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Critical density of the Universe

Using $H_0 = 70$ km s $^{-1}$ Mpc $^{-1}$ , derive the critical density $\rho_c$ from the Friedmann equation for a flat universe. Express it in kg m $^{-3}$ and in hydrogen atoms per cubic metre.

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Reading resources

Material that covers the physics behind this problem.

An Introduction to Modern AstrophysicsCarroll & Ostlie
The standard reference for olympiad-level astrophysics. Chapters 1-3 cover the celestial sphere, orbits and the magnitude system.
Textbook
Ohio State Astronomy Lecture NotesOhio State University
Clear university-level notes on stars, galaxies and cosmology with derivations you can follow line by line.
Lecture notes
Cosmology CalculatorEdward L. Wright (UCLA)
Converts redshift into distances, lookback time and scale factors. Use it to sanity-check cosmology answers.
Website