Problem 76 - College Physics

Question

\(\bullet$$\bullet\) Electrophoresis. Electrophoresis is a process used by biologists to separate dif- ferent biological molecules (such as pro- teins) from each other according to their ratio of charge to size. The materials to be separated are in a viscous solution that produces a drag force \(F_{\mathrm{D}}\) propor- tional to the size and speed of the molecule. We can express this relationship as \(F_{11}=K R v,\) where \(R\) is the radius of the molecule (modeled as being spherical), \(v\) is its speed, and \(K\) is a constant that depends on the viscosity of the solution. The solution is placed in an external electric field \(E\) so that the electric force on a particle of charge \(q\) is \(F=q E .\) (a) Show that when the electric field is adjusted so that the two forces (electrical and vis- cous drag) just balance, the ratio of \(q\) to \(R\) is \(K v / E\) . (b) Show that if we leave the electric field on for a time \(T\) , the distance \(x\) that the molecule moves during that time is \(x=(E T / k)(q / R)\) . (c) Sup- pose you have a sample containing three different biological mole- cules for which the molecular ratio \(q / R\) for material 2 is twice that of material 1 and the ratio for material 3 is three times that of mate- rial 1. Show that the distances migrated by these molecules after the same amount of time are \(x_{2}=2 x_{1}\) and \(x_{3}=3 x_{1} .\) In other words, material 2 travels twice as far as material \(1,\) and material 3 travels three times as far as material \(1 .\) Therefore, we have sepa- rated these molecules according to their ratio of charge to size. In practice, this process can be carried out in a special gel or paper, along which the biological molecules migrate. (See Figure 17.60 .) The process can be rather slow, requiring several hours for separa- tions of just a centimeter or so.

Step-by-Step Solution

Answer

(a) \( \frac{q}{R} = \frac{Kv}{E} \); (b) \( x = \left( \frac{ET}{K} \right) \frac{q}{R} \); (c) \( x_2 = 2x_1 \), \( x_3 = 3x_1 \).

Step 1: Establish force balance equation

To find the ratio \( \frac{q}{R} \), we start by setting the electric and viscous drag forces equal because they are balancing each other. Thus, we have \( F = F_{D} \). Substituting the expressions for the forces, we get \( qE = KRv \).

Step 2: Solve for charge to size ratio \( \frac{q}{R} \)

From the equation \( qE = KRv \), solve for \( \frac{q}{R} \). Rearranging the equation gives \( \frac{q}{R} = \frac{Kv}{E} \), satisfying part (a) of the problem.

Step 3: Relate distance to \( q/R \)

Assuming a constant speed, the distance \( x \) that a molecule moves in time \( T \) can be found as \( x = vT \). We substitute \( v \) from \( \frac{qE}{KR} = v \) (from Step 1), giving \( x = \frac{qET}{KR} \). Simplifying, we get \( x = \left( \frac{ET}{K} \right) \frac{q}{R} \), which answers part (b).

Step 4: Calculate and compare distances for different molecules

To demonstrate the separation, calculate \( x_1 = \left( \frac{ET}{K} \right) \frac{q_1}{R_1} \), \( x_2 = \left( \frac{ET}{K} \right) \frac{q_2}{R_2} \), and \( x_3 = \left( \frac{ET}{K} \right) \frac{q_3}{R_3} \). If \( \frac{q_2}{R_2} = 2 \frac{q_1}{R_1} \) and \( \frac{q_3}{R_3} = 3 \frac{q_1}{R_1} \), then \( x_2 = 2x_1 \) and \( x_3 = 3x_1 \) by substitution, confirming part (c).

Key Concepts

Electric Field

The electric field plays a critical role in the electrophoresis process by generating the force necessary to move charged particles through a medium. Think of the electric field as an invisible power that pushes the molecules along their path. In electrophoresis, an external electric field is applied across the solution containing the biological molecules. This field causes charged molecules to experience an electric force that drives them towards the oppositely charged electrode.
The strength of this electric force can be calculated using the formula \( F = qE \), where \( q \) is the charge of the particle and \( E \) is the electric field strength. This force directly competes with the viscous drag force to determine the migration speed of the molecules. By adjusting the electric field's magnitude, scientists can control the speed at which different molecules travel, allowing for their separation over time.

Viscous Drag

Viscous drag acts as a resistant force against the movement of molecules, much like the friction you feel when moving through water. In the context of electrophoresis, the solution in which the molecules are suspended has a certain viscosity, which generates a drag force on the moving molecules.
This drag force, denoted as \( F_D \), is proportional to the size and velocity of the molecule. It can be represented by the equation \( F_D = K R v \), where \( K \) is a constant related to the solution's viscosity, \( R \) is the radius of the molecule assuming a spherical shape, and \( v \) is its velocity. Viscous drag plays a crucial role in balancing the electric force. When the forces are in equilibrium, the molecules move at a constant speed. The relationship between viscous drag and the molecule's movement provides important insights into the charge-to-size ratio of the molecules.

Charge-to-Size Ratio

The charge-to-size ratio is a key aspect of molecule separation in electrophoresis. It essentially represents how much charge a molecule has relative to its size. A molecule's movement in the electric field is directed by this ratio because the electric force must overcome the viscous drag.
By considering the balance between electric force and viscous drag, the ratio \( \frac{q}{R} \) can be derived. It is given by \( \frac{q}{R} = \frac{K v}{E} \), which means the ratio depends on the electric field's strength, the molecule's velocity, and the viscosity of the medium. This balance allows the molecules with different \( \frac{q}{R} \) values to move at different speeds, facilitating their separation. Essentially, electrophoresis leverages this ratio to sort molecules by size and charge, effectively isolating them for further study or analysis.

Biological Molecules Separation

Separating biological molecules using electrophoresis relies on the principles we've discussed: the electric field, viscous drag, and charge-to-size ratio. When a mixture contains molecules with varied \( \frac{q}{R} \) ratios, they each migrate at different rates when subjected to the electric field.
As shown in the exercise, if molecule 2 has a charge-to-size ratio twice that of molecule 1 and molecule 3 has three times that of molecule 1, they will migrate twice and three times the distance of molecule 1, respectively. This differential migration lets scientists separate molecules based on their specific \( \frac{q}{R} \) values. This technique is particularly useful in analyzing complex mixtures of proteins, nucleic acids, or other biologically significant molecules. The process, while often slow, is highly effective at separating substances within a gel or paper medium as they move towards the electrode matching their charge.