By considering P(h) = P(0) * exp(-(mgh/kT)) - (where P(h) is the pressure result of the equation; P(0) is the pressure at the "surface" of the planet; m is the mass of the molecule that is predominant in the atmosphere (in kg); g is the gravitational acceleration present on the planet; k is Boltzmann's constant; T is mean observed temperature (in Kelvin))
We can then analyse what height is needed for P(h) to be equal to 1/e of the surface value: the scale height
h(0) = kT/mg for P(h) to be equal to 1/e, and by substituting in standard values, and assuming a predominantly hydrogen gas atmosphere, and using g=GM/r^2, we derive the result (that varies according to the different values of g, m, and T that you use):
h(0)(Jupiter) = ~20km
Have fun ;-)