The **calibration curve** It is a graph that relates two variables, which is used to verify that a measurement equipment is working properly. Regardless of the type of equipment, time, use and natural wear and tear affect the quality of the measurement.

That is why it is important to periodically verify its proper functioning. This is done by comparing the measurements provided by the equipment against those of a standard device used as a reference. This reference equipment is the most accurate.

For example, in figure 1 we have in green the output signal of an ideal device, compared to the measured magnitude, both are proportional.

On the same graph are the curves of two different instruments that are not calibrated and that have slightly different behaviors from each other and from the standard.

[toc]

**How does it work?**

For example, suppose we want to calibrate a dynamometer, which is a device used to measure forces such as the weight of objects and those that appear when an object is accelerated.

To get a spring to stretch, it is necessary to apply a force to it, which is proportional to the stretch, according to Hooke’s law.

A simple dynamometer consists of a spring in a tube with a pointer and scale to indicate stretch. At one end there is a ring to hold the dynamometer and at the other a hook to hang weights.

One way to calibrate the dynamometer is by hanging different weights, whose mass was previously determined with a balance (the reference instrument), and measuring the stretch or elongation of the spring, which is supposed to be light.

Hooke’s law applied to the spring-mass system in static equilibrium results in the following equation, which relates the length of the spring to the hanging mass:

L = (g/k) m + Lo

Where:

-L: total length of the spring

-g: acceleration due to gravity

-k: spring constant

-m: mass

-Lo: natural length of the spring.

Once you have several pairs of points *length-mass*, we proceed to graph them to build the calibration curve. Since the relationship between the length L and the mass m is linear, the curve is a straight line, where:

Slope = g/k

**How to make a calibration curve?**

These are the steps to make a calibration curve for a measuring instrument.

**Step 1**

Choose the comparison standard to use, according to the device to be calibrated.

**Step 2**

Select the appropriate range of values and determine the optimal number of measurements to perform. If we were to calibrate a dynamometer, we would first have to assess the limit of the weight that can be hung from it without permanently deforming it. If this were to happen, the instrument would be useless.

**Step 3**

Take pairs of readings: one is the reading made with the standard pattern, the other is the measurement made with the sensor being calibrated.

**Step 4**

Make a graph of the pairs of readings obtained in the previous step. It can be done by hand, on graph paper or using a spreadsheet.

The latter option is preferable, as tracing by hand may result in slight inaccuracies, while a better fit can be made using the spreadsheet.

**Examples of calibration curve**

The calibration curves are also used to convert a magnitude into another that is easy to read, through some property or law that relates them.

**Calibration of a platinum resistance thermometer**

An alternative to the use of mercury is electrical resistance. Resistance is a good thermometric property, since it varies with temperature and is also easy to measure with an ohmmeter or ammeter.

Well, in this case, a suitable standard to build the calibration curve would be a good laboratory thermometer.

You can measure temperature – resistance pairs and take them to a graph, which will later be used to determine any temperature value knowing the resistance, as long as its value is within the range of measurements that has been taken.

In the following calibration curve, the x axis is the temperature with the standard thermometer and the vertical axis is the temperature with a platinum resistance thermometer, called thermometer A.

The spreadsheet finds the line that best fits the measurements, whose equation appears in the upper right. The platinum thermometer has a shift of 0.123 ºC with respect to the standard.

**Calibration curve of a solution**

It is a method used in analytical chemistry and consists of a reference curve, where the measured quantity is the concentration of an analyte on the horizontal axis, while the instrumental response appears on the vertical axis, as shown in the following example.

The curve is used to find, by interpolation, the concentration of analyte present in an unknown sample, through said instrumental response.

The instrumental response can be an electric current or a voltage. Both magnitudes are easy to measure in the laboratory. The curve is then used to find the concentration of the unknown analyte in this way:

Let’s assume the current is 1500 mA on the calibration curve. We locate ourselves on this point on the vertical axis and draw a horizontal line to the curve. From this point we project a line vertically towards the x-axis, where the respective concentration of the analyte is read.

**solved exercise**

Build the calibration curve of a spring with elastic constant k and from the graph, determine the value of said constant, all from the following experimental data of length-mass pairs:

**Solution**

Each pair of values is interpreted as follows:

When a 1-kg mass is hung, the spring stretches to 0.32 m. If a mass of 2 kg is hung, the spring becomes 0.40 m long, and so on.

Using a spreadsheet, the length vs. mass graph is drawn up, which turns out to be a straight line, as expected from Hooke’s law, since the relationship between length L and mass m is given by:

L = (g/k) m + Lo

As explained in previous sections. The graph obtained is the following:

Below the title, the spreadsheet shows the equation of the line that best fits the experimental data:

L = 0.0713 m + 0.25

The line cut with the vertical axis is the natural length of the spring:

Lo = 0.25 m

For its part, the slope is the ratio g/k:

g/k = 0.0713

Therefore, taking g = 9.8 m/s2, the value of the spring constant is:

k = (9.8 /0.0713) N/m

k = 137.45 N/m

Having this value, our spring is calibrated and the dynamometer ready to measure forces in the following way: an unknown mass is hung that produces a certain stretch, which is read on the vertical axis.

A horizontal line is drawn from this value until it reaches the curve and at that point a vertical line is projected to the x-axis, where the value of the mass is read. Having the mass, we have its weight, which is the force that causes the elongation.

**References**

Serway, R., Vulle, C. 2011. Fundamentals of Physics. 9na Ed. Cengage Learning.

Tipler, P. 1987. Pre-College Physics. Editorial Reverté.

Tippens, P. 2011. Physics: Concepts and Applications. 7th Edition. mcgrawhill

Wilson, J. 2010. Physics Laboratory Experiments. 7th. Ed. Brooks Cole. Wikipedia. Calibration curve. Recovered from: es.wikipedia.org.