Spherical particles of a protein of density 1.4 g/cm3 are shaken up in a solution of 20°C water. The solution is allowed to stand for 1.0 h. If the depth of water in the tube is 4.0 cm, find the radius of the largest particles that remain in solution at the end of the hour.
µm
To find the radius of the largest particles that remain in the solution, we need to determine the terminal velocity of the particles and use it to calculate their size.
Step 1: Calculate the terminal velocity (Vt) of the particles
The terminal velocity can be calculated using the Stokes' Law formula:
Vt = (2/9) * [(ρp - ρf) / η] * g * r^2
Where:
- Vt is the terminal velocity
- ρp is the density of the particle
- ρf is the density of the fluid (water in this case)
- η is the viscosity of the fluid
- g is the acceleration due to gravity
- r is the radius of the particle
Given values:
ρp = 1.4 g/cm³ (density of the particle)
ρf = 1 g/cm³ (density of water at 20°C)
η = viscosity of water at 20°C (lookup value or assume 0.01 g/cm/s for simplicity)
g = acceleration due to gravity (approximately 9.8 m/s²)
Step 2: Calculate the maximum radius using the terminal velocity
The maximum radius of the particles (r_max) can be calculated using the following formula:
r_max = (Vt * t) / 2
Where:
- r_max is the maximum radius of the particles that remain in solution
- Vt is the terminal velocity
- t is the time the solution is allowed to stand (1.0 h)
Given values:
t = 1.0 h = 60 minutes
Step 3: Convert the result to micrometers (µm)
Since the given answer is in micrometers (µm), we need to convert the result from centimeters (cm) to micrometers (µm):
1 cm = 10,000 µm
Now let's calculate the values to find the radius of the largest particles remaining in solution.
Step 1: Calculate the terminal velocity (Vt):
Vt = (2/9) * [(ρp - ρf) / η] * g * r^2
= (2/9) * [(1.4 - 1) / 0.01] * 9.8 * r^2
= (8/9) * r^2
Step 2: Calculate the maximum radius (r_max):
r_max = (Vt * t) / 2
= [(8/9) * r^2 * 60] / 2
= 4/3 * r^2
Step 3: Convert the result to micrometers (µm):
r_max = 4/3 * r^2 * 10,000 µm/cm
= (40,000/3) * r^2 µm
Therefore, the radius of the largest particles remaining in solution at the end of the hour is given by (40,000/3) * r^2 µm.