## IDEAL GASSES BASIC AND TUTORIALS

An ideal gas is a hypothetical substance. The mathematic definition of an ideal gas is that the ratio pv/T is exactly equal to R at all pressures and temperatures (note that when using this equation, the pressure and temperature must be expressed as absolute quantities), otherwise it is called a real gas.

At low densities, the ratio pv/T has, by experiment, the same value of R (gas constant) for a specific gas. R is the gas constant of a gas. Different gases have different gas constants.

How close is a real gas to an ideal gas? This is an important question. It is convenient to use such a simple equation of state, but we must know how much accuracy we sacrifice for the convenience. In general a gas with a low pressure and high temperature is considered to be an ideal gas.

But high and low are relative terms. For example, 100 kPa may be considered a high pressure, if very good accuracy is required. On the other hand, 100 kPa may be considered a low pressure, if good accuracy is not required.

The relation pv=RT is known as the ideal-gas equation of state. The gas constant R is given by R=Ru/M, where Ru is the universal gas constant, which has the same value for all gases, and M is the molar mass. The value of Ru, expressed in various units, is Ru=8.314 kJ/kmol(K)=1.986 Btu/lbmol(R)=1545 ft(lbf)/lbmol(R)

Experiments by Joule show that internal energy (u) of an ideal gas is a function of temperature (T) only. Therefore, enthalpy (h=u+pv=u+RT), specific heat at constant pressure (cp) and specific heat at constant volume (cv) of an ideal gas must also each be a function of temperature (T) only.

The change of internal energy and change of enthalpy are given by the following equations:

Δu=∫ cvdT
and
Δh=∫ cpdT

The change of entropy is given by the following equations:
Δs=∫ cvdT/T + R ∫ dv/v (2.3.3)
and
Δs=∫ cpdT/T - R ∫ dp/p (2.3.4)

It is possible, but long and tedious, to calculate property changes of an ideal gas in a straight forward manner using above equations. It certainly would be convenient if tables or charts existed listing the values of the thermodynamic properties. Fortunately, tables and charts for many ideal gases are available.

The specific heats (both cp and cv) of a gas vary with temperature. The functional relationships denoting these variations are determined from experimental data. If accuracy is desired, these equations must be used.

In many applications an average value of specific heat is used (based on the temperature range under consideration) for quick results. The internal energy change, enthalpy change and entropy change of the ideal gas from state 1 to state 2 can be simplified from Eqs. (2.3.1), (2.3.2), (2.3.3) and (2.3.4) and expressed as:

Δu=cv(ΔT)
Δh=cp(ΔT)
Δs=cv [ln(T2/T1)] + R [ln(v2/v1)]
and
Δs=cp [ln(T2/T1)] - R [ln(p2/p1)]

Air is the most important gas used in engineering thermodynamic application. The gas constant (R), specific heats (cp and cv) and specific heat ratio (k) of air at room temperature have the following numerical values:

Rair=0.06855 Btu/[lbm(R)]=0.3704 [psia(ft3)]/[lbm(R)]=0.2870 kJ/[kg(K)]
(cp)air=0.240 Btu/[lbm(R)]=1.004 kJ/[kg(K)]
(cv)air=0.171 Btu/[lbm(R)]=0.718 kJ/[kg(K)]
kair=1.4

For air and many other gases, over the range of pressures and temperatures we commonly deal with, the assumption of ideal gas behavior yields a very excellent engineering approximation. However, as we get to high pressures, deviations from ideal gas behavior may be large in magnitude.

In these cases, use of the ideal gas law will depend on the degree of accuracy required for a particular problem.

The tabulation of h, u and other properties of air, using the temperature as the argument has been made and is given by Keenan and Kaye. A part of the air table is given in Table below.