## Equations describing the physical properties of moist air |

This is a formal presentation of the formulæ that describe
those properties of water vapour in air that concern conservators. The
derivation of these equations is explained concisely and the useful equations
are emphasised by **bold** type. This datasheet is intended for reference,
not as a friendly introduction to the topic! Refer to the index
for more discursive excursions into the subject of atmospheric moisture.

Use the shortcuts below to get to the relevant section, bearing in mind
that some sections rely on concepts explained earlier! If you just want
to use the equations, try the **Calculator for
the properties of moist air **(which requires a
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Saturation vapour pressure

Definition of the Pascal

Concentration of water vapour in space

Relative humidity

Dew point

Concentration of water vapour in air

Enthalpy

The psychrometer

In a closed container partly filled with water there will be some water
vapour in the space above the water. The concentration of water vapour
depends only on the temperature. It is not dependent on the amount of water
and is only very slightly influenced by the presence of air in the container.

The water vapour exerts a pressure on the walls of the container. The
empirical equations given below give a good approximation to the saturation
water vapour pressure at temperatures within the limits of the earth's
climate.

Saturation vapour pressure, p_{s}, in pascals:
**p**_{s}** = 610.78 *exp( t / ( t + 238.3 ) *17.2694 )
**where

The svp below freezing can be corrected after using the equation above, thus:

The vapour pressure curve for water, and the vapour pressure lines for 10% interval of RH, in pdf for printing.

The relationship between vapour pressure and concentration is defined
for any gas by the equation:

p = nRT/V
*p is the pressure in Pa, V is
the volume in cubic metres, T is the temperature in
degrees Kelvin (degrees Celsius + 273.15), n is the
quantity of gas expressed in molar mass ( 0.018 kg in the case of water
), R is the gas constant: 8.31 Joules/mol/m ^{3}
*To convert the water vapour pressure to concentration in
kg/m

**kg/m ^{3} = 0.002166 *p / ( t + 273.16 ) **

The Relative Humidity (RH) is the ratio of the actual water vapour pressure to the saturation water vapour pressure at the prevailing temperature.

**RH = p/p**_{s}

RH is usually expressed as a percentage rather than as a fraction.

The RH is a ratio. It does not define the water content of the air
unless the temperature is given. The reason RH is so much used in conservation
is that most organic materials have an equilibrium water content that is
mainly determined by the RH and is only slightly influenced by temperature.

Notice that air is not involved in the definition of RH. Airless space
can have a RH. Air is the transporter of water vapour in the atmosphere
and in air conditioning systems, so the phrase *"RH of the air"*
is commonly used, and only occasionally misleading. The independence of
RH from atmospheric pressure is not important on the ground, but it does
have some relevance to calculations concerning air transport of works of
art and conservation by freeze drying.

The water vapour content of air is often quoted as dew point. This is
the temperature to which the air must be cooled before dew condenses from
it. At this temperature the actual water vapour content of the air is equal
to the saturation water vapour pressure. The dew point is usually calculated
from the RH. First one calculates** p**_{s}, the saturation
vapour pressure at the ambient temperature. The actual water vapour pressure,
**p**_{a}, is:

**p**_{a}**= p**_{s}** * RH% / 100**

The next step is to calculate the temperature at which **p**_{a}
would be the saturation vapour pressure. This means running backwards the
equation given above for deriving saturation vapour pressure from temperature:

Let **w = ln ( p**_{a}**/ 610.78 )
Dew point = w *238.3 / ( 17.2694 - w )
**This calculation is often used to judge the probability of condensation
on windows and within walls and roofs of humidified buildings.

The dew point can also be measured directly by cooling a mirror until it fogs. The RH is then given by the ratio

**RH = 100 * p _{s }**

Concentration of water vapour in air

It is sometimes convenient to quote water vapour concentration
as kg/kg of dry air. This is used in air conditioning calculations and
is quoted on psychrometric charts. The following calculations for water
vapour concentration in air apply at ground level.

Dry air has a molar mass of 0.029 kg. It is denser than water vapour,
which has a molar mass of 0.018 kg. Therefore, *humid air is lighter
than dry air*. If the total atmospheric pressure is P and the water
vapour pressure is p, the partial pressure of the dry air component is
P - p** **. The weight ratio of the two components, water vapour and
dry air is:

kg water vapour / kg dry air = 0.018 *p / ( 0.029 *(P - p ) )

= 0.62 *p / (P - p )

At room temperature P - p** **is nearly equal to P, which at ground
level is close to 100,000 Pa, so, approximately:

**kg water vapour / kg dry air = 0.62 *10 ^{-5} *p **

**Thermal properties of damp air
**The heat content, usually called the

The enthalpy of dry air is not known. Air at zero degrees celsius is
*defined* to have zero enthalpy. The enthalpy, in kJ/kg, at any temperature,
t, between 0 and 60C is approximately:

h = 1.007t - 0.026 *below zero: h = 1.005t *

The enthalpy of liquid water is also defined to be zero at zero degrees
celsius. To turn liquid water to vapour at the same temperature requires
a very considerable amount of heat energy: 2501 kJ/kg at 0C

At temperature t the heat content of water vapour is:

h_{w} = 2501 + 1.84t

*Notice that water vapour, once generated, also requires more heat
than dry air to raise its temperature further: 1.84 kJ/kg.C against about
1 kJ/kg.C for dry air.
*The enthalpy of moist air, in kJ/kg, is therefore:

The final formula in this collection is the **psychrometric equation**.
The psychrometer is the nearest to an absolute method of measuring RH that
the conservator ever needs. It is more reliable than electronic devices,
because it depends on the calibration of thermometers or temperature sensors,
which are much more reliable than electrical RH sensors. The only limitation
to the psychrometer is that it is difficult to use in confined spaces (not
because it needs to be whirled around but because it releases water vapour).

The psychrometer, or wet and dry bulb thermometer, responds to the
RH of the air in this way:

Unsaturated air evaporates water from the wet wick. The heat required to evaporate the water into the air stream is taken from the air stream, which cools in contact with the wet surface, thus cooling the thermometer beneath it. An equilibrium wet surface temperature is reached which is very roughly half way between ambient temperature and dew point temperature.

The air's potential to absorb water is proportional to the difference
between the mole fraction, m_{a}, of water vapour in the ambient
air and the mole fraction, m_{w}, of water vapour in the saturated
air at the wet surface. It is this capacity to carry away water vapour
which drives the temperature down to t_{w}, the wet thermometer
temperature, from the ambient temperature t_{a} :

( m_{w} - m_{a}) = B( t_{a}- t_{w})

B is a constant, whose numerical value can be derived theoretically
by some rather complicated physics (see the reference below).

The water vapour concentration is expressed here as mole fraction in
air, rather than as vapour pressure. Air is involved in the psychrometric
equation, because it brings the heat required to evaporate water from the
wet surface. The constant B is therefore dependent on total air pressure,
P. However the mole fraction, m, is simply the ratio of vapour pressure
p to total pressure P: p/P. The air pressure is the same for both
ambient air and air in contact with the wet surface, so the constant B
can be modified to a new value, A, which incorporates the pressure, allowing
the molar fractions to be replaced by the corresponding vapour pressures:

p_{w} - p_{a}= A* ( t_{a}- t_{w})

The relative humidity (as already defined) is the ratio of p_{a},
the actual water vapour pressure of the air, to p_{s}, the saturation
water vapour pressure at ambient temperature.

**RH% **= 100 *p_{a}/ p_{s }**= 100 *( p _{w}
- ( t_{a}- t_{w})** *

*To check your program, take air at 20C and 15.7C wet bulb temperature.
The RH is 65%. The water vapour pressure is 1500 Pa. The water vapour concentration
in kg/m3 is 0.011, in kg/kg it is 0.009. The dew point is 13C.*

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