Equation of State: Ideal and Real Gases | Chemistry Optional Notes for UPSC PDF Download

The Ideal-Gas Equation of State

• Any equation that relates the pressure, temperature, and specific volume of a substance is called an equation of state. The simplest and best known equation of state for substances in the gas phase is the ideal-gas equation of state. Gas and vapor are often used as synonymous words. The vapor phase of a substance is called a gas when it is above the critical temperature. Vapor usually implies a gas that is not far from a state of condensation.
• It is experimentally observed that at a low pressure the volume of a gas is proportional to its temperature:

or

Pv = RT

• where R is the gas constant. The above equation is called the ideal-gas equation of state (ideal gas relation). Since R is a constant for a gas, one can write:
  where subscripts 1 and 2 denote two states of an ideal gas. Note that the constant R is different for each gas; see Tables A1 and A2 in Cengel book.
• The ideal gas equation of state can be written in several forms: 

where (m³/kmol) is the molar-specific volume. That is the volume per unit mole. R_u = 8.314 kJ / (kmol. K) is the universal gas constant, R = R_u /M.

• The molar mass M: is defined as the mass of one mole of a substance (in gmole or kgmol). The mass of a system is equal to the product of its molar mass M and the mole number N:
  m = MN (kg)
• See Table A-1 for R and M for several substances.
• An ideal gas is an imaginary substance that obeys the relation Pv = RT. It is experimentally observed that the ideal gas closely approximate the P-v-T behavior of real gases at low densities.

In the range of practical interest, many familiar gases such as air, nitrogen, oxygen, hydrogen, helium, argon, neon, and CO₂ can be treated as ideal gases with negligible error.

• Note: Water vapor and refrigerant vapor in refrigerators should not be treated as ideal gases.
• Note: Water vapor at pressures below 10 kPa can be treated as an ideal gas, regardless of temperature. 

Compressibility Factor

The assumption of ideal gas relation implies that:

• The gas particles take up negligible volume
• The intermolecular potential energy between particles is small
• Particles act independently of one another

However, real gases deviate from ideal gas behavior. This deviation at a given temperature and pressure can be accurately accounted for by the introduction of a correction factor called the compressibility factor Z.
or Z = v_actual / v_ideal ≤ 1. Z = 1 for ideal gases.
Gases behave very much the same at temperatures and pressures normalized concerning their critical temperatures and pressures.

Here P_R and T_R are called the reduced pressure and temperature, respectively. By curve-fitting all the data, the general compressibility chart is obtained which can be used for all gases, see Fig. A-15. 

Fig. 6: Z factor, general compressibility chart.

From the Z chart, one can conclude:

• At very low pressure (P_R << 1), the gases behave as an ideal gas regardless of temperature
• At high temperatures (T_R > 2), ideal gas behavior can be assumed.
• The deviation is highest in the vicinity of the critical point.

Example: Ideal Gas

Determine the specific volume of R-134a at 1 MPa and 50°C, using (a) ideal gas equation (b) the generalized compressibility chart. Compare the values obtained with the actual value of 0.02171 m³/kg.
Ans: From Table A-1, for R-134a, R = 0.0815 kPa.m³ /(kg.K), P_cr = 4.067 MPa, and T_cr = 374.3 K
(a) Ideal gas equation of state
Compared with the tabulated value, using ideal gas equation one would get an error of (0.02632-0.02171)/0.02171=0.212 or 21.2%.
(b) To determine the correction factor Z,
 
From Fig. A-15, Z= 0.84. Thus,
v = Z v_ideal = 0.84 (0.02632 m³/kg) =0.02211 m³/kg
The error is less than 2%.
Therefore, in the absence of exact tabulated data, the generalized compressibility chart can be used.

Other Equations of States

Van der Waals Equation of State:

where.

In this relationship, the intermolecular attraction forces and the volume occupied by molecules are considered.

Note: the accuracy of the Van der Waals equation of state is often inadequate.

Beattie-Bridgeman Equation of State
It is based on five experimentally determined constants:

where

Constants appearing in the above equation are given in Table 3-4, Cengel book.