A piece of capillary tubing was calibrated in the following manner. A clean sample of the tubing weighed 3.247 g. A thread of mercury, drawn into the tube, occupied a length of 24 mm, as observed under a microscope. The weight of the tube with the mercury was 3.489 g. The density of mercury is 13.60 g/cm³. Assuming that the capillary bore is a uniform cylinder, find the diameter of the bore.
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To find the diameter of the capillary bore, we can use the formula for the volume of a cylinder:
V = πr²h
where V is the volume, r is the radius (half of the diameter), and h is the height.
Given information:
- Weight of the tubing (empty): 3.247 g
- Weight of the tubing with mercury: 3.489 g
- Length of mercury column: 24 mm
- Density of mercury: 13.60 g/cm³
First, we need to find the volume of the mercury column.
1. Calculate the mass of the mercury:
Mass of mercury = Weight of tubing with mercury - Weight of empty tubing
Mass of mercury = 3.489 g - 3.247 g
2. Convert the length of the mercury column to cm:
Height of mercury column = 24 mm / 10
3. Calculate the volume of the mercury column:
Volume of mercury = Mass of mercury / Density of mercury
Next, we can calculate the radius (diameter) of the capillary bore.
4. Rearrange the formula for the volume of a cylinder to solve for the radius:
V = πr²h (solve for r)
r² = V / (πh)
5. Substitute the values into the formula:
r² = Volume of mercury / (π × Height of mercury column)
6. Calculate the radius:
r = √(Volume of mercury / (π × Height of mercury column))
Finally, we can calculate the diameter by multiplying the radius by 2.
7. Calculate the diameter:
Diameter = 2 × r
By following these steps, you can find the diameter of the capillary bore.