1

CHEM 465

Instrumental Analytical Chemistry

Important links: http://pubs.acs.org/journal/ancham

The Analytical Process

Sample

Stimulus

Electrons, photons, atoms, molecules, ions, heat

Qualitative analysis

Quantitative analysis

Heat, ions, molecules, atoms, photons, electrons

Response

Instruments and Components

The Physical and Chemical Domain

The Analyst’s Domain

Instrument Encodes

Data Transformation

Energy Source Transduced Information

Instrument (stimulus) Information Transducer Information Processor Readout

Photometer W lamp attenuated light photocell electrical meter scale current

beam current

Atomic flame UV/VIS PMT electrical chart recorder/computer

Emission radiation potential

2

Transducers

All modern instrumentation employs data conversion between at least three domains and often more. Each domain transformation is accomplished by a transducer.

Input Transducer — converts data from a non-electrical domain to an electrical domain.

Output Transducer — converts data from an electrical domain to a non-electrical domain.

A thermocouple generates a specific voltage at a certain temperature. It is a temperature-to-voltage input transducer.

A stepper motor running a pen on a recorder moves the pen in response to a current flow. It is a current-to-position output transducer.

Example: The pH Meter

A similar situation arises when we want to monitor and record the hydrogen ion activity (pH) of a solution and how it changes with time.

The first transducer is a pair of electrodes; one at a fixed pH and the other sampling the unknown solution. Together they produce a voltage difference. This is amplified and turned into a current to drive the pen displacement motor on a chart recorder, permanently recording the changes in pH with time.

Amplifier V - I

pH

V I Recorder

Example: Fluorometer

3

Information in the analog domain Information in time-domain

Desirable Characteristics for an Analytical Method

Numerical Criteria for Selecting an Analytical Method: Analytical Figures of Merit

Precision Concentration Range Absolute standard deviation limit of quantitation (LOQ) Relative standard deviation limit of linearity (LOL) coefficient of variation variance

Bias Selectivity absolute systematic error effects of interferences

relative systematic error coefficient of selectivity Sensitivity

calibration analytical

Detection limit blank + 3stdev of blank

A figure of merit is a number which has been derived experimentally for a given analytical instrument or technique that permits an evaluation or comparison of the technique to assess its applicability to a particular analysis problem

Table 1-3

4

Precision

• Mutual agreement of replicate measurements. The standard deviation and the variance are the most common measurements of a set of data’s precision. It is a result of random errors.

Accuracy Arises from the presence of determinate errors, or non-random errors. This shifts the measured mean value of a set of measurements away from the true value and is referred to as the error of the mean.

Three types of such errors:

Instrumental: something wrong with the instrument (batteries low, temperature effects the circuitry, calibration errors, etc.

Personal: reading the meter from the wrong angle, lack of careful technique.

Method: often a result of non-ideal chemical behaviour; slow reactions, contaminants, instability of reagents, loss of analyte by adsorption. Must use guaranteed standards (NIST).

Sensitivity • A technique’s ability to detect changes in the signal property.

• How much does the signal change for a change in the measured variable?

Two factors dictate a technique’s sensitivity: 1.  Slope of calibration curve. 2. Precision or reproducibility of measurement.

High Sensitivity

Low Sensitivity

High Precision = High Sensitivity

Low Precision = Low Sensitivity

Calibration Sensitivity

• Slope of calibration curve (most curves are made linear).

S = m C + Sbl signal

slope concentration

Blank signal (y-intercept)

• m is the calibration sensitivity.

• precision is not accounted for

5

Analytical Sensitivity

• Incorporates precision

γ = m/s Standard deviation of measurement

Slope of calibration response

Analytical sensitivity factor

• Not affected by amplification. Increase in gain, increases m and s by similar amount.

• Independent of measurement units but does depend upon concentration since ‘s’ can vary with concentration.

Signal(s), Precision, LOD, LOQ

Measured Signal Level 0

Mean Background Signal Level

Distribution of blank measurements

Detection Limit

3 sbl

Quantitation Limit

10 sbl

Detection Limit

• The smallest amount of analyte that can be reliably detected.

• Depends upon signal/noise ratio.

• Analysis signal must be larger than blank signal. How much larger?

Sd = Sbl + k sbl Minimum distinguishable analytical signal Mean blank signal

Standard deviation on blank signal

Usually taken to be 3

Quantitation Limit

The detection limit answers the question “Is this analyte present or not?” However, to actually answer the question “How much of the analyte is present?” requires a still larger signal.

The widely accepted level at which the analyte can be quantified is TEN times the standard deviation. (Detection is THREE times.)

Sq = Sbl + 10 sbl

6

Linearity Limit

As the concentration or intensity increases, at some point, every detector stops responding linearly. This identifies the upper limit of concentration to which the technique can be successfully applied.

Its origin can be electrical (the amplifier cannot produce a larger output voltage) or mechanical (the balance arm breaks under this load) in nature.

Instr

umen

t res

pons

e

Concentration

Linearity Limit

Dynamic Range

This is the region between the Quantitation Limit (LOQ -Limit of Quantitation) and the Linearity Limit (LOL - Limit of Linearity). This is the range over which the technique is useful.

To be viewed as a worthwhile, a technique should have a dynamic range of at least two orders of magnitude. Many techniques have a dynamic range of five to six orders of magnitude.

Selectivity • In every analysis, we look for a signal that comes from a specific

analyte.

• In every analysis, we obtain a signal that has a contribution from everything that is present in the sample.

• We need to minimize contributions from other species and be certain that they are negligible, or else account for their contribution by determining their selectivity coefficient.

Stotal = R ai + (R Σ kij aj+…)

Total Signal Signal of Analyte species i

Signal of other species j when in the presence of species i

Selectivity Coefficient for the detection j when trying to detect I.

Activity of each species

Signal

In all experiments, there is a signal which is derived from the output of the detector :

Sample Response: the instrument’s response when the analyte is present.

Blank Response: the instrument’s response when the analyte is absent.

The Signal: the difference between the sample and the blank response.

time

outp

ut v

olta

ge

blank

sample sample

blank

signal

7

Background or Baseline Ideally, the blank response of an instrument would be exactly 0. Then the sample response would be equal to the signal. This is never the case, though it can often be adjusted to be close to 0. There is always a residual signal associated with an instrument’s blank response. This is called the background or the baseline.

time

outp

ut v

olta

ge

blank

sample sample

blank

signal

baseline

The baseline is subtracted from both the blank and the sample response.

Drift Ideally, the baseline response is constant in time. In such a case, a constant correction factor is easily subtracted from the blank and sample to correct the signal. Invariably, however, the baseline changes slowly with time. This is called drift. Sometimes the drift is linear in time, but often it is more complex and difficult to predict.

time

outp

ut v

olta

ge

blank

sample

sample

blank signal

baseline

We need to know the value of the baseline at the time we make a measurement.

Noise • Noise is a random time-dependent change in the instrument’s output signal that is unrelated to the analyte response. These variations will tend to make the accurate measurement of sample, blank and baseline response less certain

• Noise arises from many sources. The frequency response can span the entire spectrum (see Figure 5-3)

• Measuring the intensity of the noise and comparing it to the signal is the key to determining the accuracy of a measurement and in specifying the smallest signal level one is able to measure (detection limit)

Signal-to-Noise

The determination of the magnitude of the analytical signal level requires measuring the difference between the background and the sample signal. This measure is blurred by the presence of noise. One has to account for both the signal level and the noise level in arriving at this measure.

Because of this, the important quantity is not the signal level alone nor is it the noise level alone; rather it is the ratio of the two that dictates the measurability of the signal level. This is the signal-to-noise ratio or simply S/N.

S N

mean standard deviation

x s

= =

8

100 photons A

B

SNR = 10

10,000 photons SNR ~ 100

Signal-to-Noise Peak-to-peak Noise

One measure of the amplitude of a sine wave is the peak-to-peak amplitude (this is twice the amplitude which appears in the defining equation for a sine wave).

Noise is usually specified by measuring the peak-to-peak maximum over a reasonable length of time (“reasonable” depends upon length of time needed to make a measurement).

-4

-3

-2

-1

0

1

2

3

4

0 2 4 6 8 10 12 14 16 18 20

V(peak-to-peak)

or

Vp-p

p-p Noise 2

Even though the noise is clearly not a perfect sine wave, we know it can be decomposed into a collection of sine waves and we can treat it mathematically as a sine wave.

-5

-4

-3

-2

-1

0

1

2

3

4

5

0 2 4 6 8 10 12 14 16 18 20Vp-p

Root-Mean-Square Noise For a dc signal, the magnitude of the noise, N, is defined as the standard deviation of numerous measurements of the signal strength

€

N =xi − x( )2

i∑

n −1

9

Signal-to-Noise Ratio Neither the total signal level nor noise level determine an experiment’s ability to accurately detect an analyte. Rather it is the ratio of the two that is critical. The S/N Ratio.

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4 5 6 7 8

S = 0.75

Baseline = 0.25

N = 0.035

S/N = (0.75-0.25)/0.035 = 14

Signal-to-Noise Ratio cont’d

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 1 2 3 4 5 6 7 8

Same signal level. Same baseline. S/N = 3.

Signal-to-Noise Ratio cont’d

0

0.5

1

1.5

2

2.5

3

3.5

0 1 2 3 4 5 6 7 8

In this experiment, the signal-to-noise is 1. Note how you could not make a reasonable measurement of the signal under these conditions.

100 photons A

B

SNR = 10

10,000 photons SNR ~ 100

Signal-to-Noise cont’d

10

400 600 800 1000 1200 1400 1600 1800 2000

SNR=17

SNR= 5

SNR=2.8

SNR < 2

100 ms, 50 um slit

10 ms, 50 um

10 ms, 20 um

10 ms, 10 um

Raman shift, cm-1

Raman spectra of Calcium ascorbate Types of Noise When sample is abundant, signal is high, background (baseline) is low, we hardly worry about noise. But at some point, every experiment needs to account for noise. Electrical noise can be divided into four principal sources:

• Thermal Noise (Johnson noise)

• Shot Noise

• 1/f Noise

• Environmental Noise (see Figure 5-3)

Thermal Noise Also known as white noise, Johnson noise, or Nyquist noise.

• Random motions of charge carriers buffeted by thermal motions of a solid lattice of atoms

• Arises because the atoms of a solid state conductor are vibrating at all temperatures and they bump into conductors (electrons). This imposes a new, random motion on those conductors which generates noise.

Where

Vnoise, rms is the root-mean-square noise voltage

kB is Boltzmann’s constant = 1.38 x 10-23 J K-1 (V2 s Ω-1 K-1)

T is the temperature in kelvin

R is the resistance in ohms

B is the bandwidth in Hz (s-1)

Myth: thermal noise is frequency dependent

Fact: thermal noise is white noise, analogous to white light

Vnoise,rms = 4kB T RB

Cooling to reduce Thermal Noise

Calculate the rms noise voltage of a 10 kΩ resistor at 25˚ C as it is amplified by an audio amplifier (bandwidth 15 kHz) so we can measure the voltage. What is the rms noise voltage of the resistor if it were cooled to 77K? To liquid helium (4.2 K)?

Cooling has dropped the noise originating in the resistor. We have (incorrectly) ignored noise in the amplifier itself.

V noise , rms T = 298 K ( ) = 4 k B T R B

= 4 1 . 38 x 10 - 23 ( ) 298 ( ) 10 4 ( ) 1 . 5 x 10 4 ( )

= 2 . 43 x 10 - 12 = 1 . 56 x 10 - 6 V = 1 . 56 µ V

V noise , rms T = 77 K ( ) = 0 . 80 µV

V noise , rms T = 4 . 2 K ( ) = 0 . 19 µV

11

Bandwidth Instruments respond to signal changes differently. The bandwidth or bandpass refers to the range of frequencies over which the instrument can effectively measure signals. Usually the bandwidth of an instrument can be adjusted by changing electronic filters.

Center Frequency

Bandwidth

Frequency

Sign

al S

treng

th

A simple RC circuit can be configure to act as a low pass filter, smoothing rapid changes. The low-pass RC filter allows slowly varying signals to pass “unimpeded.” The relationship between the RC time constant τ = RC and its bandwidth B is simply

B ≈ 1/4τ

Low Pass Filter

Low Pass Filter

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1 10 100 1000 10000 100000 1000000

Frequency (Hz)

Filte

r Att

entu

ation F

acto

r

A series RC circuit functions as a low pass filter, when the signal is taken as the output voltage across the capacitor. Then ac signals at low frequency pass “unattenuated.”

R

C

Vin

Vout

VoutVin

=XC

R2 + XC2

RC Filter - Tutorial

High Pass Filter

High Pass Filter

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1 10 100 1000 10000 100000 1000000

Frequency (Hz)

Filter

Att

enuat

ion F

acto

r

A series RC circuit functions as a high pass filter, when taking the output voltage across the resistor. Then ac signals at high frequency pass “unattenuated.”

R

C

Vin Vout

VoutVin

=R

R2 + XC2

RC Filter - Tutorial

Thermal Noise Reduction by Bandwidth

Consider a 10 kΩ resistor at room temperature. Pass a signal through a noiseless RC circuit (impossible, since the R in this new circuit will introduce noise, but let’s pretend, O.K.?) which has a time constant of 0.1 s. What is the expected rms noise from this filtered signal?

Noise reduction by filtering was much greater than by cooling, but we are now much more limited to the speed with which we can make a measurement and hence the rates of processes we can monitor.

Vnoise,rms T = 298K,B = 2.5s−1( )= 4 1.38 ×10−23( ) 298( ) 104( ) 2.5( )

= 2.0 ×10−8V = 20nV

B = 1/(0.1 x 4) = 2.5s-1

12

Shot Noise

Arises because charge and energy are quantized. Electrons and photons leave sources and arrive at detectors as quanta; while the average flow rate may be constant, at a given instant there are more quanta arriving than at another instant. There is a slight fluctuation because of the quantum nature of things.

q is the electron charge = 1.602 x 10-19 C

Idc is the dc current flowing across the measurement interface

B is again the measurement bandwidth in Hz

Inoise,rms = 2q Idc B

Reducing Shot Noise

What is the shot noise for a 1 amp dc current for a 15 kHz measurement bandwidth? What is it when the bandwidth is reduced to 2.5 Hz?

Again a lower noise level comes at the expense of only being able to measure slow enough processes.

Inoise,rms B = 15kHz( )

= 2 1.602 ×10−19( ) 1( ) 1.5 ×104( )= 6.9 ×10−8 A = 69nA

Inoise,rms B = 2.5Hz( ) = 8.9 ×10−10 A = 890 pA

1/f Noise

Also known as flicker noise or pink noise.

Origins are uncertain. Depends upon material, design, nature of contacts, etc. Flicker noise is determined for every measurement device. It is recognized by its 1/f dependence. Most important at low frequencies (from dc to ~200 Hz).

Long term drift in all instruments comes from flicker noise.

Measurements taken above 1 kHz can neglect flicker noise.

A narrow bandwidth makes flicker noise seem constant over that bandwidth and so it is indistinguishable from white noise.

Modulation Flicker noise, because of its 1/f behaviour, is particularly unforgiving when attempting to amplify dc signals. This is remedied by modulating the signal to a higher frequency, then amplifying, and demodulating.

Noise with a frequency characteristic different from that for the modulation-demodulation process is averaged to zero.

Two important solutions are:

• Chopper Amplifier

• Lock-in Amplifier

13

Chopper Amplifier

An input dc signal is turned into a square wave by alternately grounding and connecting the input line. This square wave is amplified and then synchronously demodulated and filtered to give an amplified dc signal that avoids flicker noise.

0

6 mV 6 mV 6 V

3 V 1500 mV

1000 x Amplifier

input output

Gain = 1500/6 = 250

Lock-in-amp - Optical Spectroscopy

Optical spectroscopy can take advantage of the lock-in technique. Using mirrors, a light source directs its emissions down two channels. Each is chopped by a rotating mechanical blade (much like a fan), producing a square wave modulation. These modulated beams produce the signal and reference that enters the four quadrant multiplier.

Light source

Chopper Assembly

Detector

Detector Sample

Monochromator

Lock-in Amplifier

Software Methods

Computers have dramatically changed the way with which we deal with noise. Many of these can help “pull the signal out of the noise”.

• Software “low pass filtering”

• Ensemble averaging

• Fourier Transform filtering

Software-based Low Pass Filtering

XPS of Au-Nanocrystals on Silicon

0

50

100

150

200

70 75 80 85 90 95 100 105 110 115

Binding Energy (eV)

Counts

An X-ray Photoelectron spectrum (XPS) of Au nanocrystals attached to a silicon surface by 3-mercaptopropyl-trimethoxysilane.

Au 4f

Si 2p

S/N = 29 on Si peak at 100 eV.

14

Software-based Low Pass Filtering

XPS - 5 Point Moving Average

0

50

100

150

200

70 75 80 85 90 95 100 105 110 115

Binding Energy (eV)

Counts

A 5 point moving average to smooth the data.

S/N = 53 on Si peak at 100 eV. Noise is decreased but so is peak amplitude. Peaks are broadened too.

Signal-to-noise enhancement via Ensemble Averaging

Noise is randomly distributed but signal is not. If we do an experiment a second time, the signal appears in the same place, but the noise will be doing something different. If we add two runs together, the signal increases, but the noise tends to smooth itself out. Signal increases as N but noise increases as √N. Hence, the S/N increases as √N.

€

Sx=

Sii=1

n

∑n

mean of the acquired signals

rms =Sx −Si( )2

i=1

n

∑n

the noise!

SN

=Sx

Sx −Si( )2

i=1

n

∑n

now, let's see what happens if we multiply by nn

SN

=nSx

nn

Sx −Si( )2

i=1

n

∑n

= n Sx

Sx −Si( )2

i=1

n

∑

i.e. we can improve the S/N by 10 if we go from 1 measurement to 100!

Signal-to-noise enhancement via Ensemble Averaging cont’d

400 600 800 1000 1200 1400 1600 1800 2000

Raman shift, cm-1

0.1 sec*

1.0 sec

30 sec

* spectra were accumulated for period indicated

Signal averaging improves S/N

15

400 600 800 1000 1200 1400 1600 1800 2000 Raman shift, cm-1

SO4-2

0.1 sec

10 sec

100 counts

1000 counts

0.1 M Na2SO4

Signal averaging is often required to increase signal above noise