Exchange of comments on “Temporal shifting: a hidden key to the skewed peak puzzle”

Journal of Chromatography A, 1148 (2007) 262–263

Discussion

Exchange of comments on “Temporal shifting:

a hidden key to the skewed peak puzzle”

Su-Cheng Pai

, Ling-Yun Chiao

a_{Division of Marine Chemistry, Institute of Oceanography, National Taiwan University, Taipei, Taiwan} b_{Division of Marine Geology and Geophysics, Institute of Oceanography, National Taiwan University, Taipei, Taiwan}

Available online 19 March 2007

Keywords: Flow injection analysis; Linear chromatography; Peak shape; Temporal shift

Soon after our paper[1]was published, a response arrived from Felinger[2]who strongly suggested that one should use the first moment to locate the correct peak position. The argument then, becomes “how can one find the correct spatial first moment from a skewed peak on a time-based recorder chart?”

In our work, we have assumed that the sample zone is spatially Gaussian and temporally carrying a tail; and we have defined a parameterΦ (in s units) which represents the shift of an apparent peak apex (t∗_p) from its true elution time (tp as calculated by tp= L/u, where L is the length of tubular channel and u the flow speed):

Φ = tp− t∗p (1)

As tpincreases, the solution forΦ gradually converges to a

fixed threshold. Thus, one may use the following approximation to evaluate the scale of the shift:

Φ ≈ D

u2 (2)

Thus, the temporal peak apex should be at

t∗ p ≈

L

u − Φ (3)

Apart from this, other parameters can also be obtained. For example, one may calculate the time for the mean peak area (denoted ast_a∗) and the temporal first moment (denoted ast∗_r). For the former, we let



c(t) dt|t=t∗ a =

At

2 (4)

∗_{Corresponding author. Tel.: +886 2 23627358; fax: +886 2 23632912.}

E-mail address:scpai@ntu.edu.tw(S.-C. Pai).

where Atis the peak area measured on the t axis (in conc-s units).

The solution forta∗can be approximated as ta∗=

L

u + Φ (5)

For the latter, we use the following equation:

tr∗=

1

At



tc(t) dt (6)

and the solution is

tr∗= L

u + 2Φ (7)

Thus,t_p∗andt_a∗lie aside the spatial first moment tpwith an

almost equal distance (Φ); whereas t_r∗is located at an even later time at tp+ 2Φ. These solutions have been verified numerically

and demonstrated inFig. 1(A–C).

Neithert∗p,ta∗nortr∗ represents the correct elution time (or

the spatial first moment), but we can use either one to construct a temporal peak after finding an empiricalΦ and compensating for the corresponding position shifts in a Gaussian equation. Thus, all three equations shown below can be used to restore an experimental peak. c(t) = √At 4πΦte −(t−t∗ p−Φ)2/(4Φt) ₍₈₎ c(t) = √At 4πΦte −(t−t∗ a+Φ)2/(4Φt) ₍₉₎ c(t) = √At 4πΦte −(t−t∗ r+2Φ)2/(4Φt) ₍₁₀₎

There is no doubt that the first one (i.e. Eq.(8)) is always the easiest and most accurate (free from baseline noise) to serve this purpose. Examples have been given in another published paper

[3].

0021-9673/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.chroma.2007.03.049

S.-C. Pai, L.-Y. Chiao / J. Chromatogr. A 1148 (2007) 262–263 263

Fig. 1. (A) A skewed temporal curve is plotted from the data of Ref.[1]Fig. 3 in which the channel length is L = 40 cm, the temporal peak area At= 1 conc-s, the flow speed u = 1 cm s−1, the expansion coefficientμt= 2 s1/2and the dispersion

coefficient D = 2 cm2_s−1_{. The true retention time is tp= L/u = 40 s, the apparent}

peak apex is found att_p∗= 38.05 s, the peak area mean is at t∗_a= 42.00 s, whereas the first moment is att_r∗= 44.00 s. The shifts from tpto eithertp∗orta∗are almost

identical in scale but opposite in signs and can be approximated byΦ = D/u2_,

the shift fort_r∗is 2Φ. (B) The position of the peak area mean (t_a∗) is obtained by taking the cross point for the half integrated peak area. (C) Integrated calculation shows that the first momentt_r∗is 44.00 s.

Felinger calculated the data provided from Fig. 3 of Ref.

[1], but he used an erroneous diffusivity of D = 4 cm2s−1, thus giving an elution time of 48 s as opposed to 40 s calculated from the spatial distribution. In that example, the temporal expansion coefficient was designated asμt= 2 s1/2., which is equivalent to a diffusivity of D = 2 cm2s−1 (i.e.D = 0.5μ2tu2). The results

of our calculation (also seeFig. 1) show that the temporal peak apex is attp∗≈ 38 s, a shift of ≈−2 s from tp; the temporal peak

area mean is attr∗= 42 s, a shift of +2 s from tp. The temporal

first moment is att_r∗= 44 s, a shift of +4 s from tp. There is no

conceptual conflict between Felinger and us that “the temporal first moment gives a larger elution time than the spatial moment,” or that “one should never use the location of the apex but the first moment.”

Felinger introduced an alternative equation (Eq. (5)of Ref

[2]) to describe the concentration profile based on temporal terms (e.g. usingσtinstead ofσL):

c(z, t) = √ A 4Dπt3e

−(z−ut)2_/4Dt

(11) He claimed that the spatial first moment of this equation is L/u, and that the temporal first moment is the same. And, therefore, the shift does not exist. However, we think it does.

The first portion of Eq.(11)is supposed to be the peak height at z = ut. But, one should be aware that the physical dimension of √

4Dπt3_{is “cm-s.” If A represents the peak area measured along}

the t axis (i.e. Atin conc-s units), then, the physical dimension

for the peak height would be “conc-cm−1, which is unrealistic. In our derivation, the peak height at (z = L, t = tp) should be h = c(L, tp)= AL

4Dπtp (12)

Since At= AL/u and u = L/tp, it can be expressed as h = c(L, tp)=  At

4Dπt3 pL−2

(13)

Therefore, the correct expression for Eq.(11)should be

c(z, t) = √ At 4Dπt3_z−2e

−(z−ut)2_/4Dt

(14) This equation is exactly the same as

c(z, t) = √AL 4Dπte

−(z−ut)2_/4Dt

(15) Therefore, the spatial first moments for both Eqs. (14)and

(15)are tp= L/u, and the temporal first moments are all equal at t∗

r = L/u + 2Φ. The shift has not “disappeared.”

Felinger also argued that the reasoning attributed to the tem-poral peak skewness is well understood and characterized. In fact, most users, except for those competent chromatographers, do normally regard the temporal peak apex as the true elution time. And, it is very difficult to explain to them and persuade them that these two times are different. Felinger’s comments and this reply can help readers further clarify the temporal shifting problem.

References

[1] S.C. Pai, L.Y. Chiao, J. Chromatogr. A 1139 (2007) 104. [2] A. Felinger, J. Chromatogr. A 1148 (2007) 260.