In physical chemistry experiments, measured values are often organized into graphs, and physical quantities are determined from fitted straight lines or fitted curves.
For example, in experiments involving reaction rates, absorbance, vapor pressure, viscosity, electrical conductivity, freezing-point depression, partition equilibrium, thermodynamic quantities, and other topics, the desired value is determined not simply by reading the measured values directly, but from the slope or intercept of a graph.
In the discussion section of a physical chemistry laboratory report, it is not sufficient simply to write that “the graph became linear,” “a fitted straight line was drawn,” or “there was an error.”
It is important to explain why the graph becomes linear, what the slope and intercept mean, how to interpret R2 and correlation, how to handle outliers, what causes errors, and how to explain differences from theoretical values.
This article clearly explains how to discuss graphs, fitted straight lines, and errors in physical chemistry laboratory reports, how to write the results, discussion examples, and points for improvement.
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 measurement methods, handling of equipment, calculation equations, graphing methods, units, and specified report format, always follow the instructions in your university’s laboratory manual and those given by your instructor or TA.
- Important Perspectives for Discussion in Physical Chemistry Laboratory Reports
- Main Items to Include in the Results
- Purpose of Creating a Graph
- How to Choose the Horizontal and Vertical Axes
- What Is a Fitted Straight Line?
- Meaning of the Slope of the Fitted Straight Line
- Meaning of the Intercept of the Fitted Straight Line
- How to Interpret the Coefficient of Determination R2
- When the Graph Is Not Linear
- Discussion of Outliers
- What Is Error?
- Random Errors and Systematic Errors
- Calculation and Discussion of Error Rate
- Comparison With Theoretical and Literature Values
- Error Caused by Temperature Control
- Error Caused by Concentration Preparation
- Reading Errors in Measuring Instruments
- Error Caused by Insufficient Instrument Calibration
- Discussion When the Points on a Graph Are Scattered
- Discussion When the Entire Graph Is Shifted
- Discussion of Logarithmic Graphs
- Discussion of Proportional Relationships
- Discussion When Using a Calibration Curve
- When a Physical Quantity Is Determined From a Fitted Straight Line
- Discussion of Units
- Discussion of Significant Figures
- Discussion of Reproducibility
- Discussion When Using Standard Deviation
- Example Discussion of a Graph in an Absorbance Experiment
- Example Discussion of a Graph in a Reaction Rate Experiment
- Example Discussion of a Graph in an Electrical Conductivity Experiment
- Example Discussion of a Graph in a Thermodynamics Experiment
- When the Graph 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 Physical Chemistry Experiment
- Summary
Important Perspectives for Discussion in Physical Chemistry Laboratory Reports
In physical chemistry experiments, laws are identified from measured values and physical quantities are determined based on theoretical equations.
Therefore, the discussion should explain not only “what the measured values were,” but also “how well the experimental results agreed with the theoretical equation” and “what caused any disagreement.”
Particularly important points are the shape of the graph, the slope of the fitted straight line, the intercept, the coefficient of determination, errors, and comparison with theoretical values.
Rather than simply listing numerical values, it is necessary to consider whether the measured values follow the theoretical equation, which measurement points deviate, and whether the deviations originate from experimental operations or the measuring equipment.
Example Discussion:
When the measured values were plotted according to the theoretical equation, an approximately linear relationship was obtained.
This suggests that the experimental results generally followed the theoretical equation.
On the other hand, some measurement points deviated from the fitted straight line, and possible causes of error include temperature control, reading of measured values, sample preparation, and instrument calibration.
Main Items to Include in the Results
In the results of a physical chemistry experiment, organize the table of measured values, calculated values, graphs, fitted straight lines, slopes, intercepts, coefficients of determination, and physical quantities obtained.
In a report, it is important to write so that it is clear what was determined from which data.
Main Items to Include in the Results
- Measurement conditions
- Table of measured values
- Units
- Equations used in calculations
- Horizontal and vertical axes of the graph
- Fitted straight line or fitted curve
- Fitted equation
- Slope
- Intercept
- Coefficient of determination R2
- Physical quantity determined from the graph
- Comparison with theoretical or literature values
- Error rate
- Presence or absence of outliers
Example of How to Write the Results:
When the measured values were plotted with concentration on the horizontal axis and absorbance on the vertical axis, an almost linear relationship was obtained.
The fitted straight line determined by the least-squares method was y = ax + b, and the coefficient of determination R2 was close to 0.99.
This suggests that there was a good proportional relationship between absorbance and concentration within the measurement range.
Purpose of Creating a Graph
The purpose of creating a graph in a physical chemistry experiment is to visually confirm the trend in the measured values and clarify the relationship with the theoretical equation.
When measured values are viewed only in a table, it may be difficult to notice the overall trend or outliers.
By graphing the data, proportional relationships, linear relationships, curved relationships, scatter, and outliers become easier to identify.
In addition, reaction rate constants, molar absorption coefficients, activation energies, partition coefficients, thermodynamic quantities, and similar values may be determined from the slope or intercept of a graph.
Therefore, a graph is not merely for appearance but is an important analytical tool for determining physical quantities.
Example Discussion:
Graphing the measured values made it possible to confirm the overall trend of the data.
Although the scatter of individual measurements is difficult to recognize from a table alone, graphing makes it easier to identify points that deviate from the fitted straight line.
In addition, because the desired physical quantity can be determined from the slope of the fitted straight line, graphing is important in the analysis of the results of this experiment.
How to Choose the Horizontal and Vertical Axes
In a graph, the choice of what to place on the horizontal and vertical axes is extremely important.
In physical chemistry experiments, the theoretical equation may be transformed so that it becomes linear before graphing.
For example, an exponential relationship appears as a curve when graphed directly, but may become linear after taking a logarithm.
If the axes are not chosen appropriately, the desired physical quantity cannot be correctly determined from the slope or intercept of the fitted straight line.
In a report, it is useful to explain why those axes were selected in relation to the theoretical equation.
Example Discussion:
In this experiment, the horizontal and vertical axes were selected so that the theoretical equation took a linear form.
As a result, the measured values were arranged approximately along a straight line, and the desired physical quantity could be determined from the slope of the fitted line.
By selecting the axes in accordance with the theoretical equation, it becomes easier to confirm whether the experimental results follow the theory.
What Is a Fitted Straight Line?
A fitted straight line is a line drawn through scattered measurement points so that it best represents the overall trend.
In physical chemistry experiments, the fitted straight line is often determined by the least-squares method.
It is used to evaluate how closely the measured values follow the theoretical equation and to determine physical quantities from the slope or intercept.
However, the fitted straight line is not the measured values themselves but a representation of the overall trend of the measured values.
Even if all points do not lie perfectly on the line, a linear relationship may still be judged to exist within the range of measurement error.
Example Discussion:
Although some scatter was observed among the measurement points, they were distributed generally along the fitted straight line.
This suggests that the linear relationship predicted by the theoretical equation generally holds within the measurement range.
Possible reasons the individual points did not lie perfectly on the line include errors in reading measured values and changes in temperature.
Meaning of the Slope of the Fitted Straight Line
The slope of the fitted straight line indicates how much the value on the vertical axis changes when the value on the horizontal axis changes.
In physical chemistry experiments, this slope may correspond to the desired physical quantity.
For example, in a calibration curve, the slope represents sensitivity, while in reaction-rate and thermodynamic experiments, values related to rate constants or energy may be determined from the slope.
The unit of the slope is obtained by dividing the unit of the vertical axis by the unit of the horizontal axis.
In a report, be sure to explain not only the numerical value of the slope but also what the slope represents.
Example Discussion:
The slope of the fitted straight line represents the change in the vertical-axis value relative to the change in the horizontal-axis value.
In this experiment, because this slope corresponds to the proportionality constant in the theoretical equation, the desired physical quantity was determined from the slope.
If the slope is close to the theoretical value, the measured values can be considered to generally follow the theoretical equation.
Meaning of the Intercept of the Fitted Straight Line
The intercept of the fitted straight line represents the value on the vertical axis when the horizontal-axis value is 0.
Even in experiments where the intercept is theoretically expected to be 0, the actual intercept may deviate from 0.
Possible causes include insufficient blank correction, zero-point offset of the instrument, errors in sample preparation, and scatter in measured values.
In some cases, the intercept has a physical meaning, while in others it indicates error or insufficient correction.
In a report, confirm the theoretical meaning of the intercept and discuss how the experimental value deviated.
Example Discussion:
Theoretically, the intercept was expected to be close to 0, but in this experiment it deviated slightly from 0.
Possible causes include incomplete zero-point correction of the measuring instrument and error in blank measurement.
However, if the deviation in the intercept is small, it is considered not to have greatly affected the overall trend of the measurements.
How to Interpret the Coefficient of Determination R2
The coefficient of determination R2 is an indicator of how well the measurement points fit the fitted straight line.
The closer R2 is to 1, the easier it is to judge that the measured values follow the fitted straight line well.
However, a high R2 does not necessarily mean that the result is theoretically correct.
R2 indicates the degree of fit to a straight line and does not guarantee that the experimental operation was correct or that the measured values contain no error.
In a report, use R2 as a reference while also considering the shape of the graph, outliers, and correspondence with the theoretical equation.
Example Discussion:
Because the coefficient of determination R2 for the fitted straight line was close to 1, the measured values are considered to follow a linear relationship well.
However, a high R2 does not mean that the measured values contain no error.
Therefore, it is necessary to consider not only the coefficient of determination but also the scatter of individual measurement points and differences from theoretical values.
When the Graph Is Not Linear
If a graph that should theoretically be linear is not linear, possible causes include measurement error, changes in experimental conditions, an inappropriate concentration range, temperature changes, limitations of the instrument, and side reactions.
Deviation from linearity may also occur when the range of applicability of the theoretical equation is exceeded.
If the graph curves, do not force the measured values to fit a straight line.
Instead, check the range over which linearity holds and which points deviate substantially.
Example Discussion:
The measured values deviated from a straight line overall and did not completely follow the theoretical equation.
Possible causes include the measurement range being too broad for the linear approximation of the theoretical equation to hold, the temperature conditions not being kept constant, and inclusion of values close to the measurement limit of the instrument.
Therefore, when evaluating linearity, the appropriateness of the measurement range and experimental conditions must also be checked.
Discussion of Outliers
An outlier is a value that deviates greatly from the trend of the other measurement points.
Outliers may be caused by reading mistakes, sample-preparation mistakes, bubbles, temperature changes, instrument instability, insufficient mixing, contamination, and similar factors.
If an outlier is present on a graph, it may greatly affect the slope and intercept of the fitted straight line.
However, outliers must not be excluded simply because they are inconvenient.
If an outlier is excluded, a clear reason must be explained.
In a report, discuss the cause of the outlier and its effect on the fitted straight line.
Example Discussion:
Some measurement points deviated greatly from the fitted straight line.
Possible causes include concentration errors during sample preparation, bubbles introduced during measurement, temperature changes, and errors in reading the instrument.
Because outliers affect the slope and intercept of the fitted straight line, they should not simply be excluded, but should be handled only after confirming the experimental cause.
What Is Error?
Error is the difference between a measured value and the true value or theoretical value.
Physical chemistry experiments contain errors in various measurements such as temperature, time, mass, volume, concentration, absorbance, voltage, current, and pressure.
Therefore, it is not unusual for experimental values not to agree perfectly with theoretical values.
In a report, error should not be treated simply as a “failure.”
Instead, discuss what caused the error and in which direction it affected the result.
Writing specific causes of error makes the discussion more persuasive.
Example Discussion:
The experimental value did not agree perfectly with the theoretical value.
This difference is considered to have resulted from the combined effects of reading errors in measuring instruments, insufficient temperature control, and concentration errors during sample preparation.
Because physical chemistry experiments involve calculations using multiple measured values, small errors at each measurement stage may have affected the final result.
Random Errors and Systematic Errors
Errors can be broadly divided into random errors and systematic errors.
Random errors are errors that cause the measured values to vary from one measurement to another.
Examples include variations in reading scales, timing differences, and small fluctuations in temperature.
Systematic errors are errors that cause measured values to shift consistently in one direction.
Examples include instrument calibration errors, zero-point offsets, errors in the concentration of standard solutions, and insufficient correction of thermometers.
When systematic error is present, even repeated measurements may produce an average value that remains shifted from the true value.
| Type of Error | Characteristic | Example |
|---|---|---|
| Random error | Measured values vary | Variation in readings, timing differences |
| Systematic error | Values shift in a constant direction | Zero-point offset, insufficient calibration, concentration-setting error |
Example Discussion:
Random error may explain why the measurement points were scattered around the fitted straight line.
On the other hand, if all measured values were shifted in a direction higher than the theoretical value, systematic errors such as a zero-point offset of the instrument or an error in the concentration of the standard solution may have contributed.
Therefore, when discussing errors, it is important to distinguish between scatter and a shift in a constant direction.
Calculation and Discussion of Error Rate
When comparing an experimental value with a theoretical or literature value, the error rate may be calculated.
The error rate expresses as a percentage how much the experimental value differs from the theoretical value.
The closer the experimental value is to the theoretical value, the smaller the error rate becomes.
Error rate (%) = |Experimental value − Theoretical value| ÷ Theoretical value × 100
When discussing the error rate, do not focus only on whether the numerical value is large or small, but explain why the difference occurred.
In addition, if the theoretical value itself applies only under specific conditions, differences in experimental conditions must also be considered.
Example Discussion:
When the experimental value was compared with the literature value, the error rate was ○○%.
Possible causes of this difference include the temperature not completely matching the literature conditions, reading errors in the measuring instruments, and errors in preparing the sample concentration.
Therefore, although the experimental value showed the same trend as the literature value, a certain deviation is considered to have occurred because of differences in experimental conditions and measurement errors.
Comparison With Theoretical and Literature Values
In physical chemistry experiments, the obtained value is compared with a theoretical or literature value.
If the experimental value is close to the theoretical value, the experimental and analytical methods may have been appropriate.
On the other hand, if the experimental value deviates greatly, the measurement conditions, equipment, sample preparation, calculation method, and range of applicability of the theoretical equation must be checked.
When comparing with a literature value, check whether conditions such as temperature, pressure, concentration, solvent, and sample purity are the same.
If the conditions differ, it is natural for the values to differ.
Example Discussion:
Because the value obtained experimentally was close to the literature value, the measurement and analysis are considered to have been generally appropriate.
However, the measurement temperature and sample conditions may not have completely matched those of the literature value.
Therefore, the difference between the experimental and literature values is considered to originate not only from measurement error but also from differences in experimental conditions.
Error Caused by Temperature Control
In physical chemistry experiments, temperature often has a large effect on the results.
Reaction rates, equilibrium constants, vapor pressure, viscosity, electrical conductivity, solubility, and similar properties depend on temperature.
Therefore, if the temperature is not kept constant, the measured values may deviate from theoretical values or the fitted straight line.
Example Discussion:
Insufficient temperature control may explain why the measured values deviated from the theoretical values.
Because the physical quantity examined in this experiment depends on temperature, changes in temperature during measurement also change the measured value.
As a result, scatter among the measurement points or deviation from the fitted straight line may have occurred.
Error Caused by Concentration Preparation
In experiments that use concentration, errors in preparing standard solutions and sample solutions affect the results.
If errors occur while using volumetric flasks, volumetric pipettes, burettes, electronic balances, and similar equipment, the actual concentration deviates from the set value.
In a graph using concentration on the horizontal axis, such errors affect the slope and intercept.
Example Discussion:
Errors in solution concentration during sample preparation may explain why some measurement points deviated from the fitted straight line.
If errors occur when weighing the sample or performing dilution operations, the set concentration and actual concentration do not agree.
As a result, the measurement points deviate from the straight line on the graph and may also affect the slope and intercept.
Reading Errors in Measuring Instruments
In physical chemistry experiments, errors may occur when reading scales or checking displayed values.
With analog scales, parallax may occur, while with digital displays, reading the value before it stabilizes may cause error.
In addition, when reading values that change rapidly, differences in reading timing also become a source of error.
Example Discussion:
Reading errors in the measuring instruments may explain the scatter in the measured values.
Small deviations may have occurred because of parallax when reading scales or because the displayed value was read before it had stabilized.
The inclusion of these errors in each measurement point is considered to have caused the scatter observed on the graph.
Error Caused by Insufficient Instrument Calibration
If a measuring instrument is not correctly calibrated, measured values may be shifted in a constant direction.
For example, calibration and zero-point adjustment are important for pH meters, spectrophotometers, conductivity meters, thermometers, pressure gauges, and similar instruments.
Insufficient calibration causes systematic error and may shift the entire set of measured values away from the theoretical values.
Example Discussion:
One possible reason the measured values were generally shifted higher than the theoretical values is insufficient calibration of the instrument.
If the zero point or sensitivity of the instrument is shifted, all measured values contain systematic error in the same direction.
Therefore, it is important to calibrate the instrument using standard samples or blanks before measurement.
Discussion When the Points on a Graph Are Scattered
If measurement points are scattered around the fitted straight line, random error may be affecting the results.
Possible causes of scatter include reading measured values, small temperature changes, slight differences in sample preparation, instrument stability, and insufficient mixing.
If the scatter is small, the reproducibility of the experiment can be considered relatively good.
If the scatter is large, the measurement method and control of experimental conditions should be reviewed.
Example Discussion:
The measurement points were scattered around the fitted straight line.
This scatter is considered to have resulted from random errors such as reading errors in the measuring instruments, small temperature changes, and sample-preparation errors.
If the scatter is large, increasing the number of measurements and using the average may reduce the effect of random error.
Discussion When the Entire Graph Is Shifted
If the measurement points are shifted parallel to the theoretical straight line, or if all points are consistently higher or lower, systematic error is suspected.
Possible causes include insufficient blank correction, errors in concentration settings, zero-point offsets of the instrument, and differences in temperature conditions.
Example Discussion:
Because the measured values were generally higher than the theoretical values, systematic error may have contributed in addition to random error.
Specifically, possible causes include insufficient blank correction, the concentration of the standard solution being higher than the set value, and an offset in the instrument zero point.
Such systematic errors cause the entire set of measured values to shift in one direction.
Discussion of Logarithmic Graphs
In physical chemistry experiments, logarithms may be taken to linearize exponential relationships.
For example, in first-order reaction-rate analysis and temperature-dependence analysis, logarithmic transformation may be used to confirm a linear relationship.
In a logarithmic graph, a relationship that appears curved in the original values may be represented as a straight line.
When logarithms are used, it is necessary to clearly state what the horizontal and vertical axes represent and which terms in the theoretical equation correspond to the slope and intercept.
Example Discussion:
When the measured values were logarithmically transformed and plotted, an almost linear relationship was obtained.
This suggests that the original measured values may follow an exponential relationship.
Because the slope of the fitted straight line corresponds to a constant in the theoretical equation, the desired physical quantity could be determined from the slope.
Discussion of Proportional Relationships
When a proportional relationship exists, the graph approaches a straight line passing through the origin.
However, in experiments, the intercept may not be exactly 0.
Possible causes include blank correction, zero-point offsets, sample-preparation errors, and measurement errors.
When discussing a proportional relationship, check not only whether the graph is linear but also whether it passes through the origin, how much the intercept deviates, and whether the proportional range has been exceeded.
Example Discussion:
The graph was generally linear, suggesting that a proportional relationship held within the measurement range.
However, the intercept of the fitted straight line was not exactly 0.
Possible causes include errors in blank correction and a zero-point offset of the instrument, so the deviation of the intercept must also be checked when evaluating proportionality.
Discussion When Using a Calibration Curve
A calibration curve is a method for determining the concentration of an unknown sample using a graph prepared from standard solutions of known concentration.
In physical chemistry and analytical chemistry experiments, calibration curves may be prepared using absorbance, electrical conductivity, and similar quantities.
In a calibration curve, it is important that the measurement points for the standard solutions are close to linear and that the unknown sample lies within the calibration-curve range.
Example Discussion:
Because the calibration curve prepared from the measured values of the standard solutions showed high linearity, it is considered suitable for determining the concentration of the unknown sample.
However, if the measured value of the unknown sample lies outside the calibration-curve range, error caused by extrapolation may become large.
Therefore, it is desirable to dilute the unknown sample so that it falls within the linear range of the calibration curve before measurement.
When a Physical Quantity Is Determined From a Fitted Straight Line
In physical chemistry experiments, the desired physical quantity is often determined from the slope or intercept of a fitted straight line.
For example, the slope may be related to a rate constant, molar absorption coefficient, heat of reaction, activation energy, partition coefficient, or similar quantity.
Therefore, it is important not simply to write the numerical value of the fitted equation but to relate it to the theoretical equation.
Example Discussion:
The slope of the fitted straight line corresponds to a constant in the theoretical equation.
Therefore, the desired physical quantity was calculated using the slope obtained from the graph.
Because the determined value was close to the literature value, the experimental results are considered to generally follow the theoretical equation and the analytical method to have been appropriate.
Discussion of Units
Handling units is extremely important in physical chemistry experiments.
Slopes and intercepts also have units, and if the units are incorrect, the meaning of the determined physical quantity changes.
In particular, conversion errors involving mL and L, °C and K, min and s, nm and m, and similar units can greatly affect the results.
In a report, use consistent units in calculations and clearly state the unit of the final physical quantity.
It is also important to confirm that the units agree with the theoretical equation.
Example Discussion:
A unit-conversion error may explain why the calculated value differed greatly from the literature value.
In physical chemistry experiments, temperature may need to be handled in K, or volume may need to be standardized in L or m3.
If the units are inconsistent, the slope and the physical quantity obtained from it are not calculated correctly, so checking units is important.
Discussion of Significant Figures
Measured values have a number of significant figures corresponding to the precision of the measuring instrument.
If a calculated result is written with excessively many digits, it may appear more accurate than the actual measurement precision.
In physical chemistry experiments, it is important to organize the significant figures of the results according to the precision of the measured values.
Example Discussion:
Although the calculation result was displayed with many digits, the significant figures must be adjusted appropriately considering the precision of the actual measured values.
Digits exceeding the smallest scale of the measuring instrument or the precision of the standard-solution concentration have no experimental meaning.
Therefore, it is important to express the results using significant figures appropriate to the measurement precision.
Discussion of Reproducibility
Reproducibility indicates whether similar results are obtained when measurements are performed under the same conditions.
If values obtained from multiple measurements are close, reproducibility can be considered high.
If the values show large scatter, there may be problems with the measurement operation or control of experimental conditions.
In physical chemistry experiments, reproducibility may be evaluated using the mean, standard deviation, and range of variation.
In a report, discuss not only the mean value but also the magnitude of the scatter.
Example Discussion:
Because the values from multiple measurements were close, the reproducibility of the experiment is considered relatively good.
On the other hand, if some measured values deviated greatly, temperature changes during measurement or errors in sample preparation may have contributed.
To improve reproducibility, it is important to keep the measurement conditions constant and ensure that the same operation can be repeated consistently.
Discussion When Using Standard Deviation
Standard deviation is a value representing the magnitude of scatter in measured values.
When the standard deviation is small, the measured values are concentrated near the mean and reproducibility is considered high.
When the standard deviation is large, the scatter in measured values is large and random error or instability in experimental operations may be contributing.
Example Discussion:
Because the standard deviation of the measured values was small, the scatter among the individual measurements was small and reproducibility is considered relatively high.
On the other hand, if the standard deviation is large, nonconstant measurement conditions, reading errors, and variation in sample preparation may have contributed.
Therefore, standard deviation is useful as an indicator for evaluating the reliability of measurement results.
Example Discussion of a Graph in an Absorbance Experiment
In experiments using absorbance, the relationship between concentration and absorbance may be graphed.
Theoretically, under constant conditions, absorbance is proportional to concentration.
A larger slope of the fitted straight line indicates a greater change in absorbance for a given change in concentration.
Example Discussion:
When the relationship between concentration and absorbance was graphed, an almost linear relationship was obtained within the measurement range.
This suggests that absorbance changed in proportion to concentration and that the theoretical equation was valid within the measurement range.
On the other hand, if the high-concentration points deviated from the fitted straight line, possible causes include loss of linearity because the concentration was too high and errors in sample preparation.
Example Discussion of a Graph in a Reaction Rate Experiment
In reaction rate experiments, changes over time are followed and a rate constant may be determined from changes in concentration or absorbance.
Depending on the reaction order, concentration, the logarithm of concentration, or the reciprocal of concentration may be graphed and linearity checked.
Example Discussion:
When the measured values were transformed and plotted against time, a fitted straight line was obtained.
This suggests that the reaction in this experiment may follow the assumed reaction order within the measurement range.
The rate constant was determined from the slope of the fitted line, but possible causes of points deviating from the line include deviation in the reaction start time, temperature changes, and errors in measurement timing.
Example Discussion of a Graph in an Electrical Conductivity Experiment
In experiments measuring electrical conductivity, the relationship between concentration, temperature, titrant volume, and conductivity may be graphed.
Because electrical conductivity depends on ion concentration and ion mobility, it provides clues for discussing changes in the state of the solution.
Example Discussion:
Electrical conductivity changed as the titrant volume increased, and a point at which the slope of the graph changed was observed.
This change in slope is considered to reflect a change in the main ionic species present in the solution.
However, insufficient cleaning of the electrode and temperature changes affect electrical conductivity, so they must be considered as possible causes of deviations in measured values.
Example Discussion of a Graph in a Thermodynamics Experiment
In thermodynamics experiments, relationships between temperature and equilibrium constants, vapor pressure, solubility, and similar quantities may be graphed.
The reciprocal of temperature or logarithms may be used to linearize the relationship, and enthalpy changes or similar quantities may be determined from the slope.
In such experiments, temperature control is particularly important.
Example Discussion:
When the temperature-related values were transformed and plotted, a fitted straight line was obtained.
A thermodynamic quantity was calculated from the slope of the fitted line, but a difference from the literature value occurred.
Possible causes include the temperature not being kept completely constant, measurement before equilibrium was reached, and calibration errors in the thermometer or measuring instrument.
When the Graph Results Can Be Considered Good
Graph results can be considered good when the measurement points follow the relationship predicted by the theoretical equation, the fitted straight line has a high R2, there are few outliers, and the physical quantity determined from the slope or intercept is close to the literature value.
However, a high R2 alone is not sufficient, and it is also necessary to confirm that the slope and intercept are physically meaningful.
Example Discussion:
The measurement points were distributed along the fitted straight line, and the coefficient of determination was also high.
In addition, the physical quantity determined from the slope of the fitted line was close to the literature value.
From these results, the measurements and analysis in this experiment are considered generally appropriate and the agreement with the theoretical equation is considered good.
Example Discussion When the Experiment Did Not Go Well
If the graph is not linear, R2 is low, there are many outliers, the result differs greatly from the theoretical value, or the intercept is unnaturally large, organize and consider the possible causes of error.
It is useful to check temperature control, concentration preparation, measuring instruments, instrument calibration, measurement range, unit conversion, and the range of applicability of the theoretical equation.
Example Discussion:
In this experiment, the measurement points showed large scatter and the fit to the fitted straight line was insufficient.
Possible causes include errors in preparing the sample concentration, insufficient temperature control, and reading errors in the measured values.
In addition, the linear relationship predicted by the theoretical equation may not have held over part of the measurement range.
Therefore, it is important to keep the measurement conditions constant and obtain data within an appropriate range.
How to Write Points for Improvement
In the discussion of a physical chemistry experiment, including not only the causes of error but also points for improvement makes the report easier to organize.
Improvements are easier to write when divided into measurement conditions, sample preparation, equipment, and graph analysis.
Improvements to Measurement Conditions
- Keep the temperature constant
- Allow sufficient time for equilibrium to be reached before measurement
- Standardize the measurement timing
- Perform sufficient stirring and mixing
- Perform multiple measurements under the same conditions
Improvements to Sample Preparation
- Measure mass and volume accurately
- Use volumetric flasks and pipettes correctly
- Check the concentration of standard solutions
- Avoid bubbles and contamination
- Perform dilution operations carefully
Improvements to Equipment and Analysis
- Calibrate the measuring instrument
- Perform blank correction and zero-point adjustment
- Use consistent units
- Check the causes of outliers
- Select axes that correspond to the theoretical equation
- Check the slope, intercept, and R2 of the fitted straight line
Example of How to Write Points for Improvement:
To reduce the scatter in the experimental values, the temperature must be kept constant and the measurement timing standardized.
In addition, accurately preparing the concentrations of standard solutions and carefully reading measuring instruments may reduce deviations among the measurement points.
Furthermore, appropriately calibrating the instrument and performing blank correction, and plotting the data using axes based on the theoretical equation, can produce a more reliable fitted straight line.
Difference Between a Superficial Discussion and a Good Discussion
In a discussion of a physical chemistry experiment, simply writing that “the graph became linear” or “there was an error” results in a superficial discussion.
Relating the theoretical equation, slope, intercept, R2, outliers, causes of error, and points for improvement produces a more persuasive discussion.
| Superficial Discussion | Good Discussion |
|---|---|
| The graph became linear. | When the measured values were graphed based on the theoretical equation, an approximately linear relationship was obtained. This suggests that the experimental results generally followed the theoretical equation within the measurement range. |
| There was an error. | Possible reasons the experimental value differed from the literature value include insufficient temperature control, concentration-preparation error, reading errors in the measuring instruments, and insufficient instrument calibration. |
| R2 was high. | Because the coefficient of determination R2 was close to 1, the measurement points are considered to fit the fitted straight line well. However, a high R2 does not guarantee that the theoretical equation is correct or that no error is present, so the physical meanings of the slope and intercept must also be checked. |
Examples of Expressions That Can Be Used in Reports
The following expressions can be used when writing the results and discussion of physical chemistry experiments.
Adjust the necessary parts according to your own experimental results.
- When the measured values were graphed based on the theoretical equation, an approximately linear relationship was obtained.
- The slope of the fitted straight line is considered to correspond to the proportionality constant in the theoretical equation.
- Possible causes of the intercept deviating from 0 include blank correction and a zero-point offset of the instrument.
- Because the coefficient of determination R2 was close to 1, the measured values fit the fitted straight line well.
- Possible causes of some measurement points deviating from the fitted straight line include sample-preparation error and temperature changes during measurement.
- The difference between the experimental and literature values is considered to originate from differences in measurement conditions and measurement error.
- Because the physical quantity depends on temperature, insufficient temperature control may have affected the results.
- Errors during concentration preparation may shift the values on the horizontal axis and affect the slope of the fitted straight line.
- If the measured values are shifted consistently in one direction, systematic error may have contributed.
- The scatter in the measured values is caused by random error, and its effect may be reduced by increasing the number of measurements.
Points to Check When Discussing a Physical Chemistry Experiment
Checking the following points before writing the report makes the discussion easier to write.
- Do the horizontal and vertical axes of the graph correspond to the theoretical equation?
- Have you written the equation of the fitted straight line?
- Have you explained the meaning of the slope?
- Have you discussed the meaning of the intercept and its deviation?
- Are you avoiding excessive reliance on the coefficient of determination R2?
- Have you considered the presence and causes of outliers?
- Have you compared the result with theoretical or literature values?
- Have you calculated the error rate?
- Have you distinguished between random and systematic errors?
- Have you considered the effects of temperature, concentration, time, and instrument calibration?
- Have you handled units and significant figures appropriately?
- Do the points for improvement correspond to the causes of error?
Summary
In physical chemistry laboratory reports, measured values are often graphed and physical quantities are determined from the slope and intercept of fitted straight lines.
Graphs are important tools for confirming trends in measured values, agreement with theoretical equations, outliers, and scatter.
Because the slope and intercept of a fitted straight line often have physical meanings, it is necessary to explain not only the numerical values but also their meanings.
In discussing errors, do not simply write that “there was an error,” but distinguish between random and systematic errors.
Temperature control, concentration preparation, reading of measuring instruments, instrument calibration, unit conversion, and outliers affect the difference between experimental and theoretical values.
In addition, even if R2 is high, the result is not necessarily theoretically correct, so it is important to judge the result together with the slope, intercept, and comparison with literature values.
In a report, rather than writing about graphs, fitted straight lines, errors, and comparison with theoretical values separately, explain how they are related.
A persuasive physical chemistry laboratory report can be produced by explaining why the measured values showed a particular trend, which errors affected the result and in which direction, and how the reliability could be improved.
