Although these quantities vary from one type of cell to another, a cell can be 2.10 ìm in diameter with a cell wall 50.3nm thick.

If the density (mass divided by volume) of the wall material is the same as that of pure water, what is the mass (in mg) of the cell wall, assuming the cell to be spherical and the wall to be a very thin spherical shell?

surface area of sphere = 4 pi r^2

volume of wall = thickness times surface area = 4 pi r^2 t
mass of wall = density * volume
= rho * 4 pi r^2 t

To find the mass of the cell wall, we need to calculate the volume of the cell wall first.

1. Calculate the volume of the cell:
The diameter of the cell is given as 2.10 ìm. Since the cell is assumed to be spherical, we can use the formula for the volume of a sphere:
Volume of a sphere = (4/3) * π * r^3

Given the diameter, we can find the radius (r) by dividing the diameter by 2:
r = diameter / 2 = 2.10 ìm / 2 = 1.05 ìm

Now we can calculate the volume of the cell:
Volume of the cell = (4/3) * π * r^3 = (4/3) * π * (1.05 ìm)^3

2. Calculate the volume of the cell wall:
The cell wall is described as a thin spherical shell, which means it has a thickness but no inner volume. We can calculate the volume of the shell by subtracting the volume of the inner sphere from the volume of the outer sphere.

The inner radius (r_inner) of the shell is found by subtracting the thickness of the wall from the radius of the cell:
r_inner = r - thickness = 1.05 ìm - 50.3nm

Now we can calculate the volume of the inner sphere:
Volume of the inner sphere = (4/3) * π * (r_inner)^3

The volume of the cell wall is then:
Volume of the cell wall = Volume of the outer sphere - Volume of the inner sphere

3. Calculate the mass of the cell wall:
Given that the density of the wall material is the same as that of pure water, we can assume a density value of 1 g/cm^3 (or 1000 kg/m^3).

We first need to convert the volume of the cell wall to cubic meters (m^3):
Volume of the cell wall (m^3) = Volume of the cell wall (ìm^3) * (1*10^-18 m^3 / 1 ìm^3)

Finally, we can calculate the mass of the cell wall:
Mass of the cell wall (kg) = Volume of the cell wall (m^3) * Density (kg/m^3)

To convert the mass to milligrams (mg):
Mass of the cell wall (mg) = Mass of the cell wall (kg) * (1*10^6 mg / 1 kg)

You can now plug in the values into the formulas and perform the calculations to find the mass of the cell wall in milligrams.

To find the mass of the cell wall, we'll need to calculate the volume of the cell and the volume of the cell wall.

1. First, let's calculate the volume of the cell:
Given, the diameter of the cell is 2.10 μm, which means the radius is half of the diameter.
Radius = 2.10 μm / 2 = 1.05 μm = 1.05 × 10^-6 m

The volume of a sphere is given by the formula:
V = (4/3)πr^3

Substituting the values:
V_cell = (4/3)π(1.05 × 10^-6)^3
V_cell ≈ 4.19 × 10^-18 m^3

2. Next, let's calculate the volume of the cell wall:
Given, the thickness of the cell wall is 50.3 nm, which means the radius of the cell wall is equal to the radius of the cell plus the thickness of the cell wall.
Radius_wall = (1.05 × 10^-6) + (50.3 × 10^-9) = 1.1003 × 10^-6 m

The volume of a thin spherical shell is given by the formula:
V_wall = 4π(r_wall^3 - r^3)

Substituting the values:
V_wall = 4π((1.1003 × 10^-6)^3 - (1.05 × 10^-6)^3)
V_wall ≈ 1.61 × 10^-24 m^3

3. Now, let's calculate the mass of the cell wall:
Given, the density of the cell wall material is the same as that of pure water, which is approximately 1000 kg/m^3.

The mass of an object is given by the formula:
m = density × volume

Substituting the values:
m_wall = 1000 kg/m^3 × 1.61 × 10^-24 m^3

To convert the mass from kg to mg:
m_wall = 1000 kg/m^3 × 1.61 × 10^-24 m^3 × 10^6 mg/g
m_wall ≈ 1.61 × 10^-15 mg

So, the mass of the cell wall is approximately 1.61 × 10^-15 mg.