Alternative Approach: Rate Theory Based on the principal of the summation of variances. σ T 2 = σ σ σ … So all contributions to the broadening of a peak can be combined in this way

σ T 2 = σ σ σ … σ 1, σ 2, etc. are individual contributions to peak width σTσT

Rate Theory Considers 3 main sources of peak broadening: 1. Eddy Diffusion (A) 2. Longitudinal Diffusion (B) 3. Resistance to Mass Transfer (C)

1. Eddy Diffusion (A): Particular pathways may differ in length. This mechanism is independent of flow rate (A)

2. Longitudinal Diffusion (B): A plug of solute in a liquid will tend to spread out into neighboring solvent. This mechanism is proportional to the inverse of the flow rate (B/u)

3. Resistance to Mass Transfer (C): Different analyte molecules may encounter more random interactions with the stationary phase. This mechanism is directly proportional to the flow rate (Cu)

Rate Theory All 3 will contribute to the height (H) of a theoretical plate depending on the rate of flow (u) H = A + B/u + Cu “Van Deemter Equation”

Historical “Van Deemter Equation” H = A + B/u + Cu u opt

Modern “Van Deemter Equation”: H = B/u + C s u + C m u

Rule of Thumb:Save time by using u ≈ 2 u opt u opt u (cm/s) u practical

Review of Chromatographic Quantities K=Partition Coefficient, Distribution Coefficient N=Total number of theoretical plates in a column H=Height Equivalent to one Theoretical Plate (mm) L=Length of a chromatographic column (mm) t r =Retention Time for a chromatographic peak (min) W=Base width of a chromatographic peak (min) σ=standard dev. in a chromatographic peak (min) F=Mobile phase flow rate (mL/min) u=linear velocity of mobile phase (mm/min)

Some Useful Derived Quantities H ≡ σ 2 /L = L/N = A +B/u + Cu = B/u + C s u + C m u N = (t r /σ) 2 = 16(t r /W) 2 = 5.54(t r /W 1/2 ) 2 k′ = K(V s /V m )= (t r – t 0 )/t 0 α = K B /K A = k′ B / k′ A = (t B – t 0 )/(t A – t 0 )where: t B >t A R s = (t B – t A ) / (W B +W A )/2 = 2(t B – t A )/Wwhere: W B ≈ W A u = L/t 0 V m = t 0 FW = 4σW 1/2 = 2.35σ