Astronomy - interstellar dust grains

[accountkiller]

Homework Statement

A typical dust grain has a radius of 10^-7 m and a mass of 10^-14 g. How many dust particles are there in a molecular cloud containing 1000*Mass_sun of dusty gas if 1% of the cloud's mass is in the form of dust grains?

Homework Equations

The Attempt at a Solution

1000*M_sun = M_cloud * 0.01
Mass_cloud = 100,000*M_sun = 2 * 10³⁸ g.

Then do I just do:
(2 * 10³⁸ g) / (10^-14 g) = 2 * 10⁵¹ dust particles.

Is that correct?

Then, it asks to find the total surface area these (spherical) grains would take up if side by side. Do I just find S.A. = 4*pi*r² = 4*pi*(10^-7 m)² = 1.2566*10^-11 m². So:
1.2566*10^-11 m² * (2 * 10⁵¹ particles) = 2.5 * 10⁴⁰ m² for the surface area. Is that how I'm supposed to do it?

 

[Redbelly98]

Hi,

I have spotted a few errors, but for the most part your methods are correct.

Homework Statement

A typical dust grain has a radius of 10^-7 m and a mass of 10^-14 g. How many dust particles are there in a molecular cloud containing 1000*Mass_sun of dusty gas if 1% of the cloud's mass is in the form of dust grains?

Homework Equations

The Attempt at a Solution

1000*M_sun = M_cloud * 0.01

I interpret the question statement differently:

M_cloud = 1000*M_sun​

and

M_dust = 1% of M_cloud​

Mass_cloud = 100,000*M_sun = 2 * 10³⁸ g.

Then do I just do:
(2 * 10³⁸ g) / (10^-14 g) = 2 * 10⁵¹ dust particles.

Is that correct?

No, but once you get a value for M_dust, your method should work.

Then, it asks to find the total surface area these (spherical) grains would take up if side by side. Do I just find S.A. = 4*pi*r² = 4*pi*(10^-7 m)² = 1.2566*10^-11 m².

Correct method, but you should redo the calculation.

So:
1.2566*10^-11 m² * (2 * 10⁵¹ particles) = 2.5 * 10⁴⁰ m² for the surface area. Is that how I'm supposed to do it?

Yes, if you had the correct surface area for one dust grain, and the correct number of dust grains, that is how you would do it.

 

[accountkiller]

Well, it says that the molecular cloud contains 1000*M_sun of *dusty gas* - so dust particles. So that's only the dust particles, there's still all the other stuff that makes up a molecular cloud that isn't included in that, so the total M_cloud mass can't just be 1000*M_sun. It says 1% of the molecular cloud's total mass is dust grains... so 1% of the molecular cloud is the 1000*M_sun of *dusty gas* - dust grains. I guess I don't understand why you get 1000*M_sun = M_cloud when you look at the question, so I'm not sure which to use. Are you sure my way is not correct?

 

[accountkiller]

Oh, interstellar dust is defined in our textbook as "tiny, SOLID particles" so then 'dusty GAS' means the whole cloud, not just the solid particles.

Okay, thank you!

 

[Redbelly98]

You're welcome