The correlation between pressure P, volume V and temperature T for a gas can be described by the ideal gas law. However, in many common situations – inside a cylinder at high pressure, in a refrigeration system at low temperature – gases do not always behave ideally. For this reason, the ideal gas equation cannot be always used; other equations should be employed instead.
Ideal behavior
A gas is defined ideal when:
- its molecules have volume that is negligible compared to the total volume of the gas. In other words, the molecules are like points and not like finite elements with an associated volume.
- the forces of attraction between the molecules of the gas are negligible. This means that the molecules just collide with each other as hard spheres, with elastic collisions.
When these conditions are respected, the behavior of the gas can be described by the equation:
PVm = RT
Where Vm is the volume for a mole of gas and R is the universal gas constant (8.314 J/mol·K).
In real life, there is no such thing as an ideal gas, as these conditions are never completely true. Despite this, at low pressure and temperature, the ideal gas equation describes the behavior of the gases pretty well.
Real behavior – Van der Waals equation
When gases are at higher pressure or lower temperature, in circumstances as described above, this equation cannot be considered valid anymore. Van der Waals equation is used instead to correlate P, V and T; the mathematical expression of this equation is shown below.
P = [RT/(Vm - b)] - (a/V²m)
The terms a and b are characteristics of each gas; they take into account the deviation from the ideal behavior. In particular:
- The term a considers the interactions between the molecules. Its value is related to the fact that the collisions between the molecules are not elastic anymore, due to their attraction forces. This will change the pressure that a gas exerts towards the walls of the vessel it is in.
- Its formula is a = 27R²T²c/64Pc
- The term b considers the volume of each molecule of gas. The molecule’s volume cannot be ignored at low temperature, when the total volume of the gas is smaller.
- Its formula is b = RTc/8Pc
Liquid-vapor equilibrium and critical conditions
The terms Tc and Pc in the formulas represent the temperature and the pressure of the gas at the critical point – critical temperature and pressure. They are characteristics of each gas, their values are reported in literature.
The critical temperature for a gas is the temperature above which the gas cannot be liquefied by just applying a pressure.
To understand this, we can consider as an example a volume of water vapor at room pressure and temperature.
- Applying a greater pressure, the vapor will be liquefied, even without lowering the temperature.
- However, above a certain temperature, the liquefaction will not take place, no matter how big the pressure applied.
- This temperature is called critical temperature Tc.
- The critical pressure Pc is the minimum pressure necessary to liquefy the gas when it is at the critical temperature.
The values of Tc and Pc can vary very much from gas to gas; for instance for water the critical temperature and pressure are 374°C and 22.06 MPa respectively, while for carbon dioxide they are 31.1°C and 7.38 MPa.
Beware of the units
When applying the Van der Waals equation and/or calculating the Van der Waals constants a and b, pay attention to the units, to be sure you do not make mistakes.
- In the equation, the temperature T is expressed in degree Kelvin, the molar volume Vm in m³/mol, the pressure P in Pascal. The same units must be used for the values of Tc and Pc.
- a has units of (pressure x molar volume squared): Pa m6/mol²
- b has units of molar volume: m³/mol
Sources
P.W Atkins, Physical Chemistry, 8th Edition, W.H. Freeman.
Clara Piccirillo - Clara Piccirillo
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