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**CHAPTER 2: Theory and Discussion**

There are several dynamic processes occurring inside a column as the mixture travels through it which affect the performance of the column and the resulting chromatogram. The first attempts to explain these processes led to the plate theory which was based on an equilibrium model. This model, however, could not explain the non-equilibrium conditions in the column nor could it explain the factors that affecting the broadening of the chromatographic peaks commonly referred to as band broadening. Another model, known as the rate theory, was developed which explained the kinetic factors effecting band broadening in 1956 for packed columns [29]. This model was later modified to fit open tubular columns as well by Golay in 1958 and is still used to measure the performance of GC columns.

This chapter introduces the fundamental theories behind GC starting with some basic definitions. The concept of plates and the rate theory are also explained which determine the efficiency of the separation. Finally, some other terms that determine separation quality are described.

**Basic Definitions**

This section explains some of the common terms and definitions that are commonly used in GC. The first of these terms is the hold-up time (*t** _{M}*), which is the time taken by an unretained component to travel the length of the column. Components like air and methane interact minimally with the stationary phase and hence are considered unretained. These unretained components can be used to measure the average linear carrier gas velocity in the column (

*ū*) or as solvents for the gas mixture samples. Retention time (

*t*

*), also known as the elution time, is the time it takes for retained components to exit the column. In many chromatographic calculations the adjusted retention time (*

_{R}*t’*

*) is taken into account which is obtained by subtracting the hold-up time from the retention time. These terms are explained graphically in Figure 2.1.*

_{R}Another important term in GC is the retention factor (

*k*), which is important in determining the efficiency of the separation. This term is calculated by dividing the adjusted retention time by the hold-up time (

*t’*

*/*

_{R}*t*

*) and is independent of the carrier gas velocity. These terms will be used extensively in defining other complex relationships in GC systems.*

_{M}**Theoretical Plate**

The GC column can be divided into a number of theoretical plates, where, a plate is the smallest section of the column where an analyte can achieve equilibrium between the two phases. The length of each plate is termed as height equivalent to a theoretical plate (HETP) [11]. The concept of a theoretical plate is explained schematically in Figure 2.2 which depicts a column with five theoretical plates. The mobile phase is represented by the white portion and the stationary phase by blue. The sample consists of two components, one of which is soluble in the stationary phase (black dots) and the other is insoluble (grey dots), and is unretained. During each step, the portion of the components in the mobile phase travels to the next plate and then equilibrium is achieved for each soluble component within a single plate. As a result of these travels and equilibrium steps, the components distribute themselves in a Gaussian profile along the column. It can be seen that in order to enhance the resolution of the two components more plates are required. This can be achieved either by using a longer column or one with a smaller plate height.

Due to the Gaussian profile of the eluted peak, the total number of plates for the column can be calculated based on the standard deviation (*σ*) and mean or the elution time (*t** _{R}*) of the peak [13]:

Here,

*w*

*and*

_{b}*w*

*are the peak widths at the base and at half height respectively. The HETP can then be calculated using the length of the column:*

_{h}The HETP is a better measure to compare efficiencies of different columns as it is independent of the length. It is desirable to achieve a large N or a small HETP for high resolution separations.

**Rate Theory**

The plate number or the height of the plate, though a good measure of efficiency, do not give a clear insight to the kinetics taking place inside the column that lead to band broadening. The original equation based on the rate theory is known as the van Deemter equation for packed columns which relates the HETP and the average linear carrier gas velocity (*ū*):

where *A*, *B*, *C* and *D* represent different phenomena that cause band broadening. The equation was later modified to the Golay equation for open tubular columns which excluded the A-term. The D-term was added later to consider the extra column effects which are prevalent in short columns with relatively high carrier gas velocities. These terms are explained in the following sections.

**The A-Term: Eddy Diffusion**

The *A*-term represents the eddy diffusion effect in packed columns. This occurs due to the non-uniform size and packing order of the columns that makes paths of different lengths along the column causing band broadening. This effect is demonstrated in Figure 2.3 where three molecules of the same component experience band broadening due to eddy diffusion [13]:

This term depends on the size of the packing material (*d** _{p}*) and the packing factor (

*λ*) which in turn is primarily affected by the uniformity in size and packing [13]:

**The B-Term: Gas Diffusion**

The *B*-term represents the longitudinal diffusion. This is when the solute in the carrier gas diffuses from high concentration to low concentration as shown in Figure 2.4 where the peak broadens as the time increments from T1 to T3 assuming zero carrier gas velocity [13].

This term is defined separately for open tubular and packed columns due to the presence of a tortuosity factor (*γ*) in packed columns that again accounts for the nature of the packed bed. The B-terms for open tubular and packed columns are given by equations (2.5) and (2.6) respectively [30].

ABSTRACT

Acknowledgements

CHAPTER 1: Introduction

1.1 : Gas Chromatography

1.2 : The GC Column

1.3 : Micro Gas Chromatography (μGC)

CHAPTER 2: Theory and Discussion

2.1 : Basic Definitions

2.2 : Theoretical Plate

2.3 : Rate Theory

2.4 : Other terms Defining Separation Quality

CHAPTER 3: Design and Fabrication

3.1 : Semi-Packed μGC Column Design

3.2 : Fabrication

3.3 : Coating Procedure

CHAPTER 4: Results and Discussion

4.1 : Experimental Setup

4.2 : Separations

4.3 : HETP Measurement

4.4 : Sample Capacity

CHAPTER 5: Conclusion

5.1 : Future Work

References

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