A calibration curve is a graph commonly used in quantitative experiments such as spectrophotometry and colorimetric analysis.
It is created from measurement values obtained from standard solutions of known concentrations and is used to determine the concentration of an unknown sample.
In a chemistry experiment report, it is important not only to create a calibration curve, but also to discuss whether the linearity is good, whether the correlation coefficient is sufficient, whether the intercept is reasonable, and why any measurement points deviate from the straight line.
In particular, when absorbance deviates from the straight line, it is necessary to consider errors in standard-solution preparation, measurement wavelength, cell contamination, blank correction, and deviations at high concentrations.
This article clearly explains the basic method for drawing a calibration curve, how to interpret the graph, causes of absorbance deviating from the straight line, and discussion examples that can be used in reports.
Note:
This article is a reference intended to assist with discussions of results obtained in chemistry experiments at universities and similar institutions.
For the actual method of creating graphs, handling of regression lines, and treatment of outliers, always follow the instructions in your university’s laboratory manual and those given by your instructor or TA.
- What Is a Calibration Curve?
- Basic Method for Drawing a Calibration Curve
- How to Determine the Horizontal and Vertical Axes
- Meaning of the Regression Line
- Why the Intercept Does Not Become 0
- Discussion of the Correlation Coefficient and Coefficient of Determination
- Main Causes of Absorbance Deviating From the Straight Line
- Deviation From the Straight Line Caused by Errors in Standard-Solution Preparation
- Deviation of Absorbance Caused by Cell Contamination or Air Bubbles
- When Blank Correction Is Insufficient
- Deviation From the Straight Line at High Concentrations
- Large Variation at Low Concentrations
- When the Measurement Wavelength Is Inappropriate
- Can Outliers Be Excluded?
- Points to Note When Determining the Concentration of an Unknown Sample
- Points to Check From the Appearance of the Graph
- When the Calibration Curve Can Be Considered Good
- Example Discussion When the Calibration Curve Did Not Work 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 Calibration Curve
- Summary
What Is a Calibration Curve?
A calibration curve is a graph showing the relationship between the concentrations of standard solutions of known concentration and their measured values.
In spectrophotometry, it is common to plot the concentration of the standard solution on the horizontal axis and the absorbance on the vertical axis.
The measurement points of the standard solutions are plotted on a graph, and a regression line is drawn through the points.
By applying the absorbance of an unknown sample to this line, the concentration of the unknown sample can be determined.
A calibration curve is not simply a graph, but a “reference for determining the concentration of an unknown sample.”
Therefore, the linearity of the calibration curve and the variation of measurement points are directly related to the reliability of the unknown concentration.
Basic Method for Drawing a Calibration Curve
When drawing a calibration curve, organize the concentrations of the standard solutions and their measured values in a table and plot them on a graph.
In spectrophotometry, the concentration of the standard solution is placed on the horizontal axis and absorbance on the vertical axis.
Items Required for a Calibration Curve
- Horizontal axis: concentration of the standard solution
- Vertical axis: measured value such as absorbance
- Plot of measurement points
- Regression line
- Equation of the calibration curve
- Correlation coefficient or coefficient of determination
- Units
- Graph title when necessary
Example of How to Write the Results:
A calibration curve was created with the concentration of the standard solution on the horizontal axis and absorbance on the vertical axis.
The obtained measurement points were generally aligned along a straight line, and the equation of the regression line was y = 0.124x + 0.006.
The correlation coefficient was 0.997, indicating high linearity between concentration and absorbance.
How to Determine the Horizontal and Vertical Axes
In a calibration curve, the quantity that acts as the cause is generally placed on the horizontal axis, while the measured quantity is placed on the vertical axis.
In spectrophotometry, changing the concentration changes the absorbance, so concentration is placed on the horizontal axis and absorbance on the vertical axis.
| Axis | Example in Spectrophotometry | Meaning |
|---|---|---|
| Horizontal axis | Concentration | Concentration of the standard solution prepared by the experimenter |
| Vertical axis | Absorbance | Value measured by the instrument |
If the units of the axes are omitted, it becomes difficult to understand what should be read from the graph.
Clearly indicate whether the concentration unit is mol/L, mg/L, ppm, or another unit.
Example Discussion:
In the calibration curve, concentration was set on the horizontal axis and absorbance on the vertical axis.
This is because the concentration of the standard solution was changed, and the absorbance changed as a result.
Clearly indicating the units of the axes makes it possible to correctly read the unknown concentration from the calibration curve.
Meaning of the Regression Line
In a calibration curve, a regression line is drawn through the measurement points.
The regression line expresses the relationship between concentration and absorbance while taking into account variation among the measurement points.
In spectrophotometry, when the Beer-Lambert law holds within the measurement range, absorbance is proportional to concentration.
Therefore, the calibration curve is expected to be linear.
y = ax + b
Here, y is absorbance, x is concentration, a is the slope, and b is the intercept.
By substituting the absorbance of the unknown sample for y, the concentration x of the unknown sample can be determined.
Example Discussion:
The regression line of the calibration curve represents the relationship between the concentrations of the standard solutions and their absorbances.
Because the measurement points were generally aligned along the regression line, a proportional relationship between absorbance and concentration is considered to have held within the measurement range.
Therefore, it is considered appropriate to use this regression line to determine the concentration of the unknown sample.
Why the Intercept Does Not Become 0
Ideally, when the concentration is 0, the absorbance is also 0, so the calibration curve appears to pass through the origin.
However, in actual experiments, the intercept may not be exactly 0.
Possible causes of deviation of the intercept from 0 include insufficient blank correction, contamination of the cell, instrument zero-adjustment errors, absorption by reagents or solvents, and errors in preparation of the standard solutions.
| Cause of Intercept Deviation | Possible Effect |
|---|---|
| Insufficient blank correction | Overall absorbance shifts higher or lower |
| Contamination or scratches on the cell | Absorption or scattering from sources other than the target component is included |
| Absorption by reagents or solvents | Absorbance may appear even when the concentration is 0 |
| Improper zero adjustment of the instrument | A systematic shift occurs in all measured values |
Example Discussion:
One possible reason the intercept of the calibration curve deviated from 0 is that blank correction was insufficient.
If blank correction is insufficient, absorption by the solvent, reagents, or cell is included in the measured value, and the absorbance may not become 0 even when the concentration is close to 0.
Therefore, the deviation of the intercept is considered to reflect the correction state of the overall measurement system.
Discussion of the Correlation Coefficient and Coefficient of Determination
The correlation coefficient and coefficient of determination are used as indicators of the linearity of a calibration curve.
The closer these values are to 1, the closer the relationship between concentration and absorbance is considered to be to a straight line.
However, a high correlation coefficient alone does not mean that the calibration curve is completely correct.
It is also necessary to check whether the concentration range of the standard solutions is appropriate, whether the absorbance of the unknown sample lies within the range, and whether there are any outliers.
Example Discussion:
The correlation coefficient of the calibration curve was close to 1, indicating high linearity between concentration and absorbance.
From this, the Beer-Lambert law is considered to have held relatively well within the measured concentration range of the standard solutions.
However, even when the correlation coefficient is high, if the absorbance of the unknown sample lies outside the range of the calibration curve, extrapolation is required and the reliability of the concentration estimate may decrease.
Main Causes of Absorbance Deviating From the Straight Line
The main causes of measurement points deviating from the straight line of a calibration curve can be broadly divided into “errors in standard-solution preparation,” “errors in measurement procedures,” “inappropriate measurement range,” and “condition of the instrument or cell.”
| Cause | What Happens | Effect on the Result |
|---|---|---|
| Error in standard-solution preparation | Actual concentration differs from the intended value | Measurement points deviate from the straight line |
| Dirty cell or air bubbles | Light is scattered or blocked | Absorbance may appear higher |
| Improper blank correction | Absorption from sources other than the target component remains | The intercept or overall values shift |
| Concentration too high | The proportional relationship breaks down | High-concentration points deviate from the straight line |
| Inappropriate measurement wavelength | Change in absorbance becomes small | Variation becomes more noticeable |
| Insufficient mixing | Solution concentration is not uniform | Measured values vary |
Deviation From the Straight Line Caused by Errors in Standard-Solution Preparation
Because a calibration curve is created using the concentrations of standard solutions as the reference, if the concentrations of the standard solutions are inaccurate, the measurement points are more likely to deviate from the straight line.
Possible causes include exceeding the calibration mark of the volumetric flask, misadjusting the liquid level in the volumetric pipette, and insufficient mixing after dilution.
Example Discussion:
One possible reason some measurement points deviated from the regression line is an error in preparation of the standard solutions.
If a volumetric flask was diluted beyond the calibration mark, the actual concentration of that standard solution would be lower than intended.
In addition, if mixing after dilution was insufficient, the concentration could differ depending on the portion sampled, causing variation in absorbance.
Deviation of Absorbance Caused by Cell Contamination or Air Bubbles
In absorbance measurements, the condition of the cell greatly affects the measured value.
If fingerprints, water droplets, or dirt are present on the outside of the cell, light may be scattered or blocked, causing the absorbance to appear higher than the actual value.
Air bubbles inside the cell can also interfere with the optical path and cause deviations in the measured value.
Example Discussion:
Possible reasons the absorbance was measured higher than the regression line include contamination or water droplets on the cell surface and air bubbles inside the cell.
If these are present in the optical path, light scattering or blocking occurs in addition to absorption by the target component, reducing the amount of transmitted light detected by the instrument.
As a result, the absorbance may have been overestimated and the measurement point may have deviated from the calibration curve.
When Blank Correction Is Insufficient
Blank correction is performed to subtract absorption by the solvent, reagents, and cell.
If blank correction is insufficient, absorption from sources other than the target component remains in the measured values, causing the entire calibration curve to shift or the intercept to move away from 0.
Example Discussion:
One possible reason the intercept of the calibration curve deviated from 0 is that blank correction was insufficient.
The blank is used to correct for absorption by the solvent, reagents, and cell other than the target component.
If correction is insufficient, absorption from sources other than the target component is included in the absorbance, causing the entire calibration curve to shift upward.
Deviation From the Straight Line at High Concentrations
Absorbance is considered proportional to concentration, but this relationship does not hold without limit.
At high concentrations, intermolecular interactions, stray light, and measurement limits of the instrument may cause the proportional relationship between absorbance and concentration to break down.
If points on the high-concentration side deviate from the straight line, that concentration range may not be suitable for use in the calibration curve.
In this case, possible improvements include lowering the concentration range of the standard solutions or diluting the sample before measurement.
Example Discussion:
One possible reason the measurement points on the high-concentration side deviated from the regression line is that the concentration exceeded the range in which the Beer-Lambert law holds.
If the concentration is too high, the proportional relationship between absorbance and concentration may break down and linearity may decrease.
Therefore, when preparing a calibration curve, it is necessary to select a concentration range in which linearity is maintained.
Large Variation at Low Concentrations
At low concentrations, absorbance is small, so the relative influence of instrument-reading error, small deviations in blank correction, and cell contamination becomes greater.
Therefore, low-concentration measurement points may deviate from the straight line.
Example Discussion:
One possible reason the low-concentration measurement points deviated from the regression line is that the absorbance was small, making the relative influence of measurement error larger.
At low concentrations, even a slight deviation in blank correction or instrument-reading error can have a relatively large effect on the absorbance.
Therefore, variation in measured values is considered more noticeable on the low-concentration side.
When the Measurement Wavelength Is Inappropriate
In spectrophotometry, measurements are made at a wavelength at which the target component absorbs strongly.
If the measurement wavelength is inappropriate, changes in concentration may produce only small changes in absorbance, making variation in the measured values appear larger.
In addition, if a wavelength absorbed by other components is selected, absorption by substances other than the target component is added, possibly reducing the linearity of the calibration curve.
Example Discussion:
If the measurement wavelength was not suitable for absorption by the target component, the change in absorbance relative to concentration may have been small, causing greater variation in the calibration curve.
In addition, if measurements were made at a wavelength absorbed by other components, absorption from substances other than the target component would be included, which could cause measurement points to deviate from the straight line.
Can Outliers Be Excluded?
If only some points deviate greatly from the calibration curve, it may be tempting to exclude them.
However, there must be a clear reason for excluding an outlier.
Rather than excluding a point simply because it does not fit the straight line, the decision should be based on observed facts such as an error in standard-solution preparation, an air bubble in the cell, or an abnormality during measurement.
Example Discussion:
Some measurement points deviated greatly from the regression line, but a clear reason is necessary when excluding outliers.
For example, if an air bubble was observed in the cell during measurement or an error occurred during preparation of the standard solution, this may provide a basis for excluding that measured value.
On the other hand, excluding measurement points simply because they do not fit the line, without identifying a cause, reduces the reliability of the results.
Points to Note When Determining the Concentration of an Unknown Sample
When determining the concentration of an unknown sample, check whether the absorbance of the unknown sample lies within the range of the calibration curve.
Using a value outside the range requires extrapolation and reduces the reliability of the concentration.
If the unknown sample was diluted before measurement, the original sample concentration must be determined by multiplying the concentration obtained from the calibration curve by the dilution factor.
Example Discussion:
Because the absorbance of the unknown sample was within the range of the calibration curve, it is considered appropriate to determine the concentration using the calibration-curve equation.
However, if the unknown sample was diluted before measurement, the concentration obtained from the calibration curve is the concentration after dilution.
Therefore, the dilution factor must be taken into account when determining the concentration of the original sample.
Points to Check From the Appearance of the Graph
After creating a calibration curve, check not only the correlation coefficient but also the graph itself.
Even if the numerical correlation coefficient is high, some points may deviate or the curve may bend on the high-concentration side.
- Are the measurement points aligned along a straight line?
- Does the graph bend on the high-concentration side?
- Is there large variation on the low-concentration side?
- Is the intercept unusually large?
- Are there any outliers?
- Is the absorbance of the unknown sample within the range of the calibration curve?
Example Discussion:
Although the correlation coefficient of the calibration curve was high, the measurement points on the high-concentration side deviated slightly from the regression line.
This suggests that the proportional relationship between absorbance and concentration may have begun to break down at high concentrations.
Therefore, when determining the concentration of an unknown sample, it is important to use a calibration curve within the range in which linearity is maintained.
When the Calibration Curve Can Be Considered Good
A calibration curve can be considered good when the measurement points are generally aligned along a straight line, the correlation coefficient or coefficient of determination is high, the intercept is not unnaturally large, and the absorbance of the unknown sample lies within the range.
Example Discussion:
In the prepared calibration curve, absorbance increased as the concentration of the standard solutions increased, and the measurement points were generally aligned along a straight line.
The correlation coefficient was also close to 1, so the linearity of the calibration curve is considered high.
In addition, because the absorbance of the unknown sample was within the range of the calibration curve, calculation of the unknown concentration using the calibration curve is considered relatively reliable.
Example Discussion When the Calibration Curve Did Not Work Well
If the calibration curve does not form a good straight line, check the preparation of the standard solutions, blank correction, cell condition, measurement wavelength, concentration range, and insufficient mixing.
Rather than simply writing that “the curve was not linear,” it is important to explain how each possible cause affected the absorbance.
Example Discussion:
Possible reasons the measurement points of the calibration curve varied from the regression line include errors in preparation of the standard solutions and contamination of the cell.
If the concentrations of the standard solutions differed from the intended values, the relationship between concentration and absorbance would be disrupted, causing measurement points to deviate from the straight line.
In addition, if the cell surface was dirty or air bubbles were present, the absorbance may have been measured higher than the actual value.
These factors are considered to have reduced the linearity of the calibration curve.
How to Write Points for Improvement
In a discussion of calibration curves, including points for improvement as well as sources of error makes the report easier to organize.
It is important to write the improvements in correspondence with the causes.
Improvements for Increasing the Linearity of the Calibration Curve
- Prepare the standard solutions accurately
- Correctly align the calibration marks of volumetric flasks and pipettes
- Mix the solutions thoroughly after dilution
- Perform blank correction correctly before measurement
- Wipe dirt and water droplets from the cell surface
- Remove air bubbles from inside the cell
- Prepare standard solutions within an appropriate concentration range
- If the unknown sample is outside the range, dilute it and measure again
Example of How to Write Points for Improvement:
To improve the linearity of the calibration curve, it is necessary to prepare the standard solutions accurately and mix them thoroughly after dilution.
In addition, dirt or air bubbles on the cell can cause the absorbance to be measured too high, so it is important to check the condition of the cell before measurement.
If linearity breaks down on the high-concentration side, improvement may be possible by narrowing the concentration range of the standard solutions or diluting the sample before measurement.
Difference Between a Superficial Discussion and a Good Discussion
In a discussion of a calibration curve, simply writing that “it became linear” or “a point deviated” results in a superficial discussion.
A persuasive discussion can be produced by relating linearity, correlation coefficient, intercept, outliers, and the range of the unknown sample.
| Superficial Discussion | Good Discussion |
|---|---|
| The calibration curve was linear. | As the concentrations of the standard solutions increased, the absorbance also increased, and the measurement points were generally aligned along a straight line. This is considered to be because a proportional relationship between absorbance and concentration held within the measured concentration range. |
| The absorbance was off. | Possible reasons some absorbance values deviated from the regression line include errors in preparation of the standard solutions, contamination of the cell surface, air bubbles, and insufficient blank correction. These factors can cause absorbance to be measured higher or lower than the actual value. |
| The correlation coefficient was high. | Because the correlation coefficient was close to 1, the linearity of the calibration curve is considered high. However, it is necessary to confirm that the absorbance of the unknown sample lies within the range of the calibration curve, because values outside the range require extrapolation and reduce reliability. |
Examples of Expressions That Can Be Used in Reports
The following expressions can be used when writing the results and discussion of a calibration curve.
Adjust the necessary parts according to your own experimental results.
- A calibration curve was created with the concentration of the standard solution on the horizontal axis and absorbance on the vertical axis.
- As the concentration increased, absorbance also increased, and an approximately linear relationship was observed between them.
- Because the correlation coefficient was close to 1, the linearity of the calibration curve is considered high.
- One possible reason some measurement points deviated from the regression line is an error in preparation of the standard solutions.
- Contamination or air bubbles on the cell surface may have caused absorbance to be measured higher than the actual value.
- If blank correction was insufficient, absorption from substances other than the target component may have been included in the measured values.
- One possible reason linearity decreased on the high-concentration side is that the concentration exceeded the range in which the Beer-Lambert law holds.
- Because the absorbance of the unknown sample was within the range of the calibration curve, the concentration calculation is considered appropriate.
- If the unknown sample is outside the range of the calibration curve, it must be appropriately diluted and measured again.
- To improve the reliability of the calibration curve, the standard-solution preparation, blank correction, and handling of the cell must be performed accurately.
Points to Check When Discussing a Calibration Curve
Checking the following points before writing the report makes the discussion easier to write.
- Are the horizontal and vertical axes set correctly?
- Are the units of the axes shown?
- Are the measurement points generally aligned along a straight line?
- Is the equation of the regression line shown?
- Have you checked the correlation coefficient or coefficient of determination?
- Is the intercept not unnaturally large?
- Are there any measurement points that deviate from the straight line?
- Have you discussed the causes of any deviating points?
- Is the absorbance of the unknown sample within the range of the calibration curve?
- Has linearity broken down on the high-concentration side?
- Was blank correction appropriate?
- Have you considered the effects of cell contamination and air bubbles?
Summary
A calibration curve is an important graph showing the relationship between the concentrations of standard solutions and measured values such as absorbance, and is used to determine the concentration of an unknown sample.
In spectrophotometry, concentration is placed on the horizontal axis and absorbance on the vertical axis, and the unknown concentration is determined from the equation of the regression line.
In a good calibration curve, the measurement points are aligned along a straight line, the correlation coefficient or coefficient of determination is high, and the absorbance of the unknown sample lies within the range of the calibration curve.
On the other hand, if absorbance deviates from the straight line, possible causes include errors in preparation of the standard solutions, contamination or air bubbles in the cell, insufficient blank correction, and breakdown of the proportional relationship at high concentrations.
In a report, do not simply write that “a calibration curve was obtained.”
Check the linearity, correlation coefficient, intercept, outliers, and range of the unknown sample, and specifically explain the sources of error and points for improvement.
Carefully discussing the calibration curve makes it possible to write a persuasive report about the reliability of the unknown concentration.
