Properties of Ideal Gases

For ideal gases the properties of temperature, pressure, volume, and others are related to each other according to two simple laws. The first is known as the ideal gas law, which can be written down in several different forms all of which are all related to each other. For example :

(1)     p V = m Rs T    or     p V = n Ru T    or     p v = Rs T

p [N/m²] = absolute pressure ( p=0 for vaccuum )

V [m³] = volume occupied by ideal gas

v [m³/kg] = specific volume = volume occupied by 1 kg of gas = inverse of density ρ

m [kg] = mass of ideal gas

Rs [kJ/(kg K)] = specific gas constants, its value depends on the gas at hand.

T [K] = absolute temperature

n [kmol] = number of moles of gas , 1 mol = 6.023×1023 molecules (Avogadro's number)

Ru = universal gas constant = 8.31434 kJ/(kmol K)

The above two forms of the ideal gas law are equivalent because :

m/n = M [kg/kmol] = molecular mass of gas , often but falsly called the molecular weight

and

Rs = Ru/M

Another property of ideal gases, much less known, is that the internal energy, u, of an ideal gas does not depend on pressure which was originally verified experimentally by the English physicist James Prescott Joule (1818-1889). In that experiment two spherical containers, A and B, are immersed in a water bath. They connected by a small tube closed off initially by a stop. Container A is filled with a gas at a high pressure while container B is empty. Everything is at the same temperature. Now the stop is opened and some gas streams from container A into B until the pressure is equal in both containers. Because of the larger volume the gas now occupies the pressure is lower. Despite that, measurements show that the temperature of the gas has not changed. Because during this process no heat has been supplied nor any work has been done. As a consequence the internal energy, u, of the gas is unchanged.
The consequences of this observation are profound for ideal gases. Firstly, because of the ideal gas law, Eq.(1), and the definition of the enthalphy h = u + p v = u + R T the enthalpy depends on temperature only as well. Furthermore, if u and h depend on temperature only so are the specific heat capacities, cv and cp. And finally :

(2)     cp - cv = Rs

For some ideal gases, specifically all the noble gases He, Ne, Ar, Kr, and Xe, the specific heat capacities are completely temperature independent and their ratio is :

(3)     κ = cp/cv = 1.667

For many diatomic gases like N2, O2, air, and CO the specific heats and their ratio are nearly temperature independent over a wide range of tempratures. A generally accepted value is :

κ = 1.4     for diatomic gases

Some Properties of selected Gases
GasFormulaM [kg/kmolRs [kJ/(kg K)] κ = cp/cv* Tcrit [K]pcrit [MPa]
HeliumHe4.003 2.07691.6675.030.23
ArgonAr39.948 0.20811.6671514.86
NitrogenN228.013 0.29681.400126.23.39
OxygenO231.999 0.25981.395154.85.08
Air 28.97 0.28701.400132.53.77
HydrogenH22.016 4.12401.40533.21.30
Carbon DioxideCO244.01 0.18891.289304.27.39
WaterH2O18.015 0.46151.327647.322.09
MethaneCH416.043 0.51821.299191.14.64
PropaneC3H844.097 0.18851.1263704.26
OctaneC8H18114.232 0.07291.04456924.9
*   at 300 K

A last question of course arises : Under which circumstances does a real life gas behave like an ideal gas ? The answer is somewhat ambiguous because the ideal gas is an abstraction which is approached by every gas as the pressure is lowered and the temperature is raised. If a few percent accuracy is sufficient then a good guideline would that if the absolute temperature of interest is more than twice that of the critical temperature ( listed as Tcrit in above table ) you are in good shape. If your temperatures are lower than that, your pressure of interest better be below 0.2 times the critical pressure ( listed as pcrit in above table). If you need a clearer answer, then you either have to get the property data of your gas ( the NIST website at http://webbook.nist.gov/chemistry/fluid/ is a very good source ) or google for the term Compressibility Factor= p V/(R T) which is of course 1 for ideal gases but ≠1 for real gases.

A last comment concerning criticial temperature and pressure, just in case you don't know these :

If you have a gas at a temperature above its critical temperature then the gas will occupy less and less volume as the pressure is increased provided you keep the temperature constant by removing heat as you compress. It will never condense as the pressure increases and we have no clear definition for when you would stop calling it a gas and start calling it a liquid.
If your temperature at which you compress and remove heat is below the critical temperature your gas will ultimately condense into a liquid at a pressure particular to your temperature and gas as you remove heat, i.e 1 atm for water at 100°C.
When condensing/boiling at a particular temperature takes place all property data of the liquid and of the gas phase have values which are unique for the gas and the given temperature, including pressure. As the boiling temperature increases the values of all properties ( like internal energy, enthalphy, specific volume, index of refraction etc. ) of the liquid approach those of the gas phase. At the critical temperatur/pressure they are equal. Here is a movie which displays the fate of a water level below the critical point temperature as the water is slowly heated from below. At the beginning ("low" temperature) you see the liquid water level as a thick black bar across the viewing window of the apparatus. The interesting part is the last quarter of the movie as the water level stays at about the same level but becomes dimmer and dimmer to lastly vanish in "mid-air". Not that this has anything to do with Stirling engines.


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Last revised: 12/04/14