article on ppmv

7/28/2019 Article on ppmv

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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, V i, 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

ρ i=

M i

V i

.

(1)

It is useful to rearrange this as

V i=

M i

ρ i

.

(2)

The ideal gas law can be written as

pivi = 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*

≅ 8.31J 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=V

iand n

i=1 . Then (3) reduces to

! Revised Friday, January 6, 2012 ! 1

Quick Studies in Atmospheric Science Copyright David Randall, 2011



pstandard

V i = R*T

standard.

(4)

It follows that the molar volume is the same for all ideal gases. With pstandard

= 100 kPa and

T standard

= 0°C , the molar volume is V i= 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

ρ i=ni M i

vi

.

(5)

Using (5), we can rewrite the ideal gas law, (3), as

pi =ρ i

M i R

*T .

(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

p j

j =1

N

∑=

ni

n j

j =1

N

∑.

(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

pi

p j

j =1

N

∑=

ρ i

M iρ j

M j j =1

N

∑.

(8)

Comparing (7) and (8), we obtain





ni

n j

j =1

N

∑=

ρ i

M iρ

j

M j j =1

N

∑.

(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:

ptot≡ p j

j =1

N

∑ , ρ tot≡ ρ

j

j =1

N

∑ .

(10)

We can write the ideal gas law for the mixture as

ptot= ρ

tot R

*

M eff

⎛ ⎝ ⎜ ⎞

⎠ ⎟ T ,

(11)

where M eff

is the “effective molecular weight” of the mixture, defined by

ρ tot

M eff

≡ρ j

M j

⎛

⎝ ⎜

⎞

⎠ ⎟

j =1

N

∑ .

(12)

For air, M eff = 28.97 g mol-1. Using (12) in (9), we obtain

ni

n j

j =1

N

∑=

ρ i

M iρ

tot

M eff

=

ρ i

ρ tot

M eff

M i,

which can be rearranged to

ρ i

ρ tot

=

M i

M eff

⎛

⎝ ⎜⎞

⎠ ⎟ ni

n j

j =1

N

∑.

(13)

This says that the mass fraction is the mole fraction times the ratio of the molecular weights.





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.



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