Surface area of N spherical droplets?

[saber1357]

I have the following problem
Assume that 30.0 cm^3 of gasoline is atomized into N spherical droplets, each with a radius of 2.00 x 10^-3 m. What is the total surface area of these N spherical droplets?
I calculated the surface area of each atom to be 5x10^-9 m^2.
I also calculated the volume of each droplet to be 3.35x10^-14 m^3.
However, my mind can't seem to relate these numbers to my task. Any help is GREATLY appreciated.

 

[G01]

First things first. Next time, please post this in the correct Homework Help section. This forum is for general academic advice not pertaining homework problems.

Ok, so you know you have N atoms each with a known surface area. If you know the surface area of one, what is stopping you from finding the total surface area of all of them?

HINT: What would be the surface area of 2 droplets? 3? 4? ...

 

[Mene]

I can understand your confusion and I am happy to help you with this problem. To calculate the total surface area of N spherical droplets, we need to use the formula for the surface area of a sphere, which is 4πr^2, where r is the radius of the sphere. In this case, the radius of each droplet is 2.00 x 10^-3 m, so the surface area of each droplet would be 4π(2.00 x 10^-3)^2 = 5.03 x 10^-5 m^2.

To find the total surface area of N droplets, we need to multiply the surface area of one droplet by the number of droplets, which in this case is N. So, the total surface area would be 5.03 x 10^-5 m^2 x N = 5.03 x 10^-5 N m^2.

In order to find the value of N, we can use the given volume of gasoline, 30.0 cm^3, and the volume of each droplet, 3.35 x 10^-14 m^3. We know that 1 cm^3 is equal to 1 x 10^-6 m^3, so we can convert the given volume to cubic meters: 30.0 cm^3 = 30 x 10^-6 m^3. Now, we can set up a proportion to find the value of N:

30 x 10^-6 m^3 / 3.35 x 10^-14 m^3 = N / 1
Solving for N, we get N = 8.95 x 10^7 droplets.

Substituting this value of N into our previous equation for total surface area, we get:
5.03 x 10^-5 m^2 x 8.95 x 10^7 = 4.50 x 10^3 m^2

Therefore, the total surface area of N spherical droplets with a radius of 2.00 x 10^-3 m is 4.50 x 10^3 m^2. I hope this helps you understand the problem better and how to approach it mathematically. If you have any further questions, please do not hesitate to ask.