Suppose that solvent 1 contains a contaminant in the form of larger protein aggregates, and the manufacturer wants to remove these aggregates by centrifugation. Solvent 1 has a density of 1000 kg/m3, and a viscosity of 1mPa s at 344K. It is known that the average protein aggregate has a radius of 0.1 μm and a density of 1100 kg/m3. Give an estimate of how many g’s the sample should be at least subjected to in order to spin down the protein aggregates.



To estimate the number of g's (relative centrifugal force, RCF) needed to spin down the protein aggregates, we can use **Stokes' law** and **sedimentation theory** for spherical particles in a viscous medium.

The sedimentation velocity \( v \) of a particle is given by:

\[
v = \frac{2r^2 (\rho_p - \rho_m) g}{9 \eta}
\]

Where:
- \( r \) = radius of the particle (protein aggregate) = \( 0.1 \, \mu m = 0.1 \times 10^{-6} \, m \)
- \( \rho_p \) = density of the particle (protein aggregate) = \( 1100 \, kg/m^3 \)
- \( \rho_m \) = density of the solvent (medium) = \( 1000 \, kg/m^3 \)
- \( g \) = acceleration due to gravity (for 1 g, \( g = 9.81 \, m/s^2 \))
- \( \eta \) = viscosity of the solvent = \( 1 \, mPa \cdot s = 1 \times 10^{-3} \, Pa \cdot s \)

We will also need the **centrifugal force** formula to convert to relative centrifugal force:

\[
RCF = \frac{r_{\text{rotor}} \omega^2}{g}
\]

Where:
- \( r_{\text{rotor}} \) is the distance from the axis of rotation (we'll assume later for a general estimate).
- \( \omega \) is the angular velocity in radians per second.

But first, we calculate the **terminal velocity** of sedimentation under standard gravity using Stokes' law.

### Step 1: Calculate the sedimentation velocity under normal gravity
\[
v = \frac{2 (0.1 \times 10^{-6})^2 (1100 - 1000) \times 9.81}{9 \times 1 \times 10^{-3}}
\]

Let’s calculate this velocity.The sedimentation velocity \( v \) of the protein aggregates under normal gravity is approximately \( 2.18 \times 10^{-9} \, m/s \).

### Step 2: Estimating the required number of g's

To effectively pellet the protein aggregates in a reasonable time, we need a much higher sedimentation velocity, which can be achieved by applying a centrifugal force. The required number of g's can be estimated by scaling up the sedimentation velocity. Assuming that a reasonable centrifugation process might aim for a velocity around \( 10^{-5} \, m/s \), we can compute the number of g's (RCF) needed:

\[
\text{Required RCF} = \frac{\text{desired velocity}}{v_{\text{normal gravity}}}
\]

Let’s compute the necessary RCF assuming a target velocity of \( 10^{-5} \, m/s \).The sample needs to be subjected to at least approximately **4587 g's** (relative centrifugal force, RCF) to effectively spin down the protein aggregates.

Would you like further details or have any additional questions?

Here are 5 related questions to expand this discussion:
1. How would increasing the temperature affect the viscosity and required RCF?
2. What role does particle shape play in centrifugation, and how would this change the calculation for non-spherical aggregates?
3. How do factors like rotor radius influence the actual angular velocity required for centrifugation?
4. How can you experimentally determine the optimal centrifugation time for protein separation?
5. What are the limitations of using Stokes' law for estimating sedimentation in real-world scenarios?

**Tip**: Always check the maximum allowable g-force of your centrifuge before setting up the spin to avoid damaging the equipment.

Suppose that solvent 1 contains a contaminant in the form of larger protein aggregates, and the manufacturer wants to remove these aggregates by centrifugation. Solvent 1 has a density of 1000 kg/m3, and a viscosity of 1mPa s at 344K. It is known that the average protein aggregate has a radius of 0.1 μm and a density of 1100 kg/m3. Give an estimate of how many g’s the sample should be at least subjected to in order to spin down the protein aggregates.

Learn how to estimate the number of g's required to spin down protein aggregates from solvent using Stokes' law and sedimentation velocity. The problem involves calculating sedimentation velocity, solvent viscosity, and determining the relative centrifugal force needed. Suitable for advanced physics and chemistry students.

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