Problem with molecular separation calculations

[emjay66]

I am attempting to complete a basic question involving molecular separation from Alonso & Finn "Fundamental University Physics" Second Edition, Chapter 2, Question 9. The Problem states
"Using the data in table 2-1 and A-1, estimate the average separation in molecules of hydrogen at STP (gas), in water (liquid), and in iron (solid)."
Now, Table 2-1 contains density values (relative to water). The relevant values are:
Iron = 7.86
Water (4 Degrees C) = 1.000
Hydrogen = $8.988\times 10^{-5}$
Table A-1 is the periodic table of the elements and the relevant values for this are:
Hydrogen = 1.00797
Oxygen = 15.9994
Iron = 55.847

I will only show my attempt to compute the answer for Hydrogen since I feel I am (probably) making the same mistake throughout.
An inspection of the answers in the back showed that for Hydrogen they obtained:
3.39*10^-7 cm for spherical geometry and 1.14 * 10^-8 cm for cubic geometry

Firstly I assume a Hydrogen "molecule" is two atoms of Hydrogen
I also assume the calculation for a mass of 1kg of the relevant substance

the molecular mass of a Hydrogen molecule = 2 * 1.00797 = 2.01594 g/mol
Number of mole of Hydrogen in 1kg = 1000/2.01594 496.047 mol
Number of Hydrogen molecules = 496.047 * (6.02*10^23) = 2.986*10^26 molecules
Number of Hydrogen molecules at STP = 2.986*10^26 * (8.988*10^(-5)) = 2.68*10^22 molecules

At this point (for a cube) I know 1kg = 1000 cubic centimetres
which means the cube root of 2.68*10^22 (= 29940610.46) would give the number of molecules along one edge (10cm)
which (to me) would mean that 10/29940610 = 3.34*10^(-7) cm, which is clearly wrong.
For the Spherical geometry I reasoned $1000 = \frac{4}{3}\pi\,r^3$ which would mean $r = (\frac{3000}{4\pi})^{1/3}$ = 6.2 cm which means the diameter would be
12.4 cm. This would give a separation of 4.4*10^(-7) cm, which is also wrong.
I understand that my thinking is clearly faulty, so some pointers in the right direction would be nice. Thank you in advance.

 

[DrClaude]

which (to me) would mean that 10/29940610 = 3.34*10^(-7) cm, which is clearly wrong.

Why is that clearly wrong?

 

[emjay66]

I only feel my approach is wrong or incomplete because my answers do not match those contained at the back of the text (also stated in the original email above). Unfortunately I have no other assertion to base it on. (ps I was a little rushed with my OP so I apologise if that wasn't entirely clear)

 

[BvU]

Did you notice that the book answers to b) and c) both have spherical < cubic ?
Perhaps you should try those and see what you find!

 

[DrClaude]

What does the book give as an answer? I get a slightly different answer than yours when considering that each molecule occupies a sphere, but you have the correct answer when considering the volume to be cubic.

See also https://www.physicsforums.com/showthread.php?t=394617

 

[emjay66]

Ok the answer for each (in the back of the book) are as follows (in spherical, cubic) order (all numbers are cm):
Hydrogen: 3.29*10^-7, 1.14*10^-8
Water: 3.85*10^-8, 6.21*10^-8
Iron: 2.82*10^-8, 4.46*10^-8

Some rethinking about the problem gave me the following:
mass of Hydrogen molecule = (2*1.00797)* 1.6606*10^-27 = 3.35*10-27kg = 3.35*10^-24g
if Density of Hydrogen at STP = 8.988*10^-5 g cm^-3 then Volume = 3.35*10^-24/8.988*10^-5 = 3.72*10^-20cm-3
Assuming the Hydrogen molecule is a sphere, then 3.72*10^-20 = 4/3 * pi * r^3 which means r (radius of Hydrogen molecule) = 2.07*10^-7 cm which means the diameter is approximately 4.14*10^-7 cm
Number of molecules of Hydrogen in 1kg = 2.684*10^19 (as before)
which is equivalent to (2.684*10^19)^(1/3) which is approximately 3*10^6 along one 10cm edge
This means that 3*10^6 molecules would occupy (with no gaps) 3*10^6*4.14*10-7 which is app 1.24 cm
leaving 8.758cm to be used as gaps for 3*10^6 molecules. This would mean that the inter molecule gap is 8.758/3*10^6 = 2.9*10^-6 cm for a cube
For a sphere I use the Wigner-Seitz approximation (with n = 3.72*10^-10cm^-3) which would give a inter molecule gap of 2.07*10^-7 cm.
I will post my other rework later.