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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
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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
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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
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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
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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
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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
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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).
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-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.
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-1
0
1
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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
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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
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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
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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
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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
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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
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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.
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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
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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