Let's carefully work through this problem step by step. --- ### **Given Data:** | Parameter | Value | Notes | |--------------|---------|--------| | Sedimentation coefficient in buffer, \( s_{buffer} \) | 5.2 S | at 20°C, in buffer with 100 mM KCl | | Temperature, \( T \) | 20°C = 293 K | Standard temperature for Svedberg calculations | | Viscosity of buffer, \( \eta_{buffer} \) | 10% less than water at 20°C | \( \eta_{water} \) at 20°C ≈ 0.001002 Pa·s | | Density of buffer, \( \rho_{buffer} \) | 20% less than water at 20°C | \( \rho_{water} \) at 20°C ≈ 998 kg/m³ | | Partial specific volume, \( \bar{v} \) | 0.65 cm³/g | or 0.65 × 10\(^{-6}\) m³/g | --- ### **Step 1: Understand what the problem asks** - **Calculate the standard sedimentation coefficient \( s_{20,w} \) in water at 20°C.** - **Estimate the molecular weight (MW) of the RNA using the sedimentation coefficient.** --- ### **Step 2: Correct the sedimentation coefficient to standard conditions** The measured \( s_{buffer} \) includes effects of buffer viscosity and density. To find the **standard sedimentation coefficient** (that is, in water at 20°C), we need to correct for these. The correction formula: \[ s_{20,w} = s_{measured} \times \frac{\eta_{buffer}}{\eta_{water}} \times \frac{1 - \bar{v} \rho_{buffer}}{1 - \bar{v} \rho_{water}} \] where: - \( s_{measured} = 5.2 \) S, - \( \eta_{buffer} \) and \( \eta_{water} \) are viscosities, - \( \rho_{buffer} \) and \( \rho_{water} \) are densities, - \( \bar{v} \) is the partial specific volume. --- ### **Step 3: Find the correction factors** **A. Viscosity correction:** Given that the viscosity of the buffer is 10% less than water: \[ \eta_{buffer} = 0.9 \times \eta_{water} \] So, \[ \frac{\eta_{buffer}}{\eta_{water}} = 0.9 \] --- **B. Density correction:** Given the density of buffer is 20% less than water: \[ \rho_{buffer} = 0.8 \times \rho_{water} \] So, \[ \frac{1 - \bar{v} \rho_{buffer}}{1 - \bar{v} \rho_{water}} \] Calculate numerator and denominator: - \( \bar{v} = 0.65\, \text{cm}^3/\text{g} = 0.65 \times 10^{-6}\, \text{m}^3/\text{g} \) - \( \rho_{water} = 998\, \text{kg/m}^3 \) - \( \rho_{buffer} = 0.8 \times 998 = 798.4\, \text{kg/m}^3 \) Calculate: \[ 1 - \bar{v} \rho_{buffer} = 1 - (0.65 \times 10^{-6} \, \text{m}^3/\text{g}) \times 798.4\, \text{kg/m}^3 \] Note: To be consistent, convert grams to kg: \[ 1\, \text{g} = 10^{-3}\, \text{kg} \] Thus: \[ \bar{v} \text{ per gram} = 0.65 \times 10^{-6}\, \text{m}^3/\text{g} = 0.65 \times 10^{-6}\, \text{m}^3 / 10^{-3}\, \text{kg} = 0.65 \times 10^{-3}\, \text{m}^3/\text{kg} \] Now: \[ 1 - \bar{v} \rho_{buffer} = 1 - (0.65 \times 10^{-3}) \times 798.4 = 1 - 0.519 \] Similarly for water: \[ 1 - (0.65 \times 10^{-3}) \times 998 = 1 - 0.649 \] Calculate: \[ \text{Numerator}: 1 - 0.519 = 0.481 \] \[ \text{Denominator}: 1 - 0.649 = 0.351 \] --- ### **Step 4: Calculate \( s_{20,w} \)** Putting it all together: \[ s_{20,w} = 5.2 \times 0.9 \times \frac{0.481}{0.351} \] Calculate numerator: \[ 5.2 \times 0.9 = 4.68 \] Calculate the ratio: \[ \frac{0.481}{0.351} \approx 1.37 \] Thus: \[ s_{20,w} \approx 4.68 \times 1.37 \approx 6.41\, \text{S} \] --- ### **Answer 1:** **The standard sedimentation coefficient \( s_{20,w} \approx \boxed{6.41\, \text{S}} \).** --- ### **Step 5: Estimate the molecular weight of the RNA** The relation between sedimentation coefficient \( s \), molecular weight \( M \), partial specific volume \( \bar{v} \), and solution properties (at standard conditions) is: \[ s = \frac{M (1 - \bar{v} \rho)}{N_A f} \] where: - \( N_A \) = Avogadro's number \( \approx 6.022 \times 10^{23} \), - \( f \) = frictional coefficient, often approximated as proportional to the shape; for a spherical particle, \( f = 6 \pi \eta r \). For an estimate, a simplified empirical relation is used: \[ M = 4200 \times s \quad \text{(for RNA and similar molecules in Svedberg units)} \] Alternatively, use the **empirical relation**: \[ M \approx 420 \times s_{20,w} \quad (\text{for RNA, approximate}) \] Using this approximation: \[ M \approx 420 \times 6.41 \approx 2692\, \text{g/mol} \] --- ### **Final estimation:** \[ \boxed{ \text{Molecular weight} \approx \mathbf{2700\, \text{Da}} } \] which suggests the RNA species is approximately 2.7 kDa, consistent with small RNA fragments or oligonucleotides. --- ## **Summary:** - **Standard sedimentation coefficient \( s_{20,w} \):** approximately **6.41 S** - **Estimated molecular weight:** approximately **2700 Da** --- **Note:** The empirical relation used here is a rough estimate; for more precise calculations, detailed frictional considerations and shape factors should be included.