!
Revised Friday, January 6, 2012!
1
ppmv
David Randall
In discussions of the carbon dioxide content of the atmosphere, the relative abundance of
CO2 is usually discussed in terms of ppmv, or “parts per million by volume.” I wrote these rather
simple-minded notes in an effort to understand what ppmv really means, and how it relates to the
mass mixing ratio.
A mole is a quantity of substance corresponding to Avogadro's number of the atoms,
molecules, electrons, or other discrete entities of which the substance is composed. Avogadro’s
number is 6.0221415 × 1023.
Consider a substance denoted by subscript i . The molar mass of the substance, M i , is the
mass of one mole of the substance. The molar volume, Vi , is defined to be the volume occupied
by one mole of the substance, at a standard temperature and pressure. It follows that the density
is
ρi =
Mi
.
Vi
(1)
It is useful to rearrange this as
Vi =
Mi
.
ρi
(2)
The ideal gas law can be written as
pi vi = ni R*T ,
(3)
where pi is the partial pressure of the gas (in general not the standard pressure); vi is the volume
occupied (in general not the molar volume); ni is number of moles present; R* ≅ 8.31 J K-1 mol-1
is the universal gas constant; and T is the temperature (in general not the standard temperature),
which is assumed to be the same for all species present.
For the special case of one mole at the standard temperature and pressure, we know that
vi = Vi and ni = 1 . Then (3) reduces to
Quick Studies in Atmospheric Science
Copyright David Randall, 2011
!
Revised Friday, January 6, 2012!
2
pstandardVi = R*Tstandard .
(4)
It follows that the molar volume is the same for all ideal gases. With pstandard = 100 kPa and
Tstandard = 0°C , the molar volume is Vi = 2.27 × 10 −2 m3 mol-1.
With ni moles, still at the standard temperature and pressure, the total volume is simply
multiplied by ni . It follows that, at the standard temperature and pressure, the mole ratio of a
gas is the same as its volume ratio. This is the origin of the expression “parts per million by
volume,” or ppmv. The true meaning is “moles per million, by volume.” Personally, I find the
term ppmv confusing and even misleading, because the gas under discussion is usually not at the
standard temperature and pressure.
Returning to the general case, the mass of the sample is the number of moles times the
molar mass, i.e., ni M i , so the density satisfies
ρi =
ni M i
.
vi
(5)
Using (5), we can rewrite the ideal gas law, (3), as
ρi *
RT .
Mi
pi =
(6)
From (3), for two gases at the same temperature and occupying the same volume vi , the
ratio of the number of moles is the same as the ratio of the partial pressures. More generally, for
N gases occupying the same volume, and sharing the same temperature, we can write
pi
=
N
ni
∑ p ∑n
j
j =1
.
N
j
j =1
(7)
This says that the ratio of the partial pressure of gas i to the total pressure is the same as the ratio
of the number of moles of gas i to the total number of moles. Similarly, from (6) we find that
ρi
pi
M
= N i .
N
ρ
∑ p j ∑ Mj
j
j =1
j =1
(8)
Comparing (7) and (8), we obtain
Quick Studies in Atmospheric Science
Copyright David Randall, 2011
!
Revised Friday, January 6, 2012!
3
ρi
ni
M
= N i .
N
ρ
∑ n j ∑ Mj
j
j =1
j =1
(9)
This could also be obtained directly from (5).
Let ptot and ρ tot , respectively, be the total pressure and total density of a mixture of gases:
N
N
j =1
j =1
ptot ≡ ∑ p j , ρ tot ≡ ∑ ρ j .
(10)
We can write the ideal gas law for the mixture as
⎛ R* ⎞
ptot = ρ tot ⎜
⎟T ,
⎝ M eff ⎠
(11)
where M eff is the “effective molecular weight” of the mixture, defined by
N ⎛
ρj ⎞
ρtot
≡ ∑⎜
.
M eff j =1 ⎝ M j ⎟⎠
(12)
For air, M eff = 28.97 g
mol-1.
Using (12) in (9), we obtain
ρi
ni
Mi
=
N
ρtot
∑ n j M eff
j =1
=
ρi M eff
,
ρtot M i
which can be rearranged to
ρi ⎛ M i ⎞ ni
=
.
ρtot ⎜⎝ M eff ⎟⎠ N
∑ nj
j =1
(13)
This says that the mass fraction is the mole fraction times the ratio of the molecular weights.
Quick Studies in Atmospheric Science
Copyright David Randall, 2011
!
Revised Friday, January 6, 2012!
4
As an example, consider the following:
•
The total mass of the atmosphere is about 5 x 1021 g.
•
The total mass of water vapor in the atmosphere is about 1.25 x 1019 g.
•
The mass fraction of water vapor in the atmosphere is about 2500 ppm.
•
From (13), the molar fraction of water vapor is about (29/18)*2500 ≅ 4000 ppmv. Note that
I have written ppmv, as is conventional, even though this is really the molar fraction.
•
For comparison, the current molar fraction of CO2 in the atmosphere is about 400 ppmv,
which is about one tenth the molar fraction of water vapor.
•
From (13), the mass fraction of CO2 is about (44/29)*400 ≅ 600 ppm. The mass fraction of
CO2 is thus about ¼ the mass fraction of water vapor.
Quick Studies in Atmospheric Science
Copyright David Randall, 2011