Math Problem Statement

A chromatographic procedure separates 4.0 mg of unknown mixture on a column with a length of 40 cm and a diameter of 0.85 cm. What size column would you use to separate 100 mg of the same mixture? column diameter = blank1 - Numeric Answer

cm If the flow rate on the original column is 0.22 mL/min, what flow rate should be used on the larger column? flow rate = blank2 - Numeric Answer

mL/min

Solution

To determine the column diameter and flow rate for the larger column:


Step 1: Scale the Column Diameter

Chromatographic separations scale proportionally with the column's cross-sectional area to handle a larger sample. The cross-sectional area is proportional to the square of the column diameter. The ratio of the sample masses gives the required ratio of the areas.

Area2Area1=Mass2Mass1\frac{\text{Area}_2}{\text{Area}_1} = \frac{\text{Mass}_2}{\text{Mass}_1}

For a column with a circular cross-section: Area=π(d2)2=πd24\text{Area} = \pi \left(\frac{d}{2}\right)^2 = \frac{\pi d^2}{4}

So, the ratio of diameters d2d_2 to d1d_1 is given by: d22d12=Mass2Mass1\frac{d_2^2}{d_1^2} = \frac{\text{Mass}_2}{\text{Mass}_1}

Substituting the values: d22(0.85)2=1004\frac{d_2^2}{(0.85)^2} = \frac{100}{4} d22=0.722525=18.0625d_2^2 = 0.7225 \cdot 25 = 18.0625 d2=18.0625=4.25cmd_2 = \sqrt{18.0625} = 4.25 \, \text{cm}

Answer for Diameter:

4.25cm\boxed{4.25 \, \text{cm}}


Step 2: Scale the Flow Rate

The flow rate scales proportionally to the column cross-sectional area. This means: Flow Rate2Flow Rate1=Area2Area1\frac{\text{Flow Rate}_2}{\text{Flow Rate}_1} = \frac{\text{Area}_2}{\text{Area}_1}

Since: Area2Area1=d22d12=18.06250.7225=25\frac{\text{Area}_2}{\text{Area}_1} = \frac{d_2^2}{d_1^2} = \frac{18.0625}{0.7225} = 25

The new flow rate is: Flow Rate2=Flow Rate125=0.2225=5.5mL/min\text{Flow Rate}_2 = \text{Flow Rate}_1 \cdot 25 = 0.22 \cdot 25 = 5.5 \, \text{mL/min}


Answer for Flow Rate:

5.5mL/min\boxed{5.5 \, \text{mL/min}}


Summary:

  • Column Diameter: 4.25cm4.25 \, \text{cm}
  • Flow Rate: 5.5mL/min5.5 \, \text{mL/min}

Let me know if you need a detailed explanation of the calculations!


Related Questions:

  1. What is the relationship between column diameter and sample capacity in chromatography?
  2. How does flow rate affect chromatographic resolution?
  3. What other factors should be considered when scaling up chromatography columns?
  4. Why does cross-sectional area play such a crucial role in chromatographic scaling?
  5. Can the same separation efficiency be maintained when scaling up a chromatographic column?

Tip: Always verify that the separation efficiency is preserved when scaling up chromatography, as larger columns might require adjustments in packing material or other parameters.

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Math Problem Analysis

Mathematical Concepts

Proportional Scaling
Area of a Circle
Flow Rate Scaling

Formulas

Area = π(d/2)^2
(d2^2) / (d1^2) = (Mass2 / Mass1)
(Flow Rate2) / (Flow Rate1) = (Area2 / Area1)

Theorems

Scaling of Chromatographic Columns

Suitable Grade Level

Grades 11-12