Precision of Fixed -Time vs. Fixed-Count Measure me nts SIR: A comparison of the precision for fixed-time and fixed-count measurements of sample minus background count rates shows that the standard deviation for fixed-time operation is always less than or equal to the standard deviation for fixed-count operation, where the sample plus background counting time is the same for both types of operation. The statistic of interest in counting problems is Zs - ZB where I s is the true count rate for the sample and ZB is the true count rate of the background. Two common modes of counting operation are fixed-time and fixedcount. In fixed-time operation, time devoted to counting the sample, ts, is the same as the time for counting background, t p . Hence, ts = t~ = t/2 where t is the total counting time, ts tB. For fixed-count operation, background is counted until the number of background counts, NB’, is equal to the number of sample counts, Ns’. The count rate corrected for background for fixed-time operation is (Ns/ts) (NB/tB) and for fixed-count operation,
+
(NS’/tS’) - ( N B ’ / t B ’ ) . Birks and Brown (1) discuss the precision of fixed-time us. fixed-count measurements. In their analysis of the problem, the number of fixed-time sample counts, Ns, was set equal to the number of fixed-count sample counts, Ns’. This is equivalent to setting only the length of time to obtain the sample counts equal for the two modes of operation, ts = ts’. A fair comparison should be made on the basis of setting the total counting time for sample plus background equal for the two techniques, ts t E = ts’ tB’ = t. Normally a selection between two techniques is made by choosing the method giving the most information (smaller standard deviation) for the same cost (time). Hence, the selection of fixed-count or fixed-time operation should be made on the basis of choosing the technique with the smaller variance (standard deviation squared) where the total counting time is the same for both techniques, t = t’.
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For fixed-time operation, the measurement of sample minus background intensity is (Ns/ts) - ( N B / ~ B ) . The variance of this quantity is QFT‘
=
[E(NS)/tS’I
+
Similarly, the variance of the sample minus background intensity for the fixed-count technique is ~ F C ’=
t8
+
tB
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(Is/ts)
=
tS’
+
tB’,
ts
+ (IB/~B)
ANALYTICAL CHEMISTRY
(1)
- t8’
=
b’ -
fB
IB 5 I s
(2)
(3)
(4)
Multiplying both sides of Equation 4 by equal quantities from Equation 3 gives (ts
LEIS
IsE(ts’) = I B E ( ~ B ’ )
+
tSIB
+
5 ~B’Is
LS‘IE
(5)
Since Kt/2)
+
(t/2)’
- is’) ( 6 ) ~= ’ t - t5’,
L ts’(t
Since ts = t B = t/2 and t substituting these values into Equation 6 gives tStB
1 ts’tB’
(7)
Dividing the left-hand side of Equation 5 by t s t B and dividing the right-hand side by a smaller quantity t&”’ gives
+ ( I s h ?I ( I s l t s ’ ) + ( I B / ~ B ’ )(8)
From Equation 1 the left-hand side of Equation 8 is U F T ~and from Equation 2 the right-hand side is U F C ~ . Hence, UFT
5
UFC
(9)
The standard deviation for fixed-count operation is always greater than or equal to the fixed-time standard deviation. For an equal amount of total counting time, the fixed-time method is the more precise technique. Conversely, for a specified precision, the expected total counting time required for the fixed-time method will be less than or equal to the expected total counting time required for the fixedcount method. For example, suppose I S = 500 C.P.S. and Z B = 200 C.P.S. Consider the solution of Birks and Brown (1) in which N s = Ns’ = N B ’ = 25,000 counts. Then, the expected counting times are E(ts) = E(ts’) = 25,000/500 50 seconds and E(tB’) = 25,000/200 = 125 seconds. From Equation 2 the variance for the fixed-count procedure is = (500/50)
+ (200/125) = 11.6
For the fixed-time procedure, t~ = ts; thus, E(tB) = E(&) = 50 seconds. From Equation 1 the variance for the fixed-time technique is UFT’
= (500/50)
+ (200/50)
= 14.0
(10)
Substituting E(tB’) = t - E(t8‘) into Equation 10 and solving for E(t.9’) gives E(ts’) = I E ~ / ( I s I E )
- ts’l2 2 0,
(Is/ts)
++
- Ls’?IB I ( t ~ -’ ~ B ) I . s
or
[E(~‘B)/~BP1
where the expected number of sample counts is E ( N 5 ) = Ists and the expected number of background counts is E ( N B ) = I B t B . Thus,
+ (IB/~B’)
The background intensity is less than the sample intensity which also includes background
up$
sprp =
(Is/ts’)
Since
Fixed-counts give a better precision, 11.6 compared to 14.0. But, by the solution of Birks and Brown ( I ) , the fixed-count technique is allowed more counting time, t’ = 50 125 = 175 seconds, compared to t = 50 50 = 100 seconds using the fixed-time method. Now, the analyst is faced with a dilemma as to which method is the more efficient based on precision and the expected counting times required. Hence, the solution in this paper %as to compare the precision of the two techniques for equal amounts of total counting times. For the fixed-count procedure the expected number of sample counts is equal to the espected number of background counts
(11)
for the fxed-count procedure where t = t’. For the above example with a total of t = 100 seconds of counting time, Equation 11 gives E(bs’) = 28.57 seconds and E(tB’) = 100 - 28.57 = 71.43 seconds. Substituting these results in Equation 2 gives UFC’
= (500/28.57)
+ (200/71.43) = 20.3
For the same amount of total counting time, the fixed-time technique gives the better precision, 14.0 compared to 20.3, illustrating the proof given here that the variance of the fixed-count procedure is never less than the variance for the fixed-time method for the same amount of total counting time. Also, the optimum allocation between sample counting time, Ts,and background counting time, TB, as given by Jarrett (2), for minimizing the standard deviation of sample minus background count rates, is TsITB =
(IS/IB)1’2
(12)
Since Is/ZB 1 1, for optimum allocation TS 2 TB. For fixed-time operation, ts = tB. The fixed-count technique results in just the opposite of optimum allocation, since Ns’ = NE‘,ZsE(ts’) = IBE(~B’)or the ratio of the espected sample to expected background counting time is E(ts’)/E(tB’)= L / I s 5 1
Using the same example as above, = 200 c.P.s., the optimum ratio of sample to background counting time is
ZS = 500 C.P.S. and Z B
T s / T B = (500/200)”2
= 1.53
From Equation 10, the ratio of the expected sample to expected background
counting time for fixed-counts is E ( t s ’ ) / E ( t ~ ’ )= 0.4. For fixed time, t s / t ~= 1.0, n-hich is closer to the optimum ratio than fixed counts. The bi\e~.l-tinieprocedure, t s / t g = 1.0, is only optimuni when Z S = I g . That is, when there is no real signal other than background, the best allocation for detecting that no signal other than background exists is to spend equal time counting sample and background. In this special case, the fixed-count procedure is equivalent to the fixed-time procedure; t , g / t B = t s ’ , / t B ’ = T ~ / T = B (ZS/IB,’” = 1 .O. Heme, the fiwd-time method is alwvay.; 3s close or closer to optimum alloc3tion of the total counting time for sninple and background counting times than fixed-counts for minimizing the standard deviation of sample minus background count rates. This is in agreement with the above proof showing that the standard deviation
for fixed-time operation is less than or equal to the standard deviation with fixed-count operation for the same amount of total counting time. If a value Knl is selected for fised-counts the background is counted until .and hT B I - lVs‘, the expected total counting time is ( N s ’ / l s ) (Ngl/lB) = t. Using this same total counting time for the fixed-time procedure (ts = t~ = t / 2 ) , the variance for the fisedtime procedure is always less than or equal to the variance for fixed-counts.
of the variance, I B / t B ’ . This time is better spent in reducing the larger component of variance, Isjts’. I t mas proved that the variance of the fixedtime method is always less than or equal to the variance of the fixed-count method for any given total counting time (ts I!B = $8’ t i s ’ ) . Conversely, for a specified precision, the expected total counting time required for the fixed-time method will be less than or equal to the expected total counting time required for the fixed-count method.
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LITERATURE CITED
SUMMARY
(1) Birks, L. S., Brown, D. M., ANAL. CHEM.34, 240 (1962). (2) Jarrett, A. A:, Mon P-126, Technical Information Division, Oak Ridge S a -
Since I S is generally much larger than I B , the fised-count procedure requires much more time to count background than to count the sample. An examination of the variance (Equation 2) shows that a large amount of time, is’, is spent reducing the smaller component
tional Laboratory (1946).
D. W. GAYLOR
General Electric Co. Vallecitos Atomic Laboratory Pleasanton, Calif.
Simultaneous Determination of Carbon and Sulfur in Organic Compounds by Gas Chromatography
SIR: In the previous work by Beuerman and Meloan ( I ) involving the determination of sulfur in organic compounds, no quantitative consideration was given to any of the combustion products other than the SO,. The emphasis was merely to ensure that the SO2 was completely separated from all of the other gases present. Further investigation has shown that under the proper conditions, COZ, and hence the carbon in the original sample, can also be separated and determined quantitatively. The previous work had shown that the halogens. nitrogen. and the oxides of nitrogen did not have retention times that would interfere with the COz peak. Therefore the main difficulties to be overcome mere the separation of 0 2 from the CO2 peak, proper trapping of the CO,, and equilibrium attainment after a change in attenuation. Separation of O,,COz, and SO2. A 20-foot column of 30y0 dinonylphthalate on Chromasorb W, conditioned for a t least one day a t 95’ C., separated the 0 2 from the CO, and provided sufficient time between the COz peak and the SOz peak to allow for electrical equilibrium when the attenuation was changed. A flow rate of 45 ml. per minute through l/d-inch tubing a t a temperature of 92” C. produced the separation shown in Figure 1. Attenuation. The detector used
consisted of a thermal conductivity cell using 2000-ohm thermistors. The sensitivity of C02 with this detector is much greater than with SO2 and, as a result, when the attenuation was set for SO2, the COz peak mas completely off scale. However, when the attenuation was changed from one peak to the other, this instrument produced a base line shift sufficient to ruin quantitative work. If at least 2 minutes were allowed after the change in attenuation, the base line came back to its original position. This problem eliminates itself-to separate Oz from C o n there are about 4 minutes until the SO2 peak comes through.
I Figure 1.
Trapping the C 0 2 . Beuerrnan and Meloan ( I ) have shown that when using 0, a t 10 to 12 ml. per minute flow rate during the combustion step, it was necessary to use liquid nitrogen to trap the SO? quantitatively. Because the liquid nitrogen temperature was well below that expected to trap the SO, [-lo’ C.) it was doubtful that the CO2, which C., would be solidifies a t -78.5’ quantitatively trapped. -4solution of Ba(0H)r placed after the trap did not produce a turbidity as would be expected if the C 0 2 was not completely trapped. This circumstance, combined with the quantitative reproducibility of the various samples tested, indicates
MINUTES
Chromatogram from a dinonylphathalate column VOL. 34, NO. 12, NOVEMBER 1962
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