(a) What is the diameter, in micrometers, of a spherical ink
droplet from an early version of an ink-jet printer if the volume of
the droplet is 200 ± 10 pL?
If the ink has a density of
1.1 g/mL, what is the mass in
milligrams of ink in a droplet
from this printer? Is
“milligrams” an appropriate
unit for describing this mass?
I would convert volume of 200 pL to mL then calculate the diameter of the sphere. Go from there to mass.
That's probably not the easiest way to do it. Remember that 1 cubic decimeter is 1 liter; therefore, convert the volume in pL to liters then to cubic decimeters, the determine the diameter. The answer will be in decimeters which you can convert to micrometers.
For the mass, the mL conversion probably is the way to go.
To find the diameter of the ink droplet, we can use the formula for the volume of a sphere:
V = (4/3) * π * r^3
Where:
V = volume
π = pi (approximately 3.14159)
r = radius of the sphere
We can rearrange the formula to solve for the radius:
r^3 = (3V) / (4π)
Now we can substitute the given volume of the droplet:
r^3 = (3 * 200 pL) / (4π)
Calculating this, we get:
r^3 ≈ 477.46 pL
Next, we take the cube root of both sides to find the radius:
r ≈ ∛(477.46 pL)
r ≈ 8.81 pL^(1/3)
The density of the ink is given as 1.1 g/mL. To find the mass of the ink droplet, we need to convert the volume from picoliters (pL) to milliliters (mL):
1 mL = 1000 pL
So, the volume of the droplet in milliliters is:
V = 200 pL / 1000
V = 0.2 mL
Now we can find the mass using the formula:
mass = volume * density
mass = 0.2 mL * 1.1 g/mL
mass ≈ 0.22 g
To convert grams to milligrams, we multiply by 1000:
mass ≈ 0.22 g * 1000
mass ≈ 220 mg
So, the mass of ink in the droplet is approximately 220 milligrams.
Yes, "milligrams" is an appropriate unit for describing this mass because it is a commonly used unit for small amounts of substances.
To find the diameter of the ink droplet, we can use the formula for the volume of a sphere:
V = (4/3) * π * r^3
where V is the volume and r is the radius of the sphere. Since the problem gives the volume of the droplet, we can rearrange the formula to solve for the radius:
r = ((3 * V) / (4 * π))^(1/3)
Substituting the given volume of 200 ± 10 pL into the equation, we get:
r = ((3 * 200) / (4 * π))^(1/3) = (600 / (4 * π))^(1/3)
Now, to find the diameter, we simply double the radius:
diameter = 2 * r = 2 * (600 / (4 * π))^(1/3)
To calculate this value, we need to know the value of π (pi), which is approximately 3.14159. Using this value, we can estimate the diameter in micrometers.
Regarding the mass of the ink droplet, we can use the density of the ink to calculate it. Density is defined as mass per unit volume. So, the mass (m) can be calculated using the formula:
m = ρ * V
where m is the mass, ρ is the density, and V is the volume. Substituting the given density of 1.1 g/mL and the volume of 200 ± 10 pL (which can be converted to mL by dividing by 1000), we get:
m = 1.1 * (200 / 1000) = 0.22 g
To convert this mass to milligrams, we can multiply it by 1000:
m = 0.22 * 1000 = 220 mg
Therefore, the mass of the ink droplet from the printer is 220 milligrams.
Using milligrams as a unit to describe the mass is appropriate in this case since it is a relatively small mass. Milligrams are commonly used to measure small quantities of substances.