizontal axis and ln[A] on the vertical axis.
4. Create a scatter plot.
5. Add a fitted straight line.
6. Display the fitted equation and R².
7. Determine the rate constant k from the slope.

Determining the Rate Constant From the Slope of the Fitted Line

The slope of the fitted line obtained from an ln concentration plot is directly related to the rate constant of a first-order reaction.
Because the first-order equation is ln[A] = ln[A]₀ − kt, the slope of the fitted line is −k.
Therefore, the rate constant k is obtained as the absolute value of the slope.

Slope of fitted line = −k

k = − slope

For example, if the slope of the fitted line is −0.025 min⁻¹, the rate constant k is 0.025 min⁻¹.
The slope is negative because as the reaction proceeds, the concentration decreases and ln concentration also becomes smaller.

How to Write the Unit of the Rate Constant

The unit of the first-order rate constant k is the reciprocal of time.
If time is plotted in seconds, the unit of k is s⁻¹.
If time is plotted in minutes, the unit of k is min⁻¹.

A common mistake in reports is to plot time in minutes but write the rate constant in s⁻¹.
The unit of the rate constant must correspond to the unit of the horizontal axis of the graph.

How to Interpret the Intercept

The intercept of an ln concentration plot theoretically corresponds to ln[A]₀.
In other words, it is the natural logarithm of the reactant concentration at t = 0.
If the experimentally determined intercept is close to ln[A]₀ calculated from the actual initial concentration, the setting of the initial concentration and reaction start time can be considered relatively appropriate.

On the other hand, if the intercept deviates greatly, possible causes include errors in preparing the initial concentration, delay in the measurement start time, insufficient mixing immediately after the reaction starts, and deviation in blank correction.

How to Interpret R2

R² is a value indicating how well the measurement points fit the fitted straight line.
The closer R² is to 1, the more linear the relationship between ln concentration and time is considered to be.
A high R² in an ln plot for a first-order reaction supports the possibility that the reaction can be analyzed as first order.

However, a high R² alone does not necessarily prove that the reaction is first order.
If the number of data points is small or the measurement range is narrow, the data may appear linear by chance.
A more careful discussion also considers the reaction mechanism and comparison with plots for other reaction orders.

Discussion When the ln Concentration Plot Is Linear

When the ln concentration plot is linear, the reactant concentration is considered to decrease according to the integrated rate equation for a first-order reaction.
In this case, the concentration decreases exponentially with time, while ln concentration decreases at a constant rate with time.

However, even when the linearity is good, it is assumed that the conversion of reactant or product concentration is correct, that the measured value is proportional to the reactant concentration, and that the temperature remains constant.

Discussion When the ln Concentration Plot Is Not Linear

If the ln concentration plot is not linear, the reaction may not be a simple first-order reaction.
Other possible causes include nonconstant experimental conditions, errors in concentration conversion, temperature changes, deviation in the measurement start time, a multistep reaction, or side reactions.

In this case, rather than forcing the data to be treated as a first-order reaction, examine the range in which linearity exists, whether deviations are larger in the initial or later stage, and what happens in plots corresponding to other reaction orders.

Difference Between a Time-Concentration Graph and an ln Concentration Graph

A time-concentration graph is used to directly observe how the reactant concentration changes with time.
In contrast, an ln concentration graph is a logarithmically transformed graph used to analyze the reaction as first order.
Because the purposes differ, distinguishing the meaning of both graphs in a report makes the discussion easier to understand.

Using Absorbance in an ln Plot

In reaction rate experiments using absorbance, an ln plot may be prepared using absorbance that is proportional to the reactant concentration.
If absorbance A is proportional to the reactant concentration, absorbance may sometimes be used for analysis instead of concentration [C].
However, correction for blank absorbance or final absorbance may be necessary.

If the absorbance of the reactant decreases, ln(absorbance) or ln(corrected absorbance) is plotted against time.
If the absorbance of the product increases, it may be necessary to convert the value to one corresponding to the amount of reactant remaining before taking the ln.

Causes of Initial Data Deviating From the Straight Line

Initial data are susceptible to effects such as deviation in the reaction start time, insufficient mixing, the time required to set the sample in the measuring instrument, and nonuniform temperature.
Particularly for fast reactions, even a slight delay before measurement begins may make the initial concentration or initial absorbance appear smaller than the actual value.

If the initial data deviate, discuss how operations immediately after the reaction began may have affected the rate constant.
The slope of the fitted line may also change depending on whether the initial point is included.

Causes of Later Data Deviating From the Straight Line

In the later stage of the reaction, the reactant concentration and absorbance become small, so even slight measurement errors have a larger effect.
Taking the ln can make errors in small values appear large.
In addition, if the reaction approaches equilibrium or side reactions become significant, the data may deviate from the straight line expected for a simple first-order reaction.

How to Handle Outliers

In reaction rate graphs, some measurement points may deviate greatly from the fitted line.
Outliers change the slope of the fitted line and affect the rate constant k.
Possible causes of outliers include errors in recording time, errors in absorbance readings, bubbles in the sample, contamination of the cell, temperature changes, and errors in concentration preparation.

A clear reason is required if an outlier is excluded.
A careful report can explain how the fitted line and rate constant change when the outlier is included and when it is excluded.

How to Select the Fitting Range

In reaction rate graphs, all measurement points may be used for fitting, or only the range with good linearity may be used for analysis.
For example, the initial stage may be affected by insufficient mixing, while the later stage may be more strongly affected by measurement error.
Therefore, linearity may be checked using the stable middle range.

However, points must not be selected simply because they are convenient.
If a fitting range is selected, the reason for using that range must be explained.

Discussion When the Rate Constant Is Large or Small

The larger the rate constant k, the faster the reaction proceeds.
In an ln concentration plot, a larger absolute value of the slope means that ln concentration decreases more rapidly with time.
Conversely, when k is small, the reaction is slow and the slope of the graph is more gradual.

When discussing the magnitude of k, relate it to conditions such as temperature, catalyst, reactant concentration, solvent, pH, and reaction mechanism.
When comparing with literature values, confirm whether the measurement conditions are the same.

Effect of Temperature on the Graph

Reaction rates are strongly affected by temperature.
If the temperature changes during measurement, the rate constant k also changes, so the ln concentration plot may not form a clean straight line.
In general, the higher the temperature, the faster the reaction and the larger k becomes.

When comparing rate constants, values measured at the same temperature must be compared.
If the experimental value differs from a literature value, temperature difference is an important source of error.

Effect of Concentration Conversion on the Graph

When concentration is determined from absorbance, titration volume, or another measured quantity, errors in the concentration conversion affect the ln concentration plot.
If the slope or intercept of the calibration curve is inaccurate, the concentration at each time is also shifted.
Because ln is then taken for these concentrations, errors in concentration conversion lead to errors in the rate constant.

Particularly in the low-concentration region, a slight conversion error may greatly affect the ln value.
If later data deviate from the line, uncertainty in the concentration conversion should also be considered.

Reading the Half-Life From the Graph

For a first-order reaction, the half-life can be calculated from the rate constant k.
However, the half-life can also be estimated from the graph by reading the time at which the concentration becomes half of its initial value.
If the measurement points are widely spaced or the initial concentration contains error, the half-life read from the graph also contains error.

t_1/2 = 0.693 / k

Common Mistakes When Creating Reaction Rate Graphs

In reaction rate graphs, mistakes are likely to occur in axis selection, units, logarithmic transformation, and treatment of the fitted line.
Before submitting a report, check the horizontal and vertical axes, units, ln calculations, and the meaning of the fitted equation.

When a Reaction Rate Graph Can Be Considered Good

A reaction rate graph can be considered good when the measurement points are appropriately distributed, the ln concentration plot is close to linear, there are few outliers, and the rate constant can be reasonably determined from the slope of the fitted line.
It is also important that the axis labels, units, fitted equation, and R² be clearly stated.

Example Discussion When the Experiment Did Not Go Well

If a reaction rate graph does not work well, possible causes can be considered from results such as measurement points deviating from the line, a low R², an unnatural slope, an intercept that does not match the initial concentration, or large deviations in the later stage.
Organizing the causes separately into reaction start time, temperature, mixing, concentration conversion, blank correction, and measurement range makes the discussion easier to write.

How to Write Points for Improvement

In a discussion of reaction rate graphs, including points for improvement as well as sources of error makes the report easier to organize.
Improvements are easier to write when divided into measurement operations, temperature control, graph creation, and data analysis.

Improvements to Measurement Operations

• Clearly define the reaction start time
• Mix the reaction solution quickly and uniformly
• Keep the measurement interval constant
• Collect initial data at short intervals
• Avoid contamination and bubbles in the cell

Improvements to Temperature and Concentration Conditions

• Keep the temperature constant using a thermostatic bath or similar equipment
• Equalize the sample temperatures before measurement
• Prepare concentrations accurately
• Measure at concentrations within the linear absorbance range
• Perform blank correction appropriately

Improvements to Graphing and Analysis

• Distinguish between the time-concentration graph and ln concentration graph
• Clearly state axis labels and units
• Display the equation of the fitted line and R²
• Check the sign of the slope when determining k
• Check the cause of outliers
• Compare with plots for other reaction orders when necessary

Difference Between a Superficial Discussion and a Good Discussion

In a discussion of reaction rate graphs, simply writing that “it became linear” or “k was determined” results in a superficial discussion.
Explaining why ln concentration is used, the relationship between the slope and k, the meaning of the intercept, and the causes of deviation from the line produces a more persuasive discussion.

Examples of Expressions That Can Be Used in Reports

The following expressions can be used when writing the results and discussion of reaction rate graphs.
Adjust the necessary parts according to your own experimental results.

• The time-concentration graph showed that the reactant concentration decreased with time.
• For a first-order reaction, a linear relationship exists between ln[A] and time t.
• 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 intercept theoretically corresponds to ln[A]₀.
• Because R² was close to 1, the measurement points fit the fitted line well.
• The deviation of the initial data may have been caused by deviation in the reaction start time or insufficient mixing.
• The deviation of the later data is considered to have occurred because the relative error became larger in the low-concentration region.
• If the temperature is not constant, the rate constant may change during measurement and reduce linearity.
• The unit of the rate constant is the reciprocal of the time unit used on the horizontal axis.

Points to Check When Discussing Reaction Rate Graphs

Checking the following points before writing the report makes the discussion easier to write.

• Have you checked the overall trend using the time-concentration graph?
• Have you prepared an ln concentration plot?
• Have you clearly stated the units of the horizontal and vertical axes?
• Have you written the equation of the fitted line?
• Have you explained that the slope corresponds to −k?
• Does the unit of the rate constant match the time unit on the horizontal axis?
• Have you related the intercept to the initial concentration?
• Have you checked linearity without relying excessively on R²?
• Have you considered the causes of deviations in the initial points, later points, and outliers?
• Have you considered temperature changes and insufficient mixing as sources of error?
• Have you considered the effects of concentration conversion and blank correction?
• Do the points for improvement correspond to the sources of error?

Summary

In reaction rate graphs, first check the change in reactant concentration using a time-concentration graph, and then use an ln concentration plot to determine whether the reaction can be analyzed as first order.
For a first-order reaction, the relationship ln[A] = ln[A]₀ − kt applies, so plotting ln[A] against time produces a straight line.

The slope of the fitted line in the ln concentration plot corresponds to −k.
Therefore, the rate constant k can be determined by reversing the sign of the slope.
Because the unit of the rate constant is the reciprocal of the time unit on the horizontal axis, it is necessary to confirm whether the graph was prepared using seconds, minutes, or hours.

In a report, discuss not only whether the graph became linear, but also deviations in the initial and later data, R², the intercept, outliers, temperature changes, reaction start time, insufficient mixing, and errors in concentration conversion.
A persuasive reaction rate experiment report can be produced by explaining the relationship between how the graph is constructed and the meaning of the rate constant.