1 (294)

Prévia do material em texto

286 8 ATOMIC STRUCTURE AND SPECTRA
where the approximation is for n ≫ 1 so that (2n+1) ≈ 2n and (n+1)2 ≈
n2
(b) For n = 100, this becomes
∆E = 2hcR̃H/1003
= 2 × (6.6261 × 10−34 J s) × (2.9979 × 1010 cms−1) × (109 677 cm−1)/1003
= 4.357 × 10−24 J
�e average radius of a H 100s orbital (Z = 1, n = 100, l = 0) is given by
the expression in Integrated activity I8.1
⟨r⟩100,0,0 =
(100)2a0
1
[1 + 1
2 (1 −
0(0 + 1)
1002
)] = 15 000 a0
�e ionization energy for an electron in the state with n = 100 is simply
minus the orbital energy
I = hcR̃H/1002
= (6.6261 × 10−34 J s) × (2.9979 × 1010 cms−1) × (109677 cm−1)/1002
= 2.179 × 10−22 J
(c) �e energy available in a collision is of the order of 12 kT (from the equipar-
tition principle), which at 298 K is 12 × (1.3806×10−23 JK−1)× (298 K) =
2.06×10−21 J.�is is well in excess of the ionization energy, so a collision
could easily result in ionization.
(d) �e minimum velocity required for ionization will be when the kinetic
energy of the H atom, Ek = 1
2mυ2 is equal to the ionization energy I;
hence υ = (2I/m)1/2
υ = (2I/m)1/2 = [2 × (2.179 × 10−22 J)/(1.0078 × 1.6605 × 10−27 kg)]1/2
= 510 ms−1
where the mass of H is taken as 1.0078 mu.
(e) �e radial wavefunction for a 100s orbital will have a �nite value at the
nucleus and 99 radial nodes.�e exponentially decaying part of thewave-
function, which dominates for su�ciently large distances, is of the form
e−r/100a0 .