Assume the average density of the universe is 0.1 of the critical density needed for closure. What is the average number of protons per cubic meter, assuming the universe is composed mostly of hydrogen?
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Final Answer

0.60 protons/m30.60 \textrm{ protons/m}^3

Solution video

OpenStax College Physics for AP® Courses, Chapter 34, Problem 29 (Problems & Exercises)

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Video Transcript
This is College Physics Answers with Shaun Dychko. This is the critical density the universe would need in order to have closure which is to say that it would need to have this much mass per cubic meter in order for gravity to stop the universe expansion and then bring it back down to a compressed singularity maybe perhaps for another Big Bang. And we are told to assume that the universe's density is actually one-tenth of this critical density so we have 0.1 times 10 to the minus 26 kilograms per cubic meter and given this assumption, how many protons must there be per cubic meter? Well, there's one proton for every 1.67 times 10 to the minus 27 kilograms and so we can see the units work out nicely here so that we have 0.60 protons per cubic meter.