**Chapter 5: Calibrations, Standardizations, and Blank Corrections**

By measuring the signal for an

unknown sample

we can use the calibration curve

To find its concentration

The equation for this line is:

The concentration of an unknown can be found by measuring its signal and solving for concentration.

Reagent

Blank

Copper Standards

**External Standards**

The slope of the line is the sensitivity

To determine the concentration, or amount, of an analyte in a sample, we must determine the relationship between the signal and concentration or amount

External standards is the simplest and most common method

Signals are obtained for a blank and standard solutions

A blank contains all the reagents and no analyte

Standards contain known concentrations of the analyte

A

calibration curve

is a plot of signal (y-axis) against concentration or amount (x-axis)

Straight line calibration curves are preferred as they are simpler to fit to a line and have constant slope (sensitivity)

External Standards

Not preferred: assumes linear relationship and any error in the standard measurement has significant effect on result

If reagent blank is measured accurately and subtracted from the standard and sample measurements (known as

blank subtraction

) the concentration of an analyte CA can be found by finding the sensitivity, k, for the standard and using it to calculate the concentration of analyte in the sample.

Single Point

The preferred method

Three or more standards and a blank are needed to accurately describe the relationship

If the calibration curve is curved (it’s called a calibration curve even when it is a straight line) we use the linear region (preferred) or fit a polynomial best-fit line

Multiple Standards

Assumes analyte in standard behaves in the similar way to analyte in samples

If analyte responds to something in the sample matrix it may give a different result, in this case the interfering matrix components should be added at a similar level to the standards (known as

matrix matching

)

Assumes we know the identity and level of interfering matrix components

Limitations

**Standard Additions**

sample

spikes

We can find this the concentration of sample (CA) by rearranging this equation to:

and using known values of the volume of sample (Vo), concentration of standard (Cstd) and x-intercept.

The x-intercept is found from the equation of the line by setting the y value equal to zero.

Used when matrix matching is not possible

A known amount of standard is added (spiked) into one or more sample solutions

Can be single addition or multiple addition (preferred)

Standard Additions

A known volume of sample (V0) is added to two flasks

Into one a known amount of standard (Vstd) of known concentration (Cstd) is added (spiked)

Both flask are diluted to the same volume (Vf)

Signals are measured for both solutions (Ssamp and Sspike)

Concentration of analyte (CA) is found by:

Single Standard Addition

Multiple standards are made by spiking increasing volumes of a standard into solutions of the sample

Volume of standard is plotted against signal

The concentration of analyte is found from the x-intercept of a straight line graph.

Multiple Standard Additions

Multiple standard additions requires a straight line graph

Assumes matric interferent acts with analyte in standard the same as analyte in sample

Uses more sample

Slow. Graph required for every sample for multiple standard additions

Limitations

Accounts for fluctuations in the sample

For example:

- Solvent evaporates causing fluctuations in the analyte concentration

- The method requires a very small volume that is difficult measure with accuracy causing fluctuations in signals

To overcome fluctuations we add a known amount of internal standard

An internal standard is not the analyte but has similar behavior to the analyte e.g. lithium might be a good internal standard for sodium (both are group I metals)

We measure the ratio of analyte signal to internal standard signal which is independent of method fluctuations

Internal Standards

We prepare a single standard with known concentration of analyte, CA, and known concentration of internal standard, CIS, and measure the signal for the analyte, SA, and internal standard, SIS. The value of K can then be determined by:

K is then used to find the concentration of analyte in a sample containing a known concentration of internal standard by measurined signals for the analyte and internal standard in the sample

Single Standard

By measuring the analyte and internal standard signals for a series of standards, we can plot a calibration curve of SA/SIS versus CA/CIS

If concentration of internal standard in samples and standards is kept constants, we can plot SA/SIS against CA

Internal standard is added to samples, SA/SIS found, and the calibration curve used to find the analyte concentration

Multiple Standards

The internal standard must have similar behavior to the analyte e.g. if the analyte reacts with a matrix component, the internal standard should react to the same extent.

Limitation

**Internal Standards**

by determining the ratio of analyte to internal standard signal in a sample we can be find using this equation to solve for x, the analyte concentration

A perfect straight line has the mathematical expression:

Due to measurement uncertainties, calibration curves are not perfect straight lines.

A linear regression is a method for fitting a straight line to a set of data in order to get good estimates for the slope and intercepts

It does this by minimizing residuals (the distances between the data point and the best-fit line in the y direction)

The R^2 value (where R is the Pearson correlation coefficient) can used as a measure of fit. The closer to 1, the better fit

Calibration curves normally:

- assume all errors affecting the x values (amount or concentration) are insignificant

- assume errors affecting y are normally distributed

- assume errors in y are independent of the value of x

Linear regression

Obtaining Analyte Concentration

A linear calibration is preferred, however a curved calibration is sometimes unavoidable.

A straight line should not be fitted to a curve

Sometimes a mathematical function can be used to generate a straight line

If this is not possible we fit a polynomial function to the curve e.g.

Solving this equation for x will give two solutions, however, normally only one will make sense.

Curvilinear and Multivariate Regression

Signal can originate from the reagents and solvent and can be corrected by subtracting the reagent blank or calibration blank

The reagent blank is the signal resulting from the reagents and solvents only i.e. in the absence of sample

The calibration blank is the y-intercept value found from the regression equation of the calibration curve

Reagent blank and calibration blank differ only by indeterminate errors

Typically analytical chemist use the calibration blank when using a calibration curve and reagent blank when using a single point standardization

Neither account for interactions between analyte and sample matrix

Blank Corrections

**Regression and Blank Correction**

Uncertainty in Regression Analysis

The error in the slope is given by:

The error in the intercept is given by:

Where:

We can use these values to provide confidence intervals:

Uncertainties are minimized by evenly spacing standards over a wide range of analyte concentrations (provided that the concentration curve is still linear)

The concentration of an analyte, CA, can be determined from the regressions equation:

To find the error in CA, we use the equation:

The confidence interval of the analyte concentration can be found by: