Firstly: Real Gases

A real gas in chemistry, physics, and mechanical engineering is a gas characterized by properties that cannot be accurately described by the ideal gas law. A real gas is not an ideal gas. These gases are encountered regularly in our practical lives, as they power engines and influence the weather. To understand the behavior of real gases, certain properties must be taken into consideration:

• They exhibit pressure effects.
• Variable specific heat.
• Van der Waals effects (where particles have sizes different from zero).
• Thermodynamic effects in non-equilibrium conditions.
• Influence on particle dissociation and chemical reactions.

Studying the thermal motion of a system takes these effects into account to obtain accurate results. The ideal gas law can be applied to obtain reasonable approximate results. On the other hand, real gas models must be applied when dealing with gas condensation near critical points at high pressures, low temperatures, and some other cases. [1]

Models for Real Gas Equations

The figure illustrates the change in pressure with volume at constant temperature for a real gas. Gases follow complex paths that thermodynamics attempts to describe with equations to facilitate calculations. Here, we will describe regions of the following curves: [2]

• Dark blue curves – changes at constant temperature (green parts – quasi-stable states).
• Parts to the left of F – liquid state.
• Point F – boiling point.
• Line FG – equilibrium between liquid and gaseous states.
• Part FA – superheated liquid.
• Part F′A – liquid under negative pressure (p < 0).
• Part AC – unrealistic expansion at constant temperature, where the system is unstable.
• Part CG – vapor under condensation.
• Point G – condensation point.
• Region to the right of point G – normal gas.
• Areas FAB and GCB are equal.
• Red curve – Critical isotherm at constant temperature.
• Point K – critical point.
• Light blue curves – supercritical states at constant temperature (very high pressure).

Van der Waals Model

We typically deal with real gases by considering their molar mass and molar volume:

Where:

• ( P ) is pressure,
• ( T ) is temperature in Kelvin,
• ( R ) is the gas constant,
• ( Vm ) is the molar volume,
• ( a ) and ( b ) are empirical parameters for each gas, sometimes determined from the critical point ( Tc ) and ( Pc ) using the relationships:

The Redlich-Kwong Model

The Redlich-Kwong equation contains two additional parameters (a and b) that are used to represent real gases. It is considered more accurate than the Van der Waals equation, and its formula is as follows:

Where:

• ( P ) is pressure,
• ( T ) is temperature,
• ( R ) is the gas constant,
• ( Vm ) is the molar volume,

( a ) and ( b ) are parameters that differ from the Van der Waals coordinates. They can be determined as follows:

Here, ( Tc ) and ( Pc ) are the critical temperature and critical pressure, respectively.

Clausius Model

The Clausius equation, named after the scientist Rudolf Clausius, is a simple equation that incorporates three coordinates used for representing real gases:

and ( Vc ) is the critical volume.

The Peng-Robinson Model

The Peng-Robinson equation contains two coordinates but is sparingly used. It is notable for its applicability not only to certain gases but also to some liquids:

Secondly: Ideal Gas

The ideal gas [3] is a thermodynamic model that describes the behavior of matter in the gaseous state. The model assumes no interaction between gas particles and considers them as point particles. Therefore, it is suitable for describing gases with low density, and it also applies to inert gases like helium, neon, and argon, where the particles do not form molecules and exist as individual atoms. This model was discovered in the 19th century.

Specifications of the Ideal Gas

This model strictly adheres to Boyle's Law and also follows Avogadro's Law. The ideal gas has the following conditions:

1. The volume of gas particles is negligible compared to the container containing them, especially under low pressure.
2. Collisions between gas particles are perfectly elastic.
3. The motion of gas particles is random and unaffected by external influences.

The ideal gas serves as a theoretical concept to facilitate dealing with various variables in thermodynamics. The three assumptions or conditions mentioned above are what make a real gas, when found under these circumstances, behave like an ideal gas.

The volume of gas particles is negligible compared to the container containing them, especially under low pressure:

A gas we study cannot exist without being in a container. If we calculate the volume of gas particles and compare it to the container's volume, we find it negligible. This holds true when the gas pressure is lower than atmospheric pressure or at room temperature. If the gas is under high pressure conditions, such as gas in a tank (like cooking gas in a cylinder), it will not behave as an ideal gas, and we cannot neglect the gas's volume concerning the tank.

Collisions between gas particles are totally elastic:

This assumes that particles do not lose any of their energy when colliding with each other.

The motion of gas particles is random and unaffected by external influences:

Random motion is movement not governed by any physical laws that can predict it. The particles move in this manner, and we assume that the motion of the particles is random.

Of course, these assumptions are fulfilled under specific conditions to approximate as much as possible the establishment of a law governing gas variables (pressure, volume, temperature).

The Ideal Gas Equation

The Ideal Gas Equation, or the thermal state equation for an ideal gas, describes the behavior of an ideal gas and its changes in temperature, for example. This equation was formulated as a result of numerous experiments (gas laws). Ludwig Boltzmann later, through probability calculations (statistical thermodynamics), provided an explanation for the gas behavior based on the molecular structure of gases.

The general gas equation describes the state functions of an ideal gas and the relationships between these functions, including temperature, pressure, and gas volume. It is written in various forms in chemistry and physics, but all have the same meaning.

In this equation:

• ( P ) is pressure,
• ( V ) is volume,
• ( T ) is temperature in Kelvin,
• ( R ) is the universal gas constant,
• ( n ) is the number of moles.

where: [4]

The general gas constant is denoted by R , and the specific gas constant is denoted as Rs. The relationship between them is given by:

Rs = R/M

Here, M is the molar mass of the gas.

By using the general gas equation along with the laws of thermodynamics, we can mathematically describe the thermal motion processes of ideal gases. Boltzmann's constant kB is related to the universal gas constant by the equation kB . N = R , where N is Avogadro's constant, representing the number of particles in one mole of substance.

In general, understanding any one of the equations mentioned above is sufficient as they all represent the general equation for gases, such as:

1- Physical Chemistry Portal, Wikipedia
2- D.-Y. Peng and D.B. Robinson, "A New Two-Constant Equaiton of State,"
3- Physics Portal, Wikipedia
4- National Institute of Standards and Technology.physics.nist.gov