# Calibration Curve Calculator

Created by Davide Borchia
Reviewed by Anna Szczepanek, PhD and Rijk de Wet
Last updated: Feb 02, 2023

• What a calibration curve is and its different types;
• When we use the standard addition method; and
• How to calculate the concentration from the calibration curve.

In addition, it will provide you with a step-by-step tutorial on how to calculate the unknown concentration based on the calibration curve. Ready?

## What is a calibration curve?

Calibration is a measurement technique to ensure that a method/instrument provides accurate results. Chemists write entire books about this topic, but we will try to keep it simple!

Analytical chemistry needs calibration: the reliability of a method of measurement relies on the correct interpretation of the relationship between the concentration of an analyte and the signal of the instrument used. A well-calibrated environment ensures that the results of an analysis will be accurate.

The process of calibration requires an understanding of the concept of calibration curve. This curve (though it is often a straight line) is obtained by testing a certain amount of samples with known concentration with the desired instrument, and then fitting the results using the mathematical model explaining the operations of the method. A plot of the curve shows the instrumental response (the so-called analytical signal) to an analyte (the substance which is measured) and allows to predict the concentration in an unknown sample.

There are many calibration curves types, differentiated by the kind of answer expected from the model:

• Linear with non-null intercept — the standard addition technique;
• Linear with zero intercept — the working curve technique;
• Logarithmic;
• Exponential;
• Power (the concentration is raised to the power of a given exponent); and
• Polynomial (you can consider this type similar to the polynomial one).

Here we will focus only on the standard addition method, which is also implemented in our calibration curve calculator: keep on reading to see if it fits your problem!

Our calibration curve calculator uses the standard-addition method to compute the value of concentration. The standard addition calibration is used when the sample comes with a matrix that gives a constant background signal in the measurement. Think of it as other solutes if their concentrations don't change, or as the signal of the solvent. Here you will learn how to use this method!

First thing: you need to build the calibration curve. One or more standards are required. A standard is a sample with a known concentration. In order to be known, a process of validation is required; this is however a pretty complex process, and it's not relevant here.

In the absence of standards, prepare a set of samples with different concentrations. Make sure that the value of concentration is included in the range of the samples. Measure your samples with the desired instrument: you will obtain a set of instrumental responses. Record them several times (usually three) — this will help reduce the uncertainty associated with the measurement process.

A linear regression model is used to fit the data. The plotted data represents the instrumental response (signal) vs. the concentration. You'll obtain two parameters, and they are fitted by the function:

$y = a \cdot x + b$

This is the calibration curve equation: here, $a$ is the angular coefficient of the line, which translates to the sensitivity of the instrument. $b$ is the intercept, and it corresponds to the background signal of the matrix. The two variables $y$ and $x$ are, respectively, the instrumental response and the concentration. As you can see. the intercept corresponds to the instrumental response for null concentration ($x = 0$).

💡 A linear fit is a regression technique that finds the line deviating the smallest amount from any sample in a set.

That's it! Now you have a calibration curve obtained by using the standard addition method. In the next section, you'll learn how to calculate the unknown concentration from the calibration curve equation.

The standard addition method finds applications in : absorption spectrometry (which uses the Lambert-Beer law), mass spectrometry, and gas chromatography are just some examples.

## How to calculate concentration from the calibration curve?

In order to calculate the unknown concentration, the equation of the linear fit is transformed into the equation:

$x = \frac{y - b}{a}$

Here you subtract the background $b$ (the effect of the matrix) from the signal $y$, and then you divide by the sensitivity of the instrument used, $a$. The result is the concentration, $x$, with units depending on the technique with which the analysis is performed.

## How to use our calibration curve calculator?

If you already have the values of the linear fit's parameters, simply insert them in the calibration curve calculator in their respective fields.

🙋 We decided to omit units from our calculator, since the signal coming from the instrument depends on the physical phenomena employed in the analysis. Remember to be consistent — finding the units of the concentration of your unknown sample won't be hard!

Now write the signal, and find out the unknown concentration.

Do you know that you can use our calculators in "reverse" too? Just fill the concentration field, and find out the expected signal!

If you don't know the parameters of your fit but you have the data from the standard samples, you can use our linear regression calculator to find these values. Check it out!

## How to calculate unknown concentration from the calibration curve? An example

• Calculate the equation which describes the calibration curve. Let’s assume that it is $y = 0.5x + 0.1$.

• Transform the above equation into $x = (y - 0.1)/0.5$.

• Measure the instrumental response of the unknown sample. Let’s assume that it is $2.1$. The units vary from experiment to experiment, and from instrument to instrument: we kept things general.

• You can calculate the unknown concentration by substituting the values:

$x = \frac{2.1 - 0.1}{0.5} = \frac{2}{0.5} = 4$

If you want to recompute concentration (for example switching from molarity and percentage concentration), you can use our concentration calculator. If you want to calculate the concentration of a diluted solution, you can use our solution dilution calculator. To convert between concentration units, use our molality calculator and molarity calculator!

## FAQ

### How do I calculate an unknown concentration from the calibration curve?

1. Choose the right calibration technique, for example, the standard addition method.
2. Measure the instrumental response (signal) from your solution.
3. Determine the parameters for the method: background and sensitivity.
4. Compute the concentration by subtracting the background from the response and dividing this difference by sensitivity.
5. That's all! Enjoy the result!

### When can I use the standard addition method?

The standard addition method is best suited for models that include a background signal coming from a matrix. This translates into the presence of an intercept in the regression curve.

### What do I need to calculate the concentration from the calibration curve?

Every calibration curve is defined by a set of parameters: in the case of linear calibration curves, they are usually:

• The slope of the line (its angle in relation to the horizontal axis); and
• The intercept.

To find out these parameters, you need to measure the signal obtained from a set of samples with known concentrations.

Lastly, measure the response from the unknown sample: that's the final quantity you need to calculate the unknown concentration.

### Which calibration curve to use for the absorption spectroscopy technique?

Since the absorption spectroscopy technique has a constant background, you need to consider it when you build the calibration curve: the best model for this technique is the standard addition method. Find out more about it at Omni Calculator's website!

Davide Borchia An example of a calibration curve, where you can see the various parameters necessary to calculate the concentration of an unknown sample. The sensitivity a of the instrument corresponds to the slope of the line, while the background signal b is equal to its intercept.
Parameters
Sensitivity (a)
Background (b)
Signal (y)
Concentration (x)
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