How big a volume of intergalactic space would equal mass of Milky Way?

[amaliablachman]

1. Most of the atoms in the universe are not in stars but in ionized gas in
intergalactic space. The Milky Way has mass in its stars equivalent to 50 billion
Suns (5 x 10^10 Msun). If the gas in the intergalactic medium only has a density of
one proton per cubic meter, about how big a volume of intergalactic space would
you need to have to equal the mass of the Milky Way? (There are about 10^57
protons in the sun.)

2. v = m/d3. Where I'm going wrong...being too tired, too frustrated, or too annoyed to figure out conversions. I know I have to find the volume, and I know I have the mass and density. But I have no idea where to begin in terms of figuring out what exactly the density is based on knowing the gas in the intergalactic medium "only has a density of one proton per cubic meter." I determined the hypothetical mass (50 billion times the mass of the sun) to be 9.9455 x 10^40, by the way...but again, am just too braindead to come close to figuring out the volume conversion. We never went over this in class, it has very little to do with what we're studying at this moment, but my professor always includes random hypotheticals on our homework because he is satan. Once I am pointed in the right direction in terms of figuring out the volume, it will be clearly very easy to solve because it's basic division, so if someone can help me figure out how to convert the density into something that's actually useful, I will cyber high five you so hard.

 

[Borek]

What is the mass of a single proton?

 

[amaliablachman]

I know the mass of a single proton is 1.67262178 × 10^-27 kg...but I honestly have no idea where to go from here. I'm completely blank and just plugging things into the wrong places. I'm sure this is way simpler than I'm making it out to be in my head but it's 6:19 am, I have been doing this insane problem set for the past 14 hours, this is the final question before I can finally go to bed...and I'm just completely unable to think about it in a way that makes sense to me.

 

[amaliablachman]

Would I just do v = m/d where m equals the mass of the Milky Way (9.9455 x 10^40) and d equals 1.67262178 × 10^-27 kg per cubic meter?

 

[amaliablachman]

...and if that's the case, then the volume needed for intergalactic space to equal the mass of the Milky Way would have to be 5.9460543 x 10^67 cm3?

 

[Borek]

Would I just do v = m/d where m equals the mass of the Milky Way (9.9455 x 10^40) and d equals 1.67262178 × 10^-27 kg per cubic meter?

Yes. I am not sure about the mass of the Milky Way, as it is not what I got multiplying numbers you gave in the original question, and data I was able to google is slightly different, but you are on the right track.

...and if that's the case, then the volume needed for intergalactic space to equal the mass of the Milky Way would have to be 5.9460543 x 10^67 cm3?

Why have you switched to cm³?

 

[amaliablachman]

Whoops, I meant m^3!

For the mass of the Milky Way, I did the mass of the sun (1.989 x10^30 kg) and then multiplied it by 50 billion, since the problem states that the mass of the Milky Way is 50 billion times the mass of the sun...right? Am I missing something there?

 

[Borek]

Whoops, I meant m^3!

OK

For the mass of the Milky Way, I did the mass of the sun (1.989 x10^30 kg) and then multiplied it by 50 billion, since the problem states that the mass of the Milky Way is 50 billion times the mass of the sun...right? Am I missing something there?

Initially you wrote about 10⁵⁷ protons, which is not equivalent to 1.989×10³⁰ kg.