(a) We can adapt Isaac Newton's formulation of
the expansion of the universe for this class:  the universe will expand 
forever if any object in that universe is expanding at a velocity at or 
above the escape velocity of all the matter in the universe holding it back.
The formula for escape velocity is something we've used often:

  v = sqrt(2GM/r)

In this case, M = (4/3) * pi * r^3 * (rho), where r is the radius of the 
universe and (rho) is the density of the universe at which v equals the
escape velocity -- known as the "critical density."

Furthermore, v = Hr as explained by the Hubble law of universal expansion,
where H is the Hubble constant.  Thus, the equation can be rewritten

            2 * G * (4/3) * pi * r^3 * (rho)
  Hr = sqrt(--------------------------------).  Square both sides, and the
                          r

"r" terms cancel completely!  So we don't even need to know how big the
universe is to compute the critical density.  We get

  H^2 = 2 * G * (4/3) * pi * (rho) 

Which, when rearranged, gives  (rho) = (3 * H^2)/(8 * pi * G).

Now, if H = 70 km/s/Mpc, we can rewrite H in units of "per second" like this:

         (70 km/Mpc)/s
  H = ------------------ = 2.26 * 10^-18  s^-1
      3.1 * 10^19 km/Mpc

So (rho) = 3 * (2.26 * 10^-18)^2 / (8 * 3.14 * 6.67 * 10^-11) 
         = 9.1 * 10^-27 kg/m^3

or, in hydrogen atoms,

   (rho) = (9.1 * 10^-27)/(1.67 * 10^-27) = 5.4 h-atoms/m^3.

(c) If the average mass density of the universe is about
2 hydrogen atoms per cubic meter, then this value is less than half the
critical density as calculated in Part (b) above.  Thus, the expansion
velocity of the universe is greater than the escape velocity.  
Based on our calculations, the universe  apparently will expand forever!