P.C. Chau (UCSD, 1999)
A word on unit conversion
One of the most vexing calculations is unit conversion, especially when we have to deal with
the (crazy) units used by the industry. There is no way to get around this problem. Make sure you
have a conversion table handy. Appendix D in Middleman's text suffices for this course. Important:
also make sure you are aware of and can use the physical property estimations in Appendix C.
Here, we do two sample calculations dealing with ppm (parts per million). One thing to keep
in mind is that when we use ppm as a unit, we usually are working with very dilute solutions. As
such, the numerical value of mole fraction is for all practical purpose the same as mole ratio.
What is a ppm?
1. What is 300 ppm SO2 in water?
First, 300 ppm means 300 parts of SO2 in 106 parts of water. By "parts" in the liquid phase,
we imply the use of mass as the measure, so 300 ppm is, say, in cgs units, 300 g SO2 in 106 g of
water.
For dilute aqueous solutions, the density is essentially that of water, 1 g/cm3, and the unit of
a ppm is roughly the same as mg/liter:
1 ppm = 10–6
g
g
mg
≈ 10– 6
=1
g H 2O
cm 3 H 2O
liter
Thus 300 ppm can be taken as 300 mg/liter. With MW = 64 for SO2, we can easily calculate the
molar concentration of SO2 to be 4.69 mmol/liter (or mM).
More formally, we should have written ppm as ppmw to denote that it is on a weight basis,
but few people do that. Also, note that the molecular weight really has units. In this example with
SO2 and cgs units, we have MW = 64 g/gmol. 1
Next, what is the mole fraction of SO2? Here, we can simply approximate it as the mole
ratio:
g SO2
18
300 ppm SO2 = 3 x 10–4
x
≈ 8.44 x 10–5 mole fraction
g H2O
64
2. What is 50 ppmv SO2 in air (at 1 atm)?
Now with gases, we often use partial pressure as a concentration unit, which is really a
volume based unit of concentration. Likewise, we use the notation ppmv to denote parts per
million by volume. For an ideal gas mixture, relative quantity measures of volume, partial
pressure and number of moles are the same. Again approximating mole fraction as mole ratio, and
with the ideal gas law, the unit of ppmv is for all practical purpose the partial pressure of SO2:
pA ≈ 50 x 10–6
cm 3 SO2
= 50 x 10–6 atm
cm 3 air
Take note that in such a dilute gas mixture, the partial pressure of air is essentially the total
pressure (1 atm in this example). With the partial pressure, we can calculate easily the molar
concentration of SO2, say, at T = 298 K, as
cA =
pA
gmol
= 2.05 x 10–9
(82.05)(298)
cm3
1
For those who have taken a "mol" for granted, it is one gram-mole (gmol) of any given
chemical species that contains 6.02 x 1023 (Avogadro’s number) molecules. In engineering, we
work with different unit systems, and it becomes important to denote the mass basis, and hence the
use of, for example, gmol and kgmol (also kmol). You'd find this convention to be the case with
how the value of the gas constant is listed in handbooks. Take note that the notation of, say, one
kgmol of a species really means the mass of that species in kg that is numerically equal to its
molecular weight. If the molecular weight of a species is M, we have the different quantities of 1
gmol = M g, 1 kg = M kgmol, 1 lb-mol = M lbm, and so forth.
1
P.C. Chau (UCSD, 1999)
This step reminds us that it is important to go back to our old chemistry text and find the gas
constant in different units. Here are three handy ones:
3
3
R = 82.05 cm atm = 8.314 m Pa = 1.987 cal
gmol.K
gmol.K
gmol.K
You may want to arm yourself with other ones which use SI units.
What is the unit of Henry's law constant?
Well, this is another unit conversion that irritates us. A common textbook definition of
Henry's law for species A is 2
pA = HCA, H [=]
atm.cm3
gmol
where H is the Henry's law constant, and pA and CA are the partial pressure and liquid phase molar
concentrations of species A. By [=] we mean "has units of." We have taken the liberty of choosing
atm as the unit for the partial pressure and the cgs molar concentration for the molar concentration.
These are "mixed," but commonly used units.
First, we may convert the molar concentration of A to mole fraction xA using the molar
density of the solution. For dilute aqueous solutions which is mostly water, we would use CT =
1/18 gmol/cm3, and approximate the mole fraction as a mole ratio. Now the Henry's constant
would have the unit of atm. This is the form that is favored by Middleman's text.
pA = HCT
CA
atm
= H'xA , H' = HCT [=] atm or
mole fraction
CT
Next, it is not uncommon that we'd find a handbook, especially in the environmental field,
which uses molar concentration in the gas phase. Now, the Henry's constant is dimensionless.
pA
H
H
=
C = H"CA , H" =
[=] 0
RT RT A
RT
Again, if the partial pressure is in atm and CA is in gmol/cm3, we would use R = 82.05
cm3.atm.gmol–1K–1 such that the converted gas phase concentration is also in cgs units of
gmol/cm3.
Finally, do not be surprised if you look up some text or a handbook and the "Henry's
constant" is reported as the reciprocal of whatever form is written above.3 You just have to be
vigilant in checking the units when you solve a problem.
What is a volume flux?
The first instance of "odd" units is in page 16 of Example 2.1.5. We usually use molar flux in
the cgs unit of gmol.cm–2s–1. For gas permeation, we can use the ideal gas law to convert the
number of moles to volume, and we'd have the unit of cm3.cm–2s–1, which is the volumetric flux
in (2.1.57).
The unit of the corresponding gas permeability coefficient K is defined accordingly in (2.1.59
and 60). Go through the equations carefully to see that they are correct.
2
For Chemical Engineering majors, you should realize that there is more to this. Consult your
thermodynamics text.
3
It is not uncommon to denote the reciprocal of Henry's constant as the solubility coefficient
bp as used in Examples 2.1.5 and 6. Check that the unit of bp in (2.1.63) is the reciprocal of the
unit of a Henry's constant.
2