Principles of Instrumentation - Skoog

category is a section that treats the effects of instrumental and environmental noise on the precision and accuracy of spectroscopic absorption measurements. Also largely new is the development of general chromatographic theory from kinetic considerations. The chapter on optical instruments brings together information that was formerly found in chapters on infrared, ultraviolet and visible, and atomic absorption spectroscopy; flame and emission spectroscopy; and fluorescence and Raman spectroscopy. Since the publication of the first edition, a host of modifications of wellestablished instrumental methods have emerged and are now included. Among these new developments are Fourier transform nuclear magnetic resonance and infrared methods, infrared photometers for pollutant measurements, laser modifications of spectroscopic measurements, nuclear magnetic resonance measurem2

Z = JR2

I I

XL ~=ilfCtanR

z· o

:lIE

.JR' +(XL -Xci' (X, - Xci arc tan -R--

FIGURE 2-11 Vector diagrams for series circuits: (a) RC circuit, (b) RL circuit, (c) series RLC circuit.

2-42 and 2-44

1 = V"

, Z

= J(SO)2

0 R

:

Za .JR2+xL2

EXAMPLE

Z

----

z

~

Because the outputs of the capacitor and inductor are 180 deg out of phase, their combined effect is determined from the difference of their reactances.

(2-45)

A similar result is obtained when Equations 2-43 and 2-45 are substituted into Equation 2-41. In one sense, the capacitive and inductive reactances behave in a manner similar to that of a resistor in a circuit-that is, they tend to impede the flow of electrons. They differ in two major ways from resistance, however. First, they are frequency dependent; second, they cause the current and voltage to differ in phase. As a consequence of the latter, the phase angle must always be taken into account in considering the behavior of cir-

XL

(2-46)

For the following circuit, calculate the peak current and voltage drops across the three components.

+X~

and for the RL circuit Z

cuits containing capacitive and inductive elements. A convenient way of visualizing these effects is by means of vector diagrams. Vector Diagrams for Reactive Circuits. Because the voltage lags the current by 90 deg in a pure capacitance, it is convenient to let q, equal -90 deg for a capacitive reactance. The phase angle for a pure inductive circuit is +90 deg .. For a pure resistive circuit, q, will be equal to 0 deg. The relationship ainong Xc, XL' and R can then be represented vectorially as shown in Figure 2-11. As indicated in Figure 2-11c, the impedance of a series circuit containing a resistor, an inductor, and a capacitor is determined by the relationship

+ (40 - 2W

1" = 10 V/53.8

n = 0.186

= 53.8 n A

VR = 0.186 x SO = 9.3 V Vc = 0.186 x 20

= 3.7 V

VL = 0.186 x 40 = 7.4 V

Note that the sum of the three voltages (20.4) in the example exceeds the source voltage. This situation is possible because the voltages are out of phase, with peaks occur-

ring at different times. The sum .of the instantaneous voltages, however, would add up to 10V. High-Pass .nd Low-Pass Fikers. Series RC and RL circuits are often used as filters to attenuate high-frequency signals while passing low-frequency components (a low-pass filter) or, alternatively, to reduce lowfrequency components while passing the high (a high-pass filter). Figure 2-12 shows how series RC and RL circuits can be arranged to give high- and low-pass filters. In each case. the input and output are indicated as the volt~es (Vp)i and (Vp) •. ; In order to employ an RC circuit as a high-pass filter, the' output voltage is taken across the resistor R. The peak current in this circuit is given by Equation 2-38. That is, I _ I' -

(Vp);

JR

2

+ (l/wC)2

Since the voltage drop across the resistor is in phase with the current,

1 = (VI'). ,

R

The ratio of the peak output to the peak input voltage is obtained by dividing the first equation by the second and rearranging. Thus,

A plot of this ratio as a function of frequency is shown in Figure 2-13a; here, a resistance of 1.0 x 10· n and a capacitance of 0.10 pF was employed. Note that frequencies below 50 Hz have been largely removed from the input signal. It should also be noted that because the input and output voltages are out of

resistor -behavior just opposite of that of the RC circuit. Curves similar to Figure 2-13 are obtaincd for RL filters. Low- and high-pass filters are of great importance in the design of electronic cirCUits.

~-----~

Resonant Circuits A resonant or RLC circuit consists of a resistor, a capacitor, and an inductor arranged in series or parallel, as shown in Figure 2-14. Series Resonant Filters. The impedance of the resonant circuit shown in Figure 2-14a is given by Equation 2-46. Note that the impedance of a series inductor and capacitor is the difference between their reactances and that the net reactance of the combination is zero when the two are identical. That is, the FIGURE 2-12 Filter circuits: (a) Highpass RC filter, (b) low-pass RC filter, (c) highpass RL filter, and (d) low-pass RL filter.

~ ~ ~ phase, the peak voltages will occur at different times. For the low-pass filter shown in Figure 2-12b, we may write

::

0." 0.2 0.0 1

100

(al

Substituting Equation 2-42 gives

(Vp). = Ip/(roC)

(VP)i

1

O.B

=

roCJR2

+ (l/roC)2

1

~

0.6

"i

;0-

0.4

= j(roCR)2 + 1 (2-48)

Figure 2-13b was obtained with this equation. As shown in Figure 2-12, RL circuits can also be employed as filters. Note, however, that the high-pass filter employs the potential across the reactive element while the low-pass filter employs the potential across the

1

1.= 21tJfC

FIGURE 2-14 Resonant circuits employed as filters: (a) series circuit and (b) parallel circuit.

0.6

(Vp). = IpXc

(Vp).,

Upon rearrangement, it is found that

(bl

Frequency. Hz

Multiplying by Equation 2-38 and rearranging yields

1 2nf.C = 2nf.L

1.0 O.B

Frequency. Hz (b)

FIGURE 2-13 Frequency response of (a) the high-pass filter shown in Figure 2-118 and (b) the low-pass filter shown in Figure 2-12b; R = 1.0 X 104 nand C = 0.10 JLF.

energy stored in the magnetic field of the inductor during the other. Thus, during a one-half cycle, the energy of the capacitor forces the current through the inductor; in the other half cycle, the reverse is the case. In principle, a current induced in a closed resonant circuit would continue indefinitely except for the loss resulting from resistance in the leads and inductor wire. The resonant frequency I. can be readily calculated from Equations 2-42 and 2-43. That is, whenf=f., XL = Xc, and

only impedance in the circuit is that of the resistor or, in its absence, the resistance of the inductor coil and ·the other wiring. This behavior becomes understandable when it is recalled that in a series RC circuit, the potentials across the capacitor and the inductor are 180 deg out of phase (see the vector diagrams in Figure 2-11). Thus, if Xc and XL are alike, there is no net potential drop across the pair. Electricity continues to flow, however; therefore, the net reactance for the combination is zero. Also, when XL and Xc are not identical, their net reactance is simply the difference between the two. The condition for resonance is XL= Xc

Here, the energy stored in the capacitor during one half cycle is exactly equal to the

The follQwing example will show some of the properties of a series resonant circuit.

i EXAMPLE Assume the following values for the components of the circuit in Figure 2-14a: (Vp)i = 15.0 V (peak voltage), L= 100 mH, R = 20 n, and C = 0200 JLF. (a) Calculate the peak current at the resonance frequency and at a frequency 75 Hz greater than the resonance frequency. (b) Calculate the peak potentials across the resistor, inductor, and capacitor at the resonant frequency. (a) We obtain the resonant frequency with Equation 2-49. Thus, 1

1.;= 2nJl00

x 10

3

x 0.2 x 10

6

= 1125 Hz

By defidition, XL = Xc at!.; therefore, Equation 2-46 becomes

To calculate I, at 1200 Hz, we substitute Equations 2-42 and 2-43 into Equation 2-46. Thus, Z = =

JR

+ [27tjL -

J202

+ [27t x

2

(1/27tfC)j2 1200 x 100 x 10

- 1/(27t x 1200 x 0.200 x lO-ti)j2 = J400

+ (754 -

663)2 = 93 0

and I, = 15.0/93 = 0.16 A. (b) At f. = 1125 Hz, we have found that I,= 0.75 A. The potentials across the resistor, capacitor, and inductor (V,)Il' (V,)L, and (J-;,)c are

(V')ll

= 0.75

x 20

= 15.0 V

= 0.75

1050 Frequency,

(V,)L = I,XL X

27t x 1125 x 100 x 10-3

= 530 V

Hz

1100 . Frequenc:y. Hz

FIGURE 2-15 Frequency response of a series resonant circuit; R = 20Q,L = l00mH, C 0.2 IlF, and (J-;,), 15.0 V.

FIGURE 2-16 Ratio of output to input voltage for the series resonant filter; L = 100 mH, C = 0.20 IlF, and (v,,)i = 15 V.

If the output of the circuit shown in Figure 2-14a is taken across the inductor or capacitor, a potential is obtained which is several times larger than the input potential (see part b of the example). Figure 2-16 shows a plot of the ratio of the peak voltage across the inductor to the peak input voltage as a function of frequency. A similar plot is obtained if the inductor potential is replaced by the capacitor potential. The upper curve in Figure 2-16 was obtained by substituting a 10-0 resistor for the 20-0 resistor employed in the example. Parallel Resonant Filters. Figure 2-14b shows a typical parallel resonant circuit. Again, the condition of resonance is that Xc = XL and the resonant frequency is given by Equation 2-49. The impedance of the parallel circuit is given by

filter (Equation 2-46). Note that the impedance in the latter circuit is a minimum at resonance, when XL = Xc. In contrast, the impedance for the parallel circuit at resonance is a maximum and in principle is infinite. Consequently, a maximum voltage drop across (or a minimum current through) the parallel reactance is found at resonance. With both parallel and series circuits at resonance, a small initial signal causes resonance in which electricity is carried back and forth between the capacitor and the inductor. But current from the source through the reactance is minimal. The parallel circuit, sometimes called a tank circuit, is widely used as for tuning radio or television circuits. Tuning is ordinarily accomplished by adjusting a variable capacitor until resonance is achieved.

=

=

Since XL = Xc at resonance, f

Part (b) of this example shows that, at resonance, the peak voltage across the resistor is the potential of the source; on the other hand, the potentials across the Inductor and capacitor are over 35 times greater than the input potential. It must be realized, however, that these peak potentials do not occur simultaneously. One lags the current by 90 deg and the other leads it by a similar amount. Thus, the instantaneous potentials across the capacitor and inductor cancel, and the potential drop across the resistor is then equal to the input potential. Clearly, in a circuit of; this type, the capacitor and inductor may have to withstand considerably larger potenti;als than the amplitude of the input voltage would seem to indicate. Figure 2-15, which was obtained for the circuit shown in the example, demonstrates that the output of a series resonant circuit is a narrow band of frequencies, the maximum of which depends upon the values chosen for L and C.

Z =

JR2 + (XC-XL XLXC

Behavior of RC )2

(2-50)

It is of interest to compare this equation with the equation for the impedance of the series

Circuits with Pulsed Inputs When a pulsed input is applied to an RC circuit, the voltage outputs across the capacitor and resistor take various forms, depend-

ing upon the relationship between the widtp of the pulse and the time constant for the circuit. These effects are illustrated in Figure 2-17 where the input is a square wave having a pulse width of 1",' seconds. The second column shows the variation in capacitor potential as a function of time, while the third column shows the change in resistor potential at the same times. In the top set of plots (Figure 2-17a~ the time constant of the circuit is much greater than the input pulse width. Under these circumstances, the capacitor can become only partially charged during each pulse. It then discharges as the input potential returns to zero; a sawtooth output results. The output of the resistor under these circumstances rises instantaneously to a maximum value and then decreases essentially linearly during the pulse lifetime. The bottom set of graphs (Figure 2-17c) illustrates the two outputs when the time constant of the circuit is much shorter than the pulse width. Here, the charge on the capacitor rises rapidly and approaches full charge near the end of the pulse. As a consequence, the potential across the resistor rapidly decreases to zero after its initial rise. When V;goes to zero, the capacitor discharges immediately; the output across the resistor peaks in a negative direction and then quickly approaches zero. These various output wave forms find applications in electronic circuitry. The sharply peaked voltage output shown in Figure 2-17c is particularly important in timing and trigger circuits.

SIMPLE ELECTRICAL

MEASUREMENTS

This section describes selected methods for the measurement of current, voltage, and resistance. More sophisticated methods for measuring these and other electrical properties will be considered in Chapter 3 as well as in later chapters.

Meter

The Ayrton Shunt. The Ayrton shunt, shown in Figure 2-19, is often employed to vary the range of a galvanometer. The example that follows demonstrates how the resistors for a shunt can be chosen.

f1o-------AJV:J

'" :\,

'-

..!!..'Aft

~ Shunt

(b)

RCO!!T.

(el RC~T.

T+l .JnLJnLJnL T+ !7l J Vi 0

1..

o

t

Vc 0

1..

--Time-

:Tl iT1

LJ U

0 --T~_

L

_

T+~' 1.._

I

-,

I

V" 0

o

-,

I

-

--Time---"

FIGURE 2-17 Output signals VR and Vc for pulsed input signal J-l; (a) time constant ~ pulse width Tp; (b) time constant ~ pulse width; (c) time co~tant yemploying a master grating as a mold for' the production of numerous plastic replicas; the products of this process, while inferior in performance to an original grating, are adequate for many applications. When a transmission grating is illuminated from a slit, ~ch groove scatters radiation and thus effedtively becomes opaque. The nonruled portions then behave as a series of closely spaced slits, each of which acts as a new radiation source; interference among the multitude of beams results in diffraction of

Figure 5-10 illustrates one ofthe important advantages of grating mopochromatorslinear dispersion along the focal plane. Resolving Power of Prism Monocbromators. The resolving .power R of a prism gives the limit of its ability to separate adjacent images having slightly different wavelengths. Mathematically, this quantity is defined as

where dl represents the wavelength difference that can just be resolved and l is the average wavelength of the two images. It can be shown that the resolving power of a prism is directly proportional to the length of the prism base b (Figure 5-8) and the dispersion of its construction material. That is,

Gl'IIting Monochromatora for Wavelength Selection Dispersion of ultraviolet, visible, and infrared radiation can be brought about by passage of a beam through a transmission grating or by

200 ~. om

I

Gl'lting 500

. 300

I

I

Glass prism

350

400

450

500

A-,j\il9£~;;; -.-,

I

I

I

200 X,om

600 800

I I II

Quartz prism

A,nm

200

250

300

I

I

I

350 400

I

A

'!

,

500600800

I

II 8

I

0

5.0

10.0 15.0 Diltance y along focal plane, em

20.0

25.0

FIGURE 6-10 Dispersion for three types ofmonochromators. The points A and B on the scale correspond to the points shown in Figure 5-7.

Monochromatic

beam It incident ongtel .

. FIGURE 6-11 Schematic diagram illustrating the mechanism of diffraction from an echellette-type grating. (From R. P. Bauman, Absorption Spectroscopy. New York, Wiley, 1962, p. 65. With permission.)

the radiation as shown in Figure 4-00 (p. 98). The angle of diffraction, of course, depends upon the wavelength. Reflection gratings are used more extensively in instrument construction than their transmitting counterparts. Such gratings are made by ruling a polished metal surface or by evaporating a thin film of aluminum onto the surface of a replica grating. As shown in figure 5-11, the incident radiation is reflected from one of the faces of the groove, which then acts as a new radiation source. Interference results in radiation of differing wavelengths being reflected at different angles r. Diffraction by a Grating. figure 5-11 is a schematic representation of an echellette-type grating, which is grooved or blazed such that it has relatively broad faces from which reflection occurs and narrow unused faces. This geometry provides highly efficient diffraction of radiation. Each of the broad faces can be considered to be a point source of radiation; thus interference among the

reflected beams 1,2, and 3 can occur. In order for constructive interference to occur between adjacent beams, it is necessary that their path lengths differ by an integral multiple n of the wavelength l of the incident beam. In Figure 5-11, parallel beams of monochromatic radiation 1 and 2 are shown striking the grating at an incident angle i to the grating normal. Maximum constructive interference is shown as occurring at the reflected angle r. It is evident that beam 2 travels a greater distance than beam 1 and that this difference is equal to (CD - AB). For constructive interference to occur, this difference must equal n).. That is, M = (CD - AB) Note, however, that angle CAD is equal to angle i and that angle BDA is identical with angle r. Therefore, from simple trigonometry, we may write

where d is the spacing between the reflecting surfaces. It is also seen that AB= -dsin

r

The minus sign, by convention, indicates that reflection has occurred. The angle r, then, is negative when it lies on the opposite side of the grating normal from the angle i (as in Figure 5-1I~ and is positive when it is on the same side. Substitution of the last two expressions into the first gives the condition for constructive interference. Thus, III =

d(sin i + sin r)

(5-5)

Equation 5-5 suggests that several values of.t exist for a given diffraction angle r. Thus, if a first-order line (D = I) of 800 nm is found at r, second-order (400 nm) and third-order (267 nm) lines also appear at this angle. Ordinarily, the first-order line is the most intense; indeed, it is possible to design gratings that concentrate as much as 90% of the incident intensity in this order. The higherorder lines can generally be removed by filters. For example, glass, which absorbs radiation below 350 nm, eliminates the higherorder spectra associated with first-order radiation in most of the visible region. The example which follows illustrates these points. EXAMPLE An echellette grating containing 2000 blazes per millimeter was irradiated with a polychromatic beam at an incident angle 48 deg to the grating normal. Calculate the wavelengths of radiation that would appear at an angle of reflection of +20, + 10,0, and -10 deg (angle r, Figure 5-11). To obtain d in Equation 5-5, we write Imm nm nm d=---x 106=500-2000 blazes mm blaze When r in Figure 5-11 equals + 20 deg , SOO(. 48 . 2) 542.6 A=sm +sm 0 =-D

n

and the wavelengths for the first-, secondo, and thUd-order reflections are 543, 271, and 181 nm, respectively. Similarly, when r = -10 deg (here r lies to the right of the grating n0"ttal)

RfIOIviDg Power of a Gntiac- It can be shown 7 that the resolving power R of a grating is given by the very simple expression

.t = SOO[sin 48 + sine-10)] = 284.7

where n is the diffraction order and N is the number of lines illuminated by the radiation from the entrance slit. Thus, as with a prism (Equation 5-4~ the resolving power depends upon the physical size of the dispersing element.

.t

R = .u = aN

(5-7)

For the grating, we employ Equation 5-7

.t -=aN .u For the first-order spectrum (D = I) N

n

n

Wavelength (nm) for r

20 10 0 -10

n=1 543 458 372 285

n=2 271 229 186 142

n=3 181 153 124 95

A concave grating can be produced by ruling a spherical reflecting surface. Such a diffracting element serves also to focus the radiation on the exit slit and eliminates the need for a lens. Dispersion fJl GntiDp. The angular dispersion of a grating can be obtained by differentiating Equation 5-5 while holding i constant; thus, at any given angle of incidence

EXAMPLE ~mpare the size of (I) a 6O-deg fused silica' prism; (2) a 6O-deg glass prism; and (3) a grating with 2000 lines/mm, that would be required to resolve two lithium emission lines at 460.20 and 460.30 nm. Average values for tlje dispersion (dn/d.t) of fused silica and glass ~n the region of interest are 1.3 x 10- 4 and 3.6 x 10-4 nm-I, respectively. For the two prisms, we employ Equation 5-4

.t

.u =b-d)'

.t

I

.u

dn/d.t

b-- x-dr D -=-d)' dcosr

(5-6)

Note that the dispersion increases as the distance d between rulings decreases or as the number of lines per millimeter increases. Over short wavelength ranges, cos r does not change greatly with .t, so that the dispersion of a grating is nearly linear. By proper design of the optics of a grating monochromator, it is possible to produce an instrument that for all practical purposes has a linear dispersion of radiation along the focal plane of the exit slit. Figure 5-10 shows the contrast between a grating and a prism monochromator in this regard.

dn

R=-

460.25

I

------

460.30 - 460.20

x--

dn/d)'

For fused silica, b = 460.25 nm x I x 10-7 em 0.10 nm 1.3 x 10-4 nm-I om = 3.5 em and f