8/7/2019 Atmospheric_thermodynamics_Lecture
1/33
Atmospheric Thermodynamics
The Empirical Gas Laws and the
Ideal Gas Law
8/7/2019 Atmospheric_thermodynamics_Lecture
2/33
Empirical Gas Laws
The empirical gas laws form the basis for
the analysis and forecasting of
thermodynamics processes in theatmosphere.
Empirical means that the laws were
developed through experimentation and
observation of gases.
8/7/2019 Atmospheric_thermodynamics_Lecture
3/33
Empirical Gas Laws (Cont.)
According to Boyles law at a constant
temperature the pressure of a gas is
inversely proportional to its volume.
p w 1/V
8/7/2019 Atmospheric_thermodynamics_Lecture
4/33
Empirical Gas Laws (Cont.)
According to Gay-Lussacs law at a
constant pressure the temperature of a
gas is directly proportional to its volume.
T w V
8/7/2019 Atmospheric_thermodynamics_Lecture
5/33
Ideal Gas
An ideal gas is any gas in which
thermodynamic processes can be
described exactly by Boyles Law andGay-Lussacs Law.
8/7/2019 Atmospheric_thermodynamics_Lecture
6/33
Ideal Gas Law
Boyles Law and Gay Lussacs Law can be
combined into a single equation called the
ideal gas law (or the equation of state).
pV/T = constant
This form of the ideal gas law applies to a
gas composed a single type of molecule.
8/7/2019 Atmospheric_thermodynamics_Lecture
7/33
Ideal Gas Law (Cont.)
We need to modify this form of the ideal gas
law to account for the different gases that
make up the composition of a mixture ofgases like we find in the Earths
atmosphere.
8/7/2019 Atmospheric_thermodynamics_Lecture
8/33
Empirical Gas Laws (Cont.)
Avagadros Law states that at a specifictemperature and pressure one mole of agas will occupy the same volume as one
mole of any other gas.
A mole of a gas contains as many
elementary units (molecules, atoms, etc.)as there are C atoms in exact 0.012 kg ofC12.
8/7/2019 Atmospheric_thermodynamics_Lecture
9/33
Ideal Gas Law (Cont.)
Based on this observation the ideal gas lawis sometimes written as
pV = nR*T
where
n is the number of moles of a gas, and
R* is the universal gas constant
R* = 8.314 J mol-1 K-1
8/7/2019 Atmospheric_thermodynamics_Lecture
10/33
Ideal Gas Law (Cont.)
Since
n = m/Mwhere
m is the mass of the gas, and
M is the molecular mass (weight) of the gas,
8/7/2019 Atmospheric_thermodynamics_Lecture
11/33
Ideal Gas Law (Cont.)
we can rewrite the ideal gas law as
pV = mR*T/M
If we divide both sides by the volume, we
getp = mR*T/(MV)
8/7/2019 Atmospheric_thermodynamics_Lecture
12/33
Ideal Gas Law (Cont.)
p = mR*T/(MV)
Since density, , is mass divided by volume,
= m/V
we can write the ideal gas law as
p = R*T/M
8/7/2019 Atmospheric_thermodynamics_Lecture
13/33
Ideal Gas Law (Cont.)
p = R*T/M
The specific gas constant, R, for any gasis defined as
R = R*/M
Substitution producesp = RT
8/7/2019 Atmospheric_thermodynamics_Lecture
14/33
Ideal Gas Law (Cont.)
This form of the ideal gas law is valid for a
single gas, because it contains this
specific gas constant, R, for a single gas.
We need to modify R, if we want to create
an ideal gas law that we can use for the
mixture of gases in the Earths
atmosphere.
8/7/2019 Atmospheric_thermodynamics_Lecture
15/33
Empirical Gas Laws (Cont.)
According to Daltons Law the total
pressure exerted by a mixture of gases is
equal to the sum of the partial pressures
exerted by the individual gases that
comprise the mixture.
p = pi = p1 + p2 + p3 + + pn
8/7/2019 Atmospheric_thermodynamics_Lecture
16/33
Ideal Gas Law (Cont.)
p = pi
Using a form of the ideal gas law we hadearlier we can substitute forpi and write
Daltons Law as
p = ((miR*Ti)/(ViMi)
8/7/2019 Atmospheric_thermodynamics_Lecture
17/33
Rule for Summations
Anything that is a constant can be pulled
outside the summation symbol and
multiplied after summing the remaining
terms.
8/7/2019 Atmospheric_thermodynamics_Lecture
18/33
Assumptions About a Mixture of
GasesIf the mixture of gases is well mixed by the
winds, then all of the individual gases in
the mixture will occupy the same volume
(i.e. V1 = V2 = V3, etc.)
8/7/2019 Atmospheric_thermodynamics_Lecture
19/33
Assumptions About a Mixture of
Gases (Cont.)If the mixture of gases is well mixed by the
winds, then molecular collisions will
transfer internal energy and all of the
gases in the mixture will eventually have
the same temperature (i.e. T1=T2=T3, etc.)
By definition the universal gas constant, R*,
is a constant.
8/7/2019 Atmospheric_thermodynamics_Lecture
20/33
Ideal Gas Law (Cont.)
p = ((miR*Ti)/(ViMi)
Thus, we can pull R*, T, and V out of thesummation and write
p = (R*
T/V) (mi/Mi)
8/7/2019 Atmospheric_thermodynamics_Lecture
21/33
Ideal Gas Law (Cont.)
p = (R*T/V) (mi/Mi)
Multiply the right hand side by
(mi/mi)
to getp = (R*T)(mi/V)((mi/Mi)/(mi)
8/7/2019 Atmospheric_thermodynamics_Lecture
22/33
Ideal Gas Law (Cont.)
Since mi = m, total mass of all of the
gases in the mixture, then
(mi/V) =
8/7/2019 Atmospheric_thermodynamics_Lecture
23/33
Ideal Gas Law (Cont.)
and
p = (R*T)(mi/V)((mi/Mi)/(mi))
becomesp = (R*T)(mi/Mi)/(mi)
8/7/2019 Atmospheric_thermodynamics_Lecture
24/33
Mean Molecular Mass (Weight)
Meteorologists define a mean molecular
mass (weight), M, such that
1/M = ( (mi/Mi))/(mi)
8/7/2019 Atmospheric_thermodynamics_Lecture
25/33
Dry Air
In thermodynamics the term dry airrefers to
the normal mixture of gases observed in
the Earths atmosphere, except for the fact
that it contains no watervapor.
8/7/2019 Atmospheric_thermodynamics_Lecture
26/33
Major Components of Dry Air
Gas M (kg/mol) Mass Fraction R (J kg-1 K-1)
Nitrogen (N2) 0.028 0.755 296.8
Oxygen (O2) 0.032 0.232 259.8
Argon (Ar) 0.040 0.013 208.1Carbon
dioxide (CO2) 0.044 0.0005 188.9
8/7/2019 Atmospheric_thermodynamics_Lecture
27/33
Mean Molecular Mass of Dry Air
Meteorologists define the mean molecular
mass of dry air, Md based on the mass
fractions and molecular masses of the
individual gases in the table.
If we assume we have one kilogram of dry
air and plug the appropriate numbers into
the equation for 1/M, we get
8/7/2019 Atmospheric_thermodynamics_Lecture
28/33
Mean Molecular Mass (Weight)
Meteorologists define a mean molecular
mass (weight), M, such that
1/M = ( (mi/Mi))/(mi)
8/7/2019 Atmospheric_thermodynamics_Lecture
29/33
Mean Molecular Mass for Dry Air
(Cont.)
0.755 kg + 0.232 kg + 0.013 kg + 0.0005 kg1/Md = 0.028 kg mol
-1 0.032 kg mol-1 0.040 kg mol-1 0.044 kg mol-1
0.755 kg + 0.232 kg + 0.013 kg + 0.0005 kg
Md = 0.029 kg mol-1
8/7/2019 Atmospheric_thermodynamics_Lecture
30/33
Md = 0.029 kg mol-1
Gas M (kg/mol) Mass Fraction R (J kg-1 K-1)
Nitrogen (N2) 0.028 0.755 296.8
Oxygen (O2) 0.032 0.232 259.8
Argon (Ar) 0.040 0.013 208.1Carbon
dioxide (CO2) 0.044 0.0005 188.9
Since approximately 75% of the mass of dry air is N2
, itshould make sense that our Md is just a little larger thanM for N2.
8/7/2019 Atmospheric_thermodynamics_Lecture
31/33
Ideal Gas Law for Dry Air
p = (R*T)(mi/Mi)/(mi)
If we let
1/Md = ( (mi/Mi))/(mi)
then we can substitute to get
p = (R*T)/Md
8/7/2019 Atmospheric_thermodynamics_Lecture
32/33
Gas Constant for Dry Air
Meteorologists often define as gas constantfor dry air, Rd, as
Rd = R*/Md
Rd = 8.314 J mol-1 K-1 / 0.029 kg mol-1
Rd = 287 J kg-1 K-1
8/7/2019 Atmospheric_thermodynamics_Lecture
33/33
Ideal Gas Law for Dry Air (Cont.)
p = (R*T)/Md
and
Rd = R*
/Mdso substitution produces
p = RdT
which is the most common form of the idealgas law for dry air.