On moist air and dew points
Daidzic, N. E. (
On moist air and dew points
Nihad E. Daidzic Ph.D.
AAR Aerospace Consulting
Dynamics of moist air and condensation of water vapor in terrestrial
atmosphere is responsible for many essential weather phenomena, some of them
also quite hazardous. To understand many atmospheric phenomena and weather
dynamics, we need to tour the field of thermodynamics and more specifically
atmospheric thermodynamics and psychrometry. Thermodynamics is
wellestablished fundamental scientific discipline essential to many engineering and
atmospheric sciences, but is rather absent from the aviation/aeronautics
curriculums. Hence, to understand the basics of moist air, atmospheric temperature
lapse rates (ALR), dew points (and frost points), lifting condensation levels (LCL),
etc., we need to comprehend fundamental thermodynamic processes first.
In aviation sciences and operations, moist air also affects aircraft
performance, pressurization and air-conditioning systems (HVAC, HLS, and ECS
systems), etc. Density altitude (DA) is a weak, nevertheless not negligible function
of air moisture content. This is especially important for jet- and non-turbocharged
piston-engines operations in warm climates and weather conditions with expected
reduced performance in high air humidity. Air conditioning for use in aircraft?s
pressurized vessels necessarily has to remove large quantities of moisture to prevent
ice formation and fuselage structural damages. Even such relatively dry cabin air
will lead to swift cooling and condensation in the case of explosive/rapid
decompressions filling the cabin with mist.
The main purpose of this review article is the desire to summarize and
present fundamental thermodynamic considerations of moist air,
atmosphericwater phase transitions, and dew points in a clear way. Of course, like in everything
else it is not possible to cover every aspect of the topic presented. The goal is also
to provide consistent derivations with necessary mathematical rigor. In addition,
we provide some practical working equations for computation of specific- and
relative-humidity (RH) from measured (dry) air and dew point (DP) temperatures
and vice versa. Intent is also to provide deeper understanding of atmospheric
thermodynamics and stress its importance and consequences, directly or indirectly,
to flight operations. Thermodynamics is of essential important to meteorologists,
but it is also important to aviation and aeronautics professionals, practitioners,
pilots, operators, aviation/aeronautics education and air transportation in general.
To spare an average reader from simultaneously reading and consulting
many expert books in theoretical, applied, and statistical thermodynamics, kinetic
theory of gases, quantum statistics, and hundreds of articles dealing with the
fundamental problems of thermodynamics, air motion and instability, we have
summarized basic concepts. We tried to present this complex topic in a coherent
uniform way using standard terminology. Nomenclature is provided at the end and
just before references. Hence, parts of this lengthy article can be seen as short
tutorials in important elements of atmospheric and engineering thermodynamics
intended for professionals and students of aeronautics, aerospace, and aviation. In
addition to this basic tailored review, we are also providing some new alternative
working relationships for useful computations.
We consulted several expert books in equilibrium and non-equilibrium
theoretical and applied (engineering) thermodynamics, statistical
physics/mechanics, molecular physics, and kinetic theory of gases to verify our
derivations. Among these are
de Groot and Mazur (1984
, Lay (1963), Reif (1965), Saad (1966),
, and Wannier (1987). From books with specific
emphasis on atmospheric thermodynamics we used texts by
Iribarne and Godson
Bohren and Albrecht (1998)
. We also used
fundamentals manual with the focus on HVAC engineering and applications
where physics of moist air is of paramount importance. Of the atmospheric
science/physics, cloud physics, and dynamic meteorology books we consulted
Fleagle and Businger (1980),
Iribarne and Cho
, Rogers (1979),
Wallace and Hobbs
. Popular meteorology and weather books by Forrester (1981) and
were also consulted. Aviation and pilot-oriented meteorology and weather
texts, such as, FAA (2016) do not provide much information on this matter. Many
articles dealing with the specific topic discussed here were used and will be cited
where and when appropriate.
This article is structured so that the basic thermodynamic theory, principles
and properties of dry and moist air mixtures are summarized first. Next, derivation
of phase transition equations and analytical integration of various phase transitions
is presented. Finally, we introduce the concept of dew point and present some
practical equations for computations of DPs and RH. This article primarily serves
as a review of essential important atmospheric thermodynamic processes and
provides a list of most relevant references that readers may use in further learning
and research. We are also clarifying some common and widespread misconceptions
about the moist air.
In this section, we present basic thermodynamic theory of dry and moist air
and treat it as an ideal gas. While dry air in lower atmosphere behaves closely as an
ideal gas, the fact is that moist air having a condensable phase shows larger
departure from an ideal gas approximation.
Dry Air as an Ideal Gas
Homosphere?s atmospheric air is a dilute mixture of gases (overwhelmingly
diatomic molecular Nitrogen and Oxygen). Dry air is a mixture of non-condensable
gases, i.e., gases that will not change phase (e.g., liquefy) in terrestrial atmospheric
conditions. Gas per se is defined as a state-of-the-matter at temperatures above
respective critical point (CP) temperature. Atmospheric dry air exists at
temperatures well above its critical temperatures. No amount of pressure increase
will liquefy gas, while vapors will liquefy (condense) under increasing pressures
(or reduced temperatures). Humid or moist air contains variable amount of water
vapor that could change into liquid (condensate) and/or solid (ice). For most
practical applications, air is considered an ideal- or perfect-gas as the pressures are
relatively low and temperatures not too high:
pV = n?T = mRdT
p ? v = ?T
p ? v =
= R T
Mass specific volume v , molar specific volume v , dry air gas-constant Rd,
mass m, amount of mass (number of moles) n , molar mass M , and the molecular
mass M are:
m ??m3 kg?? v =
n ??m3 kg ? mol?? Rd =
?J kgK? m = nM = nN0M ?kg? N0 =
In ideal-gases, the internal energy u and the enthalpy h are both functions
of temperature alone. An ideal-gas is also sometimes called thermally perfect, while
if additionally, the specific heat capacities are constant (temperature independent)
then it is also calorically perfect
. Important physical constants are
Avogadro number N0, which defines the number of molecules per g-mol (or
kilogram-mol) of substance, the universal gas-constant ?, which is related to
specific gas-constant through molecular mass per mole, and the Boltzmann?s
constant kB, given as:
N0 = (6.022045 ? 0.000031) ?1026 molecules
kg ? mol
? = 8314.41
kg ? mol ? K
= 1.3807 ?10?23
molecule ? K
The molar mass of dry air is 28.9644 (kg/kg-mol). Vapors are defined as
gaseous phases below critical point?s (CP) temperature (TCP) close enough to the
gas-liquid equilibrium curves where increasing pressure alone will liquefy the
vapor. Vapors do not necessarily follow ideal-gas laws. Since the partial pressure
of water vapor at room temperatures is very low in moist-air mixture, it is behaving
closely to an ideal gas. Small deviations are taken care by using an enhancement
factor, which is a weak function of pressure and temperature. Often, such
corrections are omitted in practice other than for very accurate computations. Hence
a kinetic model of air is represented as a system of many discrete particles
(monoatomic, di-atomic, tri-atomic, etc., molecules), which are widely separated and each
particle is essentially unaffected by the presence of others. No molecular forces
exist due to large distances, other than during brief perfectly elastic collisions in
which linear-momentum between particles is exchanged. An air-parcel is an
imaginary small volume of air containing huge number of molecules. The mean
free-path (MFP) or the average distance a molecule travels before colliding with
another molecule can be calculated using the classical kinetic theory of gases
(Hansen, 1976; Kennard, 1938; Sears, 1953; Saad, 1966; Tribus, 1961)
2? d 2n
2? d 2 p
p ? molecules ?
kBT ?? m3 ??
A typical Oxygen (O2) or Nitrogen (N2) molecule scattering cross-section
areas are on the order of 10-10 m (0.1 nm). At SL ISA atmospheric pressure and
temperature, the number of molecules per cm3 (0.06 inch3) is about 2.54 x 1019. At
one standard absolute atmosphere (1 ata or 760 torr or 1013.25 hPa/mbar or 29.92
inches Hg) and 15oC, MFP is approximately 10-8 m (10 nm) or two-orders of
magnitude larger than the representative molecular linear size. It is also about one
order-of-magnitude larger than the average linear spacing between molecules. In
fact, molecular near-misses are more frequent then than collisions. The collision
frequency of air molecules in a cm3 at standard SL ISA conditions is about 109
collisions per second. Although, a large number, the fact is that molecules miss
each other much more often than collide. An air-parcel volume can be defined using
linear scale dimension of about 1 to 10 micrometers (at least two-orders of
magnitude larger than MFP) for that matter. The ?individuality? of molecules is
completely lost in an air parcel and thermodynamic pressure, temperature, density,
specific volume, etc., are defined within the realm of the continuum mechanics.
More advanced MFP models involve, for example, Lennard-Jones potential
implying distant molecules are slightly attracted, while closer particles are first
attracted more as the distance between particles is reduced and then powerfully
repelled when ?too? close. Dry air behaves practically as a perfect-gas up to high
pressures of about 200 bar (2,900 psi). Beyond that pressure, the molecules are
?squeezed? so tightly that real-gas effects and inter-molecular forces should be
taken into account. For the non-interacting non-condensable ideal-gas mixture, we
p = ? pi = ? fi p
M = ? fiM i
V = ?Vi = ? fiV
m = ? mi = ? niM i
? fi =
? ni =
? pi =
?Vi = 1
These are the famous Dalton (partial pressures) and Amagat-Leduc (partial
volumes) laws and apply to dilute gas mixture
(Holman, 1980; Lay, 1963; Saad,
. Here, fi?s are molal or volume fractions and differ from mass fractions.
Dalton law implies that various gases coexist independently. The molar masses M
and volume (concentration) fractions of the most abundant gases in the atmosphere
are given in (National Oceanic and Atmospheric Administration, 1976). The
average molar mass and the gas constant of dry air mixture up to about 86 km is
? fiMi = Md = 28.9644 [kg/kmol]
= 287.056 [J kg-1 K-1]
It is common to include all nonreactive trace gases (Argon, CO2, etc.) with
the molecular nitrogen and call it atmospheric Nitrogen (Saad, 1966). While most
of the atmospheric Oxygen is in diatomic form, some O3 and atomic Oxygen also
exist. Practically, atmospheric dry air consists of inert atmospheric Nitrogen,
Oxygen, and variable amount of atmospheric water. At standard SL atmospheric
pressure and 273 K (about 0oC) conditions one kg-mol of air occupies volume of
22.4 m3. For an ideal gas, the specific heats at constant-pressure and
? ?h ?
cp = ? ?
? ?T ? p
? ?u ?
cv = ? ?
? ?T ?v
cp ? cv = R
cp ? cv = ?
? = p
cp = ?
Classical and statistical thermodynamics shows that for diatomic gases (N2,
O2) at moderate temperatures
cp ? cv = R
We could model specific heat capacity (at constant pressure or constant
volume) bit more generally and accurately with a polynomial fit:
cx (T ) =? x + ? xT +? xT 2 +? xT 3 +
For example, in
we find a correlation (originally due to Kobe,
K. A.) for dry air valid for temperature range 273-1,800 K with maximum error of
0.72% and with coefficients converted in SI units here:
cp (T ) = a0 + a1T + a2T 2 + a3T 3 ?J/kgK?
where: a0 = 970.177
Enthalpy of dry air using Eq. (6) is:
The isentropic process (ds=0) for an ideal gas is described as:
pv? = const.
? ? p =
( dp p )
( dv v)
d (ln p )
d (ln v)
Using the ideal-gas law (Eq. 1) in logarithmic form and keeping in mind
that dry-air gas-constant is a constant, we obtain:
d (ln v) = d (ln T ) ? d (ln p)
p1 = ?? v2 ?? = ?? T1 ??? ?1
p2 ? v1 ? ? T2 ?
The ideal-gas dry-air isentropic relations follow from Eqs. (8) and (9):
Isentropic relationships (Eq. 10) are used to define potential temperatures
or Poisson?s equation
Rd = ? d ?1 =
cpd ? d
It is a temperature that air parcel will have at reference pressure of 1,000
mbar (or hPa) if isentropically expanding or compressing. As
Iribarne and Cho
state - air ascending isentropically (hence also adiabatically) in the
atmosphere will have T and p varying according to Poisson?s equation (Eq. 11). If
air is moved adiabatically to reference 1,000 mbar pressure level, it will acquire
potential temperature according to Eq. (11). This fact is used in construction of dry
adiabats in meteorological thermodynamic diagrams. Selected physical and
thermodynamic properties of most abundant gases found in terrestrial atmosphere
are given in Table 1
(Holman, 1980; Saad, 1966; Sears, 1953)
Often vapors and gases do not follow ideal-gas model faithfully. For
example, water vapor deviates from the ideal-gas behavior. In general, real gases
can be described using the virial equation of state
(Holman, 1980; Lay, 1963; Saad,
1966; Sears, 1953)
= Z ( p,T ) = 1+ A
= 1+ A' p + B' p2 + C' p3 +
Selected Gas Properties at 15 to 20oC and Atmospheric Pressure of 1 ata
cp ? cv = cp ? cv
Additionally, many specialized real-gas equations/models of state exist,
such as, already mentioned perturbative virial equation, Van der Waals equation,
? d ?? 1
Beattie-Bridgman equation, Dieterici equation, two-parameter Redlich-Kwong
model, Peng-Robinson model, etc. Many gases in homogeneous dry air mixture,
such as N2, show that compressibility factor stays almost constant and very close
to one for not too high pressures. Unfortunately, that is not the case with water
vapor. Non-ideal gas behavior of water vapor (e.g., Mollier?s steam tables and
diagram) is typically taken into account through experimentally obtained correction
factors, which are weak functions of pressure and temperature
Eskridge, 1996; Buck, 1981; Hyland, 1975; Hyland & Wexler, 1983a, 1983b)
Theoretical water-vapor real-gas models based on modeling of virial coefficients
of moist air over liquid and/or ice are given, for example, in
Hyland and Wexler
Wexler and Greenspan (1971)
, and Wexler (1976, 1977).
Properties of Moist Air
Unsaturated moist or humid air is a two-component dilute gas mixture of
inert dry air and superheated water vapor (Saad, 1966). Psychrometry is a branch
of applied thermodynamics focusing on the properties and behavior of moist air
and has many engineering applications, such as, in HVAC systems. We omitted
enhancement factor for simplicity in this work. Absolute humidity of moist air is
equal to water vapor mass density in an ideal-gas:
We borrow the terminology mostly from meteorology. Partial pressure of
water vapor in moist air e follows Dalton and Amagat-Leduc ideal-gas mixture laws
to large extent (but not fully). The mixture ratio for two-component moist air
mixture in thermodynamic equilibrium is:
The ratio of molar masses of dry air and water vapor is:
M v = Rd ? 18.016
M d Rv
Total mass of humid (moist, wet) air is thus:
m = md + mv
= 1+ r ? 1
Up to 40 g (0.04 kg) H2O per kg of dry air or 40 parts-per-thousand (?)
sometimes occur. In mid-latitude this is typically maximum of 15 to 20 g
atmospheric H2O per kg of dry air or 15 to 20 ?. The specific humidity defines the
amount of water vapor per unit mass of moist air:
q ? v =
mv + md
= v =
? r ? r 2 + r3 ? r 4 +
Relative humidity (RH) is temperature dependent and defined as:
? RH ?
? = ?? 100 ?
e (T )
es (T ) rs
Atmospheric RHs (and other parameters) are measured with radiosondes
(FAA, 2016; Lester, 2007; Saucier, 1989)
. If we assume water
vapor under low partial pressures to behave as an ideal-gas, we can write for
thermodynamic equilibrium (Tv=Td=T):
= ?v RvTv
? d RdTd
e = ?v Rv = r
pd ? d Rd ?
Tv = Td = T
Now we have for mixture ratio:
Similarly, it can be shown:
p ? e
r = ? ?
? ? e
p ? (1?? ) ? e
Moist air gas constant is derived from the mixture model:
mRm = md Rd + m R
On the other hand:
Mass density of moist air mixture follows directly:
?m ? ?d ? (1? 0.61? q) ? ?d
Alternatively, we can write first-order approximation for the moist air
mixture mass density using the ideal-gas law and the molar masses of water and
non-condensable dry air mixture:
? m =
M v pv + M d ( p ? pv ) ? M d p ? ??1? 0.61?
?T ?T ?T ?
? ? ? d (1? r )
Clearly, humid air is less dense than dry air occupying the same volume,
which directly affects the density altitude (DA). In many cases, r is on the order of
10-2 and the moist-air gas-constant is only very slightly larger (1-2%) than for dry
air, while the moist-air density is about 0.6% lower. In meteorology, it is preferable
to keep dry-air gas-constant and use virtual temperatures (TV) instead
2002; Rogers, 1979; Saucier, 1989; Tsonis, 2007)
p = ?mRmT = ?mRd (1+ 0.61? q)T ? ?mRdTV
TV = T (1+ 0.61? q) ? T ?K?
In general, TLCL < TDP < TWB < T < TV. Specific heats at constant pressure
and volume for moist air are
cpm = 1+ r cpd + 1+ r cpv = cpd ? (1+ 0.86 ? q ) ? cpd
cvm = 1+ r cvd + 1+ r cvv = cvd ? (1+ 0.96 ? q ) ? cvd
Average values of specific heats at constant pressure and constant volume
for water vapor and dry air are given in Table 1. Enthalpy of water vapor can be
modeled similarly as for dry air using cubic fitting polynomials
specific enthalpy of moist air is a sum of dry air and water vapor enthalpies:
mhm = md hd + m h
(1+ r ) hm = hd + r ? hv
hm = r hd + q ? hv
The isentropic (and adiabatic) coefficient of expansion/compression of moist air is:
? m ? cpm ? cpd ? (1+ 0.86 ? q)
cvm cvd ? (1+ 0.96 ? q)
? cpd (1+ 0.86 ? q) ? (1? 0.96 ? q) ? ? d (1? 0.1? q)
As specific humidity of moist air is approximately 10-2, the reduction of
isentropic coefficient for moist air is only about 1? of dry air value. It is thus
acceptable to use dry air value in many applications. Another coefficient of frequent
use in pseudo-adiabatic (saturated adiabats) processes of moist air is:
? m ?1
= cpd ? (1+ 0.86 ? q)
Rd (1+ 0.61? q)
? ?? ? d ?? (1+ 0.25? q) ? 7 ? (1+ 0.25? q)
? ? d ?1 ? 2
Isentropic process of ideal-gas moist air is described with:
? ? m ?1 ?
d (ln T ) = ? ? d (ln p)
? ? m ?
Clapeyron and Clausius-Clapeyron Equations
Clapeyron (C) and the special case Clausius-Clapeyron (C-C) equation
applied to ideal-gases describe phase transitions (Lay, 1963). C-C equation is
normally used in gas-liquid phase transition computations, but can also be used for
gas-solid (deposition/sublimation) transitions. A C-equation is used for liquid-solid
transitions (solidification, melting, fusion, freezing) since no condensed phase
(liquid or solid) behaves as an ideal gas. We will use the C-equation to compute the
liquid water-ice transition.
provides a very elegant derivation of the
C-equation, which will be replicated for convenience and importance. We will use
two approaches in derivation, one involving Carnot process and the other Gibbs
free-energy thermodynamic functions.
The most efficient thermodynamic cycle is the Carnot cycle, which consist
of two isotherms during which heat is entering and leaving the system, and two
adiabats defining isentropic compression and expansion. The efficiency of the
Carnot cycle is:
= Qin ? Qout = T2 ? T1 = 1? T1
Qin T2 T2
If we assume reversible Carnot-cycle engine operating in a
singlecomponent two-phase gas-liquid (vapor-liquid) phase transition region between
two heat reservoirs (source and sink) and differing infinitesimally in temperatures
The heat entering the system from the source is used for phase change
(liquid v?? to vapor v???) of working fluid of mass m, and the infinitesimal (but not
a total differential) thermodynamic work obtained by gas-liquid (?gl?) phase
change and phase-transition expansion, are:
Q = Lgl = mlgl
? W = m(v??? ? v??) dpgl
Carnot-cycle efficiency is now:
m (v??? ? v??) dpgl = dT
m ? lgl
T (vgs ? vls )
T (v??? ? v??)
The C-equation defining the slope of the p-T phase transition readily yields:
The C-equation for gas-liquid phase transitions can be extended to other
phase transitions such as gas-solid (sublimation, deposition) and liquid-solid
(fusion, solidification, melting, freezing, etc.). Another, and perhaps more elegant
way to derive the C-equation is by using the Gibbs (free-energy) function
1966; Sears, 1953)
, and defined in
(Holman, 1980; Lay, 1963; Rief, 1965; Saad,
1966; Sears, 1953; Tribus, 1961)
G ? H ? T ? S
= h ? T ? s
According to the First Law-of-Thermodynamics, Gibbs function remains
constant during liquid-vapor phase transition
, i.e., g?? = g??? . Using
the same Carnot-cycle process in the liquid-vapor region, we can write Gibbs
function for a reversible process as:
dg = dh ?T ? ds ? s ? dT = ?s ? dT + v ? dp
(? q = T ? ds = dh ? vdp)
Since, the Gibbs function stays constant for the vapor and liquid phases and
temperature and pressure differential are equal in both phases
, we can
?s??? dT + v??? dp = ?s???? dT + v???? dp
(v??? ? v??) dp = (s??? ? s??) dT
The change of entropy during phase transition is related to the specific latent
heat of condensation-evaporation:
?s = ( s??? ? s??) = gl
T (v??? ? v???) T (vgs ? vls ) T ? ?V
Finally, the Clapeyron equation derived using the Gibbs functions becomes:
The C-C equation is the special case of the C-equation when the working
gas/vapor is an ideal-gas. The C-C equation can be used for gas-liquid
(condensation, evaporation) and gas-solid (deposition, sublimation) phase
transitions. Since the specific volume of the gas is normally much larger than the
specific volume of the liquid (condensed) phase, we have in the saturation region
where the vapor behaves as an ideal gas:
Thermodynamic diagrams used in meteorology and aviation weather
product will show isobars, isotherms, isochores, dry-adiabats, pseudo-adiabats,
isohumes etc. Some popular thermodynamic diagrams used for analysis of moist
air are Emagrams, Tephigrams, Skew-T log-p diagrams, St?ve diagrams, etc. For
more details and methods of construction, consult
. In fact, Reichmann (1993) uses St?ve diagrams for planning
crosscountry soaring flights.
Theoretical and Semi-Empirical Models of Phase Transitions
The Clapeyron (C) equation can be simplified and integrated easily in the
case when the working fluid is (or approximately is) an ideal gas. Since the specific
volume of saturated gas is normally much larger than of saturated liquid, we can
write C- and C-C equations as:
T (vs???? vs??)
des = lv ? es
dT Rv ?T 2
vs??? ?? vs??
This differential equation is integrated directly when the latent heat of vaporization
lv is constant:
The vapor pressure can also be presented in a modified p-T diagram (Lay, 1963):
es l T ? 1 ?
? d (ln es ) = ? v ? d ? ? ? ln
Rv TTP ? T ?
es = lv ?? 1
Rv ? TTP
As the temperature increases so does the average molecular kinetic energy
and less energy is required for phase transition. In condensation process, the latent
heat is released (less energy is needed to keep molecules close together in
condensed state than in a gas). However, in the first approximation and for limited
temperature range of interest in meteorology and aviation sciences, we assumed
that the latent heat of phase transition is constant in Eq. (45). The reference point
in C-C equation is based on the well-defined triple-point (TP) of water (all three
phases coexisting) with the equilibrium pressure of about 611.657 Pa or 6.11657
mbar or hPa
(Guildner et al., 1976)
and water TP-temperature of 273.16 K (0.01oC).
Hence, for vaporization (vapor-liquid) transition:
provides derivation of the TP conditions and shows by
analytical considerations that its temperature is about 0.01 K higher than the
icepoint temperature. For a vapor-solid transition (sublimation, deposition) and
constant latent-heat of phase transition, we have similarly:
? lsg ? 1
eS (T ) = 611.657 ? exp ?
? Rv ?? 273.16
1 ?? ?
?? = 611.657 ? exp ? 22.50 ?
T ?? ?
lsg = 2,836 kJ/kg, T ? ???50o C, 0o C?
In the case of liquid-solid transition, we have to use C-equation.
Interestingly, water is just one of the very few substances in nature, which has solid
phase (ordinary ice) with a larger specific volume (lower density) than the
corresponding liquid phase. Water expands upon freezing resulting in a negative
melting/fusion slope, which is so important in nature:
So far, we have assumed latent heat to be constant in given temperatures
zone. That may be acceptable for vapor-solid water transition, but not sufficiently
accurate for vapor-liquid transition. Indeed, latent heat of vaporization is function
of temperature. Dutton (2012) delivers temperature-dependent linear fit of the
latent heat of evaporation in terms of cal/g (1 cal/g=4.186 kJ/kg) as:
lv = 597 ? 0.57 ? (T ? 273) ?cal/g?
or in SI units
lv = 2, 499 ? 2.386 ? (T ? 273) ?kJ/kg?
dlv = ?0.57 ? 0
v = ?2.386 ? 0
T ??233, 313?
Based on thermodynamic considerations and varying enthalpies and
specific heat capacities of saturated liquid and saturated vapor,
lv = lv0 + (cpv ? cpw )(T ? T0 ) = 2,500 ? 2.368? (T ? 273) ?kJ/kg?
lv0 = 2.50?106 J/kg, cpv = 1.850?103 J/kgK, cpw = 4.218?103 J/kgK
Dutton?s (2002) linear approximation agrees well with the tabulated water
vapor saturation pressures provided in Rogers (1979),
. The C-C integral with temperature-dependent latent heat of condensation
ps l T ??1? a ? (T ? 273)?? dT
p?TP d (ln evs ) = Rv0v T?TP T 2
The integration (temperature in K) yields:
e l ? ? 1
ln s = v0 ? ?(1+ a ? 273) ? ?
eTP Rv ? ? TTP
1 ? ? T ??
? ? a ? ln ? ??
T ? ? TTP ??
We could have also used more accurate higher-order polynomial
representation of the latent heat as a function of temperature. However, the fact
remains that C-C equation is only an approximation of phase transition
thermodynamics for ideal gases. Water vapor does not behave exactly as an ideal
gas and the presence of the non-condensable dry-air component complicates the
situation a bit. More accurate water vapor saturation pressure correlations exist.
They are semi-empirical based on measurements and have been continuously
refined over the last 200 years. We will only mention relatively simple, yet
reasonably accurate Magnus-type equations.
The Clausius-Clapeyron equation was derived based on an existence of
pure substance, e.g., water vapor and liquid in equilibrium. However, most air
contains dry non-condensable or inert fraction and the variable water content in all
three phases. The question arises on how does pressure of inert gas components
affect the vapor pressure. In fact,
and Saad (1966), provide answer to
that question. Utilizing Gibbs function equality in the presence of the inert gas
component. In moist air, the water vapor changes phases in the presence of dry air.
The pressure of the inert (indifferent) dry air affects the equilibrium vapor pressure
in moist air mixture to a small degree. True, the effect is small, but has been
included in many other studies for higher accuracy. For example,
Alduchov and Eskridge (1996)
provide equations for the enhancement factor f,
which is defined as the ratio of the saturation water vapor pressure of moist air to
that of pure water vapor and solid water, i.e., ice:
fv = v
fi = eii
While the equilibrium (saturation) vapor pressure is a function of
temperature only, the enhancement factor is a weak function of both, pressure and
(Alduchov & Eskridge, 1996; Buck, 1981)
. This can be observed from
the virial equation of state given by Eq. (12).
Dew Point and Relative Humidity Computations
DP is temperature achieved by isobaric cooling of moist air until
equilibrium vapor pressure is reached (so-called ?saturation? vapor pressure). This
process can be observed in the p-T diagram shown in Fig. 1. While not the most
suitable diagram to show atmospheric thermodynamic processes, it will suffice for
our purpose. Using definition of RH from Eq. (18), we write:
ln? (T ) = ln e(T ) ? ln es (T )
For example, RH can increase by increasing vapor pressure (adding water
vapor or humidifying to air) as it is done in HVAC applications and occurs in some
meteorological phenomena or by cooling air parcels at constant pressures (e.g.,
radiation fog or advection fog). As seen in Fig. 1, isobaric cooling of air parcel
moves the state from the starting point A to the point C located on the saturation
line. Since the process is isobaric and we have closed material system, the vapor
pressure at point A with temperature T has the same vapor pressure as the point C
or the DP. There is no exchange of mass/matter with the surroundings, but there is
energy/heat exchange (cooling) and thus it is not an adiabatic process. Following
the isoenthalpy line from point A we arrive at point D on the liquid-vapor
equilibrium curve. Point D defines the WB-temperature of state A. Accordingly,
for unsaturated region (e.g. point A), the relation DB > WB > DP is valid. RH can
be calculated from DB and WB temperature measurements
Additionally, DPs are related to RH:
e (T ) = es (TDP )
RH = es (TDP )
es (T )
Using the C-C equation derived in Appendix D, we may write:
es (T ) = esTP (TTP ) ? exp ?? lv ?? 1
? Rv ? TTP
? ?? = C1 ? exp ?? ? lv Rv ?
T ?? ? T ?
? lv Rv ?
C1 = esTP (TTP ) ? exp ? ?
? TTP ?
Eq. (57) defines DP at constant DB air temperature T and is designated in
Fig. 1 with point B. The saturation pressure at point B (isothermal heating) is higher
than the saturation pressure at point C (TDP) achieved by isobaric cooling of an air
parcel. Hence, RH is less than one at point A (with associated mixture ratio and
specific humidity). The vapor pressure at point A with unsaturated air and the
saturation pressure at DP for condition C are equal:
? l ? 1
es (TDP ) = esTP (TTP ) ? exp ? v ?
? Rv ? TTP
1 ?? v ?
?? = C1 ? exp ?? ? lv R ?
? TDP ?
If DPs and air temperatures are known, RH is computed as:
? l ? 1
? = exp ? v ?
? Rv ? T
1 ?? ? ? T ? TDP ????
?? ? exp ??5417 ? ?
TDP ?? ? ? T ?TDP ??
Assuming all pseudo-constants remain unchanged during the isobaric cooling, we
TDP (T , RH ) = T ?1?
? RH ??
? ln ? ??
? 100 ??
= T ?1?
? ln? ?
This is the same result as obtained by Lawrence (2005). Related but not
identical to DP is the WB-temperature
. RH of air can be
computed or by DB and WB temperatures using psychrometer (with some
corrections) or by dry-bulb (DB) air temperature and hygrometers (e.g.,
chilledmirror dew point hygrometer) measuring DPs directly. Using DP depression, we
obtain for RH directly from Eq. (59):
? 5417 ? D ?
RH (T ,TDP , D) = 100 ? exp ? ? ?
? T ?TDP ?
T (TDP , RH ) = TDP ???1+ 5T4D1P7 ? ln ??? 1R0H0 ??????
While usually easiest to measure using common (DB) thermometers, in case
DP and RH are only known, the air temperature estimate is:
As reported by
Alduchov and Eskridge (1996)
, and Lawrence
(2005) among many other authors, a large number of expressions for computation
of water and ice saturation vapor pressures, DPs, and RH exist. Of many models,
we will only mention popular Magnus-type formulas for saturation vapor pressures
for water and ice
(Alduchov & Eskridge, 1996; Buck, 1981; Lawrence, 2005)
Lawrence (2005) refers to Magnus-formulas as more appropriately
August-RocheMagnus type of formulas. Essentially, the same form of the expression was used
with ever improving coefficients based on more accurate water vapor pressure
measurements. Even more accurate correlations for the region of interest in
meteorology exist, but will not be addressed here. For more details on various
models, the reader is encouraged to consult Buck (1981) and
. An August-Roche-Magnus empirical formula for water saturation
vapor pressure is
(Buck 1981; Lawrence, 2005)
? A ?? ?
es (? ) = C ? exp ? ? ?Pa ?
? B +? ?
A = 17.625 ???
B = 243.04 ?? o C?
C = 610.94 ?Pa ?
? 40o C ? ? ? 50o C
The coefficients are from
Alduchov and Eskridge (1996)
. DPs are now:
? DP =
? e ?
B ? ln ? ?
? C ? =
A ? ln ?? e ?
? C ?
? ? RH ?
B ? ?ln ? ? +
? ? 100 ?
A ?? ?
B +? ??
A ? ln ?? RH ?
? 100 ?
?? o C?
Linearized expression for DPs when RH > 50%, was proposed by Lawrence (2005):
? DP =? AIR ?
(100 ? RH)
?? o C?
RH ? 50%
Lawrence (2005) also provides simple estimate of RH from air and DP
RH=100 ? 5?(? AIR ?? DP )
RH ? 50%
Results and Discussion
Numerical values of saturated water vapor pressure in equilibrium with
liquid water is shown in Fig. 2 on semi-logarithmic plot. The C-C equation
solutions using constant latent heat of evaporation (Eq. 46) and linearly varying
latent heat of evaporation (Eq. 53) in the range of interest are plotted against the
experimental data from the Smithsonian Meteorological Tables as reported in
Rogers (1979) and
. More accurate measurements of the saturation
vapor pressure (over liquid and ice) are given, for example, in
. Those data are actually based on the measurements and
correlations from Wexler (1976, 1977) and Hyland and Wexler (1983a, 1983b).
Additionally, we used Magnus-type semi-empirical correlation with the constant
coefficients coming from
Alduchov and Eskridge (1996)
as given in Eq. (63). This
semi-empirical correlation provides no real advantage over theoretical C-C
relationships for the given temperature range. In fact, the C-C model using linearly
changing latent heat of evaporation provides best fit to experimental/measured data.
Water saturated vapor pressures in equilibrium with super-cooled liquid
water and solid water (ice) is shown in Fig. 3. We used C-C theoretical models in
both cases. The latent heat of sublimation for water vapor-ice is fairly constant over
the temperature range. For the vapor-liquid equilibrium, we also used C-C model
with latent heat of evaporation changing linearly for the given temperature range.
The experimental data over ice are taken from
Wexler (1977), and
Bielska et al. (2013)
. The experimental data from
Rogers (1979) and
were used for vapor-liquid equilibrium in
supercooled (metastable) region. All these comparisons were also used to verify vapor
pressure models for subsequent analysis of LCL.
Saturation values for mixture ratio (green squares) and specific humidity
(red solid line) are shown in Fig. 4 together with the vapor mass as a function of air
temperature (black solid line). Note rapid increase in water vapor amount with
higher air temperatures (above about 10oC). The difference between the mixture
ratio and specific humidity is small indeed and barely observable until high air
temperatures. Highest recorded DPs were around 36oC. That corresponds to about
maximum amounts of 40-gram water per kg of dry air. Apparently, small amount,
but the energy released per kg of dry air in condensation is very significant.
Computations of DPs versus DB air temperatures for constant parametric
RHs (40-100%) is shown in Fig. 5. Besides results from model used (Eq. 60), we
also presented DP computations based on Magnus-type formula given with Eq. (63)
and Lawrence?s estimate from Eq. (65). Computations of DPs using simple
Magnus-type formulas with optimized
Alduchov and Eskridge (1996)
does not deviate much from computations based on simple C-C model. Errors using
Lawrence?s model (2005) are large below RH of 50% and increase somewhat with
higher air temperatures for any RH at and above 50%. Computations of RHs from
known DB air temperatures and DP depressions is shown in Fig. 6. Again, we added
Lawrence?s (2005) simple model given with Eq. (66). Clearly, errors are large in
Lawrence?s model for RH < 50%, which the author clearly stated. We also observe
that in Lawrence?s model RH, while on average is a decent approximation for
higher RHs, is a constant over the entire range of temperatures for constant DP
depressions. Hence, it can serve as a good first approximation or as a rule-of-thumb
only. For more accurate computations of RH we recommend using Eq. (61).
The largest source of uncertainties in our model comes from the deviations
of moist air from the ideal-gas law and by assuming that thermodynamic parameters
remain constant in a given temperature region. Simultaneously tackling all these
additional variables would considerably complicate the analysis and would require
much more space and effort. Additionally, the effects of Henry?s and Raoult?s laws
(Bohren, 2001; Lay, 1963; Pauling, 1988)
were not considered in phase transitions.
In fact, atmospheric air will dissolve into liquid water (precipitation) as governed
by the Henry?s law.
As we have seen, relatively small amount of water vapor co-exists with
atmospheric dry air circulating primarily in troposphere. However, latent heat of
vapor-liquid phase transition is relatively large (about 2,500 kJ/kg). A typical
midlatitude T-storm (CB) can contain at any moment about 10 million kg of water
vapor, which upon condensation would release about 10 TJ of energy. That would
be equivalent to about metric 2.4 kiloton (TNT) energy release. Of course, T-storms
are extremely complex and dynamic thermal-fluid phenomenon and moist air
(Tstorm?s ?fuel?) is entrained for the significant part of their lifetime, so the total
energy released can be significantly higher. Luckily, all that energy is released over
relatively long time (temporal scale on the order of an hour) and large space (spatial
scale of a km or a mile). Nevertheless, CBs are violent and present real hazard in
commercial and personal air transportation.
One of the persistent common misconceptions is that air somehow ?holds?
or ?contains? variable amount of water vapor depending on its own temperature
and that upon reaching certain limit it cannot ?hold? more water vapor ? and the
excess must condense. Such notion is false and is also repeatedly stated in some
aviation weather texts. Such misconception has been rebutted often. For example,
gives an eloquent no-formula explanation of the
?air-absorbingwater-vapor? fallacy. Explanations of the actual physics can be found in many
expert references cited here. In ideal gas, the molecules of dry air and water vapor
co-exist in the entire volume being practically ?ignorant? of each other. Dry air
cannot ?count? and ?manage? water-vapor molecules. Hence, dry air has no
?knowledge? how much water molecules are there and therefore cannot control it.
This issue has been stated long time ago by the English chemist and meteorologist
John Dalton. Dalton was indeed conducting phase transitions experiment. He is
most famous for introducing atomic theory in chemistry and research in gas laws.
The maximum amount of water vapor and the saturation of equilibrium
between vapor and liquid phases is governed by the Clapeyron equation. True, the
presence of non-condensable (inert) dry air mixture does affect the water vapor
equilibrium pressure, but to a very small degree. For example,
Saad (1966) show that increase in the partial pressure of inert gas (dry air) does
affect the saturation (phase equilibrium) pressure of the condensable component by
increasing the pressure in the liquid phase which then increases evaporation rate
thus increasing vapor saturation pressure and vapor content. However, this effect is
very small. Thermodynamic equilibrium is assumed to take place instantly. The
effect is stronger as the equilibrium temperatures and pressures increase. In
addition, one has to consider that water vapor does not follow the ideal-gas law
faithfully. For example,
includes a deviation from the ideal-gas law
factor in defining moist air parameters. Hence, combined effects due to non-ideal
gas behavior and the presence of non-condensable dry air component is taken into
account through, the so called, enhancement factor ?f?, which is a very weak
function of temperature and pressure. In a sense, the enhancement factor is an
inverse of the compressibility factor Z (? 1) from the virial equation, which is a
function of pressure and temperature and different for every gas or vapor
1980; Hyland, 1975; Lay 1963; Wexler, 1976, 1977; Wexler and Greenspan, 1971)
While Clapeyron equation is exact, the Clausius-Clapeyron equation is only an
approximation of the Clapeyron equation when condensable component (e.g.,
water vapor in air) mimics ideal gas in its gaseous phase. Hence, the equilibrium
liquid-vapor pressure values are not truly exact using C-C equation. Therefore,
many empirical correlations have been developed for water vapor pressure in the
past 200 years or so.
The basic thermodynamic theories of dry and moist air are summarized and
most important relationships were derived for clarity and better understanding.
Atmospheric water vapor-liquid, vapor-ice and liquid-ice phase transitions
described by the Clapeyron and Clausius-Clapeyron equations are also presented.
While moist air exhibits real-gas behavior for many practical applications not
requiring very high accuracies, it can be regarded as an ideal-gas. Some common
misconception regarding moist air have been addressed. Basic theory and several
original practical expressions to compute dew points and/or relative humidity are
given and compared to existing ones. Density altitude corrected for humidity can
be easily computed from presented moist air thermodynamic relationships, but that
has not been presented in this article. Understanding thermodynamics and
dynamics of atmospheric moist air and phase transitions is of fundamental
importance for safety, economy, and performance of flight operations.
Dr. Nihad E. Daidzic is president and chief engineer of AAR Aerospace Consulting,
L.L.C. He is also a full professor of Aviation, adjunct professor of Mechanical
Engineering, and research graduate faculty at Minnesota State University. His
Ph.D. (Dr.-Ing.) is in fluid mechanics and Sc.D. in mechanical engineering. He was
formerly a staff scientist at the National Center for Microgravity Research and the
National Center for Space Exploration and Research at NASA Glenn Research
Center in Cleveland, OH. He also held various faculty appointments at Vanderbilt
University, University of Kansas, and Kent State University. His current research
interest is in theoretical, experimental, and computational fluid dynamics,
microand nano-fluidics, aircraft stability, control, and performance, mechanics of flight,
piloting techniques, and aerospace propulsion. Dr. Daidzic is an ATP AMEL and
FAA ?Gold Seal? CFI-A/CFI-IA/MEI/CFI-RH/CFI-G/AGI/IGI instructor for
several decades with flight experience in airplanes, helicopters, and gliders.
? ??kg/m3 ??
? ??m2 ??
c ?kJ/kg K?
g ??m/s2 ??
s ?kJ/kg K?
v ??m3 /kg??
Ratio of molecular masses for humid air.
Mean free path (MFP).
Molecular cross-section area.
Dew point depression (spread).
Latent heat (vaporization, fusion, etc.).
R ?J/kg K?
? ?J/kg K?
V ??m3 ??
Radius (average) of spherical Earth approximation.
Gas constant (gas specific).
Work (thermodynamic or technical).
Compressibility factor (real gases).
Constant pressure process.
Constant volume process.
American Society of Heating,
Cumulonimbus cloud (T-Storm).
Dew point depression (spread) [K].
Density Altitude [m, ft].
Dew Point [K].
Environmental Control Systems.
Environmental Lapse Rate.
Federal Aviation Administration (2016). Aviation weather (AC 00-6B).
Department of Transportation (DOT), Washington, DC: Author.
Fleagle, R. G., & Businger, J. A. (1980). An introduction to atmospheric physics
(2nd ed.). New York, NY: Academic Press.
Forrester, F. H. (1981). 1001 questions answered about the weather. New York,
International Civil Aviation Organization (1993). Manual of the ICAO standard
atmosphere (Doc 7488-CD, 3rd ed.). Montreal, Canada: Author.
Lawrence, M. G. (2005). The relationship between relative humidity and the
dewpoint temperature in moist air: A simple conversion and applications.
Bulletin of the American Meteorology Society, 86, 225?233.
Lay, J. E. (1963). Thermodynamics: A macroscopic-microscopic treatment.
Columbus, OH: Charles E. Merill Books.
National Atmospheric and Oceanic Administration. (1976). U.S. standard
atmosphere, 1976 (NOAA-S/T 76-1562). Washington, D.C.: U.S.
Government Printing Office.
Pauling, L. (1988). General chemistry. Mineola, NY: Dover.
Reif, F. (1965). Fundamentals of statistical and thermal physics. New York, NY:
Reichmann, H. (1993). Cross-country soaring (7th ed.). Stuttgart, Germany:
Motorbuchverlag (English translation by the Soaring Society of America
Rogers, R. R. (1979). A short course in cloud physics (2nd ed.). Oxford, UK:
Romps, D. M. (2008). The dry-entropy budget of a moist atmosphere. Journal of
Atmospheric Science, 65, 3779-3799, doi:10.1175/2008JAS2679.1
Saad, M. A. (1966). Thermodynamics for engineers. Englewood Cliffs, NJ:
Wannier, G. H. (1987). Statistical physics. Mineola, NY: Dover.
Alduchov , O. A. & Eskridge , R. E. ( 1996 ). Improved Magnus form approximation of saturation vapor pressure . Journal of Applied Meteorology , 35 , 601 - 609 . doi: 10 .2172/548871.
ASHRAE. ( 2001 ). Fundamentals handbook (SI) . Atlanta, GA: Author.
Bielska , K. , Havey , D. K. , Scace , G. E. , Lisak , D. , Harvey , A. H. , & Hodges J. T. ( 2013 ). High-accuracy measurements of the vapor pressure of ice referenced to the triple point . Geophysical Research Letters , 40 , 6303 - 6307 , doi:10.1002/2013GL058474
Bohren , C. F. , & Albrecht , B. A. ( 1998 ). Atmospheric thermodynamics . Oxford, UK: Oxford University Press.
Bohren , C. F. ( 2001 ). Clouds in a glass of beer: Simple experiments in atmospheric physics . Mineola, NY: Dover.
Bolton , D. ( 1980 ). The computation of equivalent potential temperature . Monthly Weather Review , 108 , 1046 - 1053 , doi:10.1175/ 1520 - 0493 ( 1980 ) 108 , 1046 :TCOEPT.2.0.CO;2.
Buck , A. ( 1981 ). New equations for computing vapor pressure and enhancement factor . Journal of Applied Meteorology , 20 , 1527 - 1532 . doi: 10 .1175/ 1520 - 0450 ( 1981 ) 020 < 1527 : NEFCVP>2.0 .CO; 2 .
Daidzic , N. E. ( 2015a ). Efficient general computational method for estimation of standard atmosphere parameters . International Journal of Aviation Aeronautics and Aerospace , 2 ( 1 ), 1 - 37 . doi: 10 .15394/ijaaa. 2015 .1053
Dutton , J. A. ( 2002 ). The ceaseless wind: An introduction to the theory of atmospheric motion . Mineola, NY: Dover.
Guildner , L. A. , Johnson , D. P. , & Jones , F. E. ( 1976 ). Vapor pressure of water at its triple point . Journal of Research of the Natural Bureau of Standards, 80A(3) , 505 - 521 , doi:10.6028/jres.080A. 054 .
de Groot , S. R. , & Mazur , P. ( 1984 ). Non-equilibrium thermodynamics . Mineola, NY: Dover.
Hansen , C. F. ( 1976 ). Molecular physics of equilibrium gases (NASA SP- 3096). Washington, DC: National Aeronautics and Space Administration .
Hill , T. L. ( 1987 ). Statistical mechanics: Principles and selected applications . Mineola, NY: Dover.
Holman , J. P. ( 1980 ). Thermodynamics (3rd ed.) . Singapore, Singapore: McGrawHill.
Houghton , J. ( 2002 ). The physics of atmosphere (3rd ed.) . Cambridge, United Kingdom: University Press.
Hyland , R. W. ( 1975 ). A correlation for the second interaction virial coefficients and enhancement factors for moist air . Journal of Research of the Natural Bureau of Standard, 79A(4) , 551 - 560 . doi: 10 .6028/jres.079a. 017 .
Hyland , R. W. , & Wexler , A. ( 1973 ). The Second Interaction (Cross) Virial Coefficient for Moist Air . Journal of Research of the Natural Bureau of Standard., 77A(1) , 133 - 147 . doi: 10 .6028/jres. 077a. 007
Hyland , R. W. , & Wexler , A. ( 1983a ). Formulations for the thermodynamic properties of the saturated phases of H2O from 173.15 K to 473.15 K. ASHRAE Transactions , 89 ( 2A ), 500 - 519 .
Hyland , R. W. , & Wexler , A. ( 1983b ). Formulations for the thermodynamic properties of dry air from 173.15 K to 473.15 K , and of saturated moist air from 173.15 K to 372.15 K, at pressures to 5 MPa. ASHRAE Transactions , 89 ( 2A ), 520 - 35 .
International Organization for Standardization ( 1975 ). Standard atmosphere (ISO 2533:1975/Add. 2: 1997(en)) . Geneva, Switzerland: Author.
Iribarne , J. V. , & Cho , H.-R. ( 1980 ). Atmospheric physics. Dordrecht , Holland: D. Reidel Publishing Company.
Iribarne , J. V. , & Godson , W. L. ( 1981 ). Atmospheric thermodynamics (2nd ed .). Dordrecht, Holland: D. Reidel Publishing Company.
Kennard , E. H. ( 1938 ). Kinetic theory of gases, with an introduction to statistical mechanics . New York, NY: McGraw-Hill .
Saucier , W. J. ( 1989 ). Principles of meteorological analysis . Mineola , NY: Dover.
Sears , F. W. ( 1953 ). Thermodynamics, the kinetic theory of gases, and statistical mechanics (2nd ed ., 5th printing 1964 ). Reading, MA: Addison-Wesley.
Stull , R. ( 2016 ). Practical meteorology: An algebra-based survey of atmospheric science . http://www.eos.ubc.ca/books/Practical_Meteorology/
Tribus , M. ( 1961 ). Thermostatics and thermodynamics . Princeton, NJ: Van Nostrand.
Tsonis , A. A. ( 2007 ). An introduction to atmospheric thermodynamics (2nd ed .). Cambridge, UK: University Press.
Tufty , B. ( 1987 ). 1001 questions answered about hurricanes, tornadoes and other natural disasters . New York, NY: Dover.
Wallace , J. M. , & Hobbs , P. V. ( 2006 ). Atmospheric science: An introductory survey (2nd ed .). Burlington, MA: Academic Press.
Wexler , A. , & Greenspan , L. ( 1971 ). Va pressure equation for water in the range 0 to 100? C. J. Res. Nat. Bur. Stand. , 75A(3) , 213 - 230 . doi: 10 .6028/jres.075a. 022 .
Wexler , A. ( 1976 ). Vapor pressure formulation for water i n range 0 to 100?C. A revision . J. Res. Nat. Bur . Stand., 80A(5 and 6) , 775 - 785 . doi: 10 .6028/jres.080a. 071 .