# Humid Air vs. the Ideal Gas Law

## Pressure, temperature and volume in a perfect ideal gas like moist air (air with water vapor).

In a perfect or ideal gas the correlations between pressure, volume, temperature and quantity of gas can be expressed by the **Ideal Gas Law.**

The **Universal Gas Constant, _{Ru}** is independent of the particular gas and is the same for all "perfect" gases, and is included in of The Ideal Gas Law:

p V = n R

_{u}T (1)

where

p = absolute pressure (N/m

^{2}, lb/ft^{2})V = volume (m

^{3,}ft^{3})n = is the number of moles of the gas present

R

_{u}= universal gas constant (J/mol^{o}K, lb_{f}ft/(lb mol^{o}R) = 8.3145 J/mol K = 0.08206 L atm/mol K = 62.37 torr /mol KT = absolute temperature (

^{o}K,^{o}R)

For a given quantity of gas, both n and R_{u} are constant, and Equation (1) can be modified to

p

_{1}V_{1 }/ T_{1}= p_{2 }V_{2 }/ T_{2}(2)

expressing the relationship between different states for the given quantity of the gas.

Equation (1) can also be expressed as

p V = N k T (3)

N =number of molecules

k = Boltzmann constant = 1.38066 10

^{-23}J/K = 8.617385 10^{-5}eV/K

- One mole of an ideal gas at STP occupies 22.4 liters.

The Ideal Gas Law express the relation between pressure, temperature and volume in an ideal or perfect gas.

The Ideal Gas Law expessed by the Induvidual Gas Constant

The Ideal Gas Law can be expressed with the Individual Gas Constant as

p V = m R T (4)

where

p = absolute pressure (N/m

^{2}, lb/ft^{2})V = volume of gas (m

^{3,}ft^{3})m = mass of gas (kg, slugs)

R = individual gas constant (J/kg

^{o}K, ft lb/slugs^{o}R)T = absolute temperature (

^{o}K,^{o}R)

Density can be expressed as

ρ = m / V (4b)

where

ρ = density (kg/m

^{3}, slugs/ft^{3})

and equation (4) can be modified to

p = ρ R T (4c)

### The Individual and Universal Gas Constant

The Individual Gas Constant can be expressed with the Universal Gas Constant and the molecular weight of a gas like

R = R_{u}/ M_{gas}(2)

where

M_{gas}= molecular weight of the gas

R_{u}= universal gas constant (8314.47J/(kmol K))

The Molecular weight and the Individual Gas Constants for air and water vapor are listed below:

Gas | Individual Gas Constant - R | Molecular Weight ( kg/kmole) | |

Imperial Units ( ft lb/slug)^{ o}R | SI Units ( J/kg K) | ||

Air | 1716 | 286.9 | 28.97 |

Water vapor | 2760 | 461.4 | 18.02 |

### Air Pressure

Daltons Law states that

*the total pressure exerted by a mixture of gases is the sum of the partial pressures of the individual gases*

The total pressure in moist air can therefore be expressed as

p_{t}= p_{a}+ p_{w}(3)

where

p_{t}= total pressure (kPa)

p_{a}= partial pressure dry air (kPa)

p_{w}= partial pressure water vapor (kPa)

#### Dry Air Partial Pressure

By using *(1)* and *(2)*, the dry air partial pressure can be expressed as

p_{a}=ρ_{a}(286.9J/kg K)T(4)

#### Water Vapor Partial Pressure

The water vapor partial pressure can be expressed as

p_{w}= ρ_{w}(461.5J/kg K)T(4b)

Unlike other gases in air, water vapor may condense under common conditions. Since the boiling point of water at normal atmospheric pressure (*101.3 kPa)* is *100 ^{o}C,* the vapor partial pressure of water is low compared to dry air partial pressure in moist air. Common values for vapor pressure in moist air are in the range

*0.5*to

*3.0 kPa*.

Maximum vapor pressure before water vapor start to condense at an actual temperature is called saturation pressure - *p*_{ws}.