Air expands when heated and contracts when cooled.  The formula for
this phenomenon can be described with the following air density
equation:

D = D0 x (T0/T) x (P/P0), where

T0 = 545.69ºR (86ºF) or 303.15ºK (30ºC),
P0 = 14.7 psi (= 1 atm = 760 mm Hg = 29.92 in. Hg = 1.033 kg/cm2 =
1.013 bar = 33.9 ft H2O), and
D0 = 1.1649 g/L = 32.986 g/CF = 0.072751 lb/CF.
T, P, T0, and P0 must be in absolute temperature and pressure. Values
for T0, P0, and D0 were taken from tables in the CRC Handbook of
Chemistry and Physics.

I mention this formula only as a point of reference so you can loosely
see where I'm coming from.  OK, now for the question...

What formula would describe the exhaust gas density characteristics
for the exhaust gases of a four stroke gasoline engine?

Question will be considered answered when:
1) A formula is presented with underlying citations.
2) Each variable in the formula is sufficiently defined.
3) Source of all constants has been cited.

I presume that there is a standard gasoline formulation used in this
field of study and it is this "standard" that I expect will be used in
your answer.  You should assume a stoichiometric air/fuel ratio of
14.7:1 (lambda=1.0).

I will double the original question $$ amount if the formula you provide
allows for air/fuel ratios where lambda equals other than 1.0.

Note that a $5 tip will be given for web citations that include an
online "calculator" that allows one to calculate exhaust gas density
given varying lambda, temperature and pressure conditions.

An additional $2 tip will be given for web citations that include a
visual graph of the density curves for the exhaust gases of gasoline
engines.

Thanks and happy answering...  - John

OK, here's a start.

Assuming complete combustion; a fuel of formula CxHy will burn as follows

CxHy + x+(y/4) O2 -> xCO2 + y/2 H2O.

Both CO2 and H2O are likely to be gases at the temperature of
combustion, though of course the water will then probably condense as
it cools to ambient.

So you can perhaps assume that for every mole of CxHy you'll get x
moles of CO2 exhaust gas. A typical fuel for this purpose might be
octane (as per octane ratings etc).

Octane has the formula C8H18, and so you'll get 8 moles of CO2 per
mole of octane (114g), or ~ 4 UK ounces. It has the density of ~ 0.9
or so (http://www.simetric.co.uk/si_liquids.htm), so 114g has the
volume of around 125 ml or roughly a quarter of a pint.

Is this the sort of reasoned argument you're looking for?

S