LAB TECHNIQUE - CALIBRATION CURVES & BEER'S LAW


Introduction

A calibration curve is a method used in analytical chemistry to determine the concentration of an unknown sample solution. It is a graph generated by experimental means, with the concentration of solution plotted on the x-axis and the observable variable — for example, the solution’s absorbance — plotted on the y-axis. The curve is constructed by measuring the concentration and absorbance of several prepared known solutions, called calibration standards. Comparing the measured values of known compounds to measurements from samples where we do not know the amount is the idea behind a calibration curve.  

Creating a Calibration Curve

Spectrophotometry can be useful in determining the concentration of an unknown solution. For example, if a researcher has a sample of river water and wants to know its lead content**, he or she can determine it by using a spectrophotometer to plot a calibration curve. First, the researcher creates several standard solutions of lead, ranging from less to more concentrated. These samples are placed into the spectrophotometer, which records a different absorbance for each one.

The experimentally determined absorbance values are plotted on a graph against the known concentration of each calibration standard. A set of points is created, which in the case of absorbance should be roughly linear due to Beer’s law. A line is drawn to connect these data points, forming the calibration curve. In almost every case, the data points will not be mathematically exact, so the line should be drawn to intercept the maximum number of points — it is a line of best fit. Although the relationship of absorbance to concentration is linear, this is not always true for other experimentally determined variables, and occasionally curves must be employed to describe the relationship.




Comparing Unknown

At this stage, the unknown solution can be analyzed. The sample is inserted into the spectrophotometer, and its absorbance is measured. Since this sample is being measured against several standards containing the same compound, its absorbance and concentration must fall somewhere along the calibration curve for that compound. This means that once the solution's absorbance is known, its concentration can be deduced mathematically (or graphically).

A horizontal line can be drawn from the unknown solution’s y-value — its absorbance, which has just been measured. The equation for the line (y = mx + b) of the calibration curve and the calculated slope & y-intercept is used to mathematically determine the solution's concentration (x in the equation).


Beer's Law (Background Info)

Beer's Law relates the experimental absorbance value for a chromophore (a substance that absorbs light) to the concentration of the chromophore in solution. Beer's law has many forms, the most common is

Aλ = ε l C 

In this equation A is the measured absorbance of the chromophore at the wavelength (usually at a peak maximum, or max), determined from a spectrum spanning ultaviolet and/or visble wavelgnths of light). The Greek letter epsilon, e, stands for the molar extinction coefficient (M cm), an experimentally determined constant for the specific chromophore at the same wavelength (this wavelength is always specified). The molar extnction coeffivient is a quantiative measure of the light absorbance by the chromophore at that wavelength for a one molar solution and one centimeter path lenght. The valuye of l is the path length, or the stance the light travels through the solution in the cuvette (continer) used for the absorbance measurement. Lastly, C is the molar cocnetration of the chromophore (mol/L) used for the measurement.

Beer's law says that the relationship between the absorbance of the chromophore and its concentraiton is linear, allowing construction of a standard curve by plotting absorbance versus concentration, such as shown in Figure 1. Notice all of the units cancel upon multiplying elC, consistent with A being a unitless quantity.



Footnotes:


**It is important to note that you cannot measure the concentration of lead in a water sample directly by using a spectrophotometer because lead in water is colorless. The species that you are trying to measure using a spectrophotometer MUST have a color. There is way you can use a spectrophotometer to measure the concentration of lead in water, but it is not a direct as this "example" makes it seem.