A researcher in Urumqi drops a rhodium ball of diameter 4.20 mm through a dilute aqueous solution of a high molar mass natural gum contained within a glass tube which has four graduations marked 10 cm apart. It is 18 C in the laboratory and the ball takes 1.86 seconds to fall between the first and second marks, 1.74 seconds to fall between the second and the third, and 1.74 seconds to fall between the third and the fourth. What is the viscosity of the liquid? What other information should be provided to allow you to answer the question more accurately?
To calculate the viscosity of the liquid, we can make use of Stokes' law, which relates the drag force on a sphere moving through a viscous fluid to its velocity and the properties of the fluid.
Stokes' law states that the drag force (F) acting on a sphere moving through a viscous fluid is given by the equation:
F = 6πηrv
Where η is the dynamic viscosity of the fluid, r is the radius of the sphere, and v is the velocity of the sphere.
Since we are given the diameter of the rhodium ball (4.20 mm), we can calculate its radius as half of the diameter:
r = 4.20 mm / 2 = 2.10 mm = 0.0021 m
The velocity of the sphere can be determined by taking the distance between the marks and the time it takes for the ball to fall between them. From the information given:
- Between the first and second marks: t = 1.86 s
- Between the second and third marks: t = 1.74 s
- Between the third and fourth marks: t = 1.74 s
To calculate the velocity, we divide the distance by the time:
- Between the first and second marks: v₁ = (10 cm) / (1.86 s) = 5.38 cm/s = 0.538 m/s
- Between the second and third marks: v₂ = (10 cm) / (1.74 s) = 5.75 cm/s = 0.575 m/s
- Between the third and fourth marks: v₃ = (10 cm) / (1.74 s) = 5.75 cm/s = 0.575 m/s
Now, let's solve for the viscosity (η). Since we have three sets of measurements, we can take the average of the three viscosity values calculated using the equation:
η = (F₁/F₂ + F₃)/2
To calculate the drag force (F) for each measurement, we rearrange Stokes' law equation:
F = 6πηrv
Substituting the known values:
- For the first measurement:
F₁ = 6πη(0.0021 m)(0.538 m/s)
- For the second measurement:
F₂ = 6πη(0.0021 m)(0.575 m/s)
- For the third measurement:
F₃ = 6πη(0.0021 m)(0.575 m/s)
By substituting the drag force values into the viscosity equation, we can calculate the average viscosity of the liquid.
In order to answer the question more accurately, we would need additional information such as the density of the rhodium ball, the density of the liquid, or the exact dimensions of the glass tube. These additional details would help in providing a more precise calculation of the viscosity of the liquid.