1a) We are initially given a large amount of data which is in
non-standard units. Before attempting the main part of the
question, it is useful to convert these quanitites into standard
units:
b) Since the volume is doubled and $R$, $T$ and $n$ are constant,
the pressure must halve.
$P_3 = \frac{P_2}{2} = 4.71 \times 10^4$ Pa
c) If the gas is not expanding against a vaccuum, then it will do
work against the external pressure. Thus, the energy of the gas
will decrease, and so its temperature will fall. Since $T$
decreases, this means that $P$ will have to decrease an additional
amount. The final pressure is lower than in the previous
scenario..
3a) We are back to the original scenario again. The volume and
temperature are kept constant, and so we can write:
4) This part of the problem is obviously quite open ended, in that
a number of different approaches may be taken. We could model such
a situation by assuming that the gas as compressible until the
particles are all touching. Additionally, we model each particle as
a cube:
2.47 moles of carbon atoms contains 14.87 x 10$^{23}$
molecules.
The radius of a carbon atom is ~ 77pm, and so each carbon atom can
be modelled as a cube of side 154pm.
Therefore the volume of a carbon atom is $3.63 \times 10^{-30}
$m$^3$, and so the total volume occupied by the carbon atoms is
$5.43 \times 10^{-6} $m$^3$
5 moles of air is composed of 30.11 x 10$^{23}$ molecules. Air is
largely composed of nitrogen, and so as a broad assumption, we can
assume that all of these molecules are nitrogen.
The radius of an N is 75pm, but since nitrogen exists as N$_2$, we
should model this as a rectangular prism, with two sides of length
150pm, and one of length 300pm.
Thus, the volume of a nitrogen molecule is $6.75 \times 10^{-30}$
m$^3$
Therefore, the overall volume occupied by nitrogen is $2.03 \times
10^{-5}$ m$^3$.
Summing the volumes for carbon and for nitrogen gives the total
occupied volume, which is $2.6 \times 10^{-5}\ $m$^3$. Using this
model, this is roughly the smallest volume that the gas could be
compressed to.