SME
Analytical Chemistry · Problem type
Best flow speed from the van Deemter equation
Use this for “A column has A = 0.050 cm, B = 0.40 cm²/s and C = 0.010 s. Find the flow speed that gives the smallest plate height, and that height.”
Chart
Local minimum at (≈ 6.32, ≈ 0.176)
H(10) = 0.19
This is an example. Type your own values to start a new problem.
uₒₚₜ (cm/s)
=
B (cm²/s)
C (s)
Hₘᵢₙ (cm)
=
A (cm)
+
2
B (cm²/s)
×
C (s)
u
Flow speed
example · cm/s
H
Plate height at that speed
calculated · cm
Formulas
H
=
A
+
B
u
+
C
×
u
0.19
=
0.05
+
0.4
10
+
0.01
×
10
uₒₚₜ
=
√(
B
C
)
6.325
=
√(
0.4
0.01
)
Hₘᵢₙ
=
A
+
2
×
√
B
×
C
0.1765
=
0.05
+
2
×
√
0.4 × 0.01
Assumptions
•
Van Deemter: H = A + B ÷ u + Cu. A is eddy diffusion (paths of different length), B ÷ u is diffusion along the column, and Cu is the slow transfer between phases.
•
A smaller plate height H means more plates in the same column, so sharper peaks.
•
The least H is where the falling B ÷ u and the rising Cu balance: uₒₚₜ = √(B ÷ C).
Step-by-step
WE KNOW
A = 0.05 cm, B = 0.4 cm²/s, C = 0.01 s, u = 10 cm/s
FIND
plate height at that speed (H), best flow speed (uₒₚₜ), least plate height (Hₘᵢₙ)
Step 1 · Find plate height at that speed (H)
Use H = A + B ÷ u + C × u
Add the three terms at this flow speed.
H
=
0.05
+
0.4
10
+
0.01
×
10
H
=
0.05
+
0.04
+
0.1
H
=
0.19
cm
Step 2 · Find best flow speed (uₒₚₜ)
Use uₒₚₜ = √(B ÷ C)
Where
dH/du
=
−
B
u
2
+
C
is
0:
u
2
=
B
C
.
uₒₚₜ
=
√(
0.4
0.01
)
uₒₚₜ
=
√
40
uₒₚₜ = 6.325 cm/s
Step 3 · Find least plate height (Hₘᵢₙ)
Use Hₘᵢₙ = A + 2 × √(B × C)
At
uₒₚₜ
the
terms
B
u
and
Cu
are
equal,
each
√
BC
:
add
both
to
A
.
Hₘᵢₙ
=
0.05
+
2
×
√
0.4 × 0.01
Hₘᵢₙ
=
0.05
+
2
×
√
0.004
Hₘᵢₙ = 0.05 + 2 × 0.06325
Hₘᵢₙ = 0.05 + 0.1265
Hₘᵢₙ = 0.1765 cm
Check
Put the numbers back in. Both sides should match.
0.19
=
0.05
+
0.4
10
+
0.01
×
10
✓
6.325
=
√(
0.4
0.01
)
✓
0.1765
=
0.05
+
2
×
√
0.4 × 0.01
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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