Deviation from Ideal Behaviour by Gases

Van der Waalsinglequotes equation and deviation from the ideal behaviour:

  At high  temperatures and low pressures (what we may call as normal conditions)  most of the gases follow ideal behaviour and the factor (PV/RT ) is 1 for one mole of a gas ,and the graph of ( PV/RT ) vs P is close to 1.

     The factor ( PV/RT ) is called compressibility factor and is denoted by Z.

Where, P is the pressure, V is the molar volume, R is the gas constant and T is the temperature.Its value is 1 for ideal gases by definition. Its very useful in taking into account the behaviour of real gases. The value of Z normally increases with an increase in pressure and decreases with decrease in temperature.As the assumptiion in the ideal gas beahviour is that intermolecular interactions are negligible but at low temperatures and at high pressures they become significant.High pressures increases their frequency of collison and at low temperatures they are moving rather slowly, due to which the attractive forces between the molecules have a noticeable effect on each other and the real volume of the gases decreases from their ideal volume , hence pulling the value of Z down from 1. On the other hand at low pressures or high temperatures the repulsive forces become more dominant and the value of Z becomes more than 1.In fact, the value of Z deviates most from 1 whenver the gas is close to phase change either at low temperatures or high pressures.The variation from the ideal behavior can be gauged from the following graphs:

 Thus, the value of compressibilty factor is not one in a lot practical situations ,and the ideal gas equation also gets modified for real gases. In other words , we need to apply some corrections in the pressure and the volume terms, because at high pressures the volume occupied by the gas molecules compared to the volume of the container is not insignificant and cansinglequotet be ignored.Similiarly, at low temperatures and high pressures attractive forces between molecules is significant and their is a correction required in the  pressure term.Thus the size of the molecules and their interactive forces can not be neglected alltogether.Though the two factors tend to cancel each other out and the product PV tends to be equal to the product of singlequoteactualsinglequote pressure and singlequoteactualsinglequote volume, one of theses factors become more dominant than the other and the value of Z becomes different from one. For example in the figure above the curve at 200K intersects the line Z =1 at pressure of approximately 400K. Hence the ideal gas equation is replaced by the equation containing these corrections in the pressure and volume terms.

Its called Van Der Waals equation and is represented as:

Where, p is the pressure of the fluid or the gas,

V is the total volume of the container containing the gas,

R is the gas constant,

T is the temperature,

"a" and "b" are called van der waals constants , "a" is a measure of attractions between the molecules and "b" is measure of volume excluded by one mole of the particles of the gas.

, the factors "n²a/V²" is pressure correction owing to the intermolecular attractive forces and,

 the factor  "nb" is the volume correction due to finite size of the molecules.

The unit of a is atm L² mol^-2 and the unit of b is L mol^-1.

  The  value of the constant "b" is = 4 X volume ocupied by the molecules in 1 mole of gas.

 Critical phenomenon and the liquefication of a gas:

  The phenomenon of the changing the state of the gas to liquid form is called liquefication. Any gas can be liquified if we increase the pressure suffuiciently or decrease the temperature or both.But there is limit on both pressure and temperature which should be there to liquefy the gas.As we reduce the temperature the gas , the kinetic energy of the gas decreases and the molecules do not posses sufficient energy to overcome the intermolecular attractive forces , hence the gas condenses into liquid.But , for each substance , there is minimum lowering a temperature is needed.That particular temperature above which the gas cannot be liquefied is called critcal temperature and is denoted by T_c. So, no matter how high the pressure is if the temperature is above the critical temperature the liquefication of gas is impossible.

Similarly, the minimum pressure required and below which the liquefication of gas is not possible is called critical pressure. Its denoted by P_c.

Similarly , there is a term called critical volume (V_c), which is defined as the volume occupied by one mole of a substance at critical temperature and critical pressure.

• The constants T_c, P_c,V_c are specific to a particular gas, which means they are different for different gases.
  
• The numerical value of T_c, P_c,V_c are calculated from the van der waal gas equation.The values are:
                                                    T_c = 8a/27Rb ; P_c = a/ 27b² and V_c = 3b
  
• If we substitute the value of all the critical constants we get the numerical value of RT_c/P_cV_c = 8/3 ( Note, that in ideal conditions the RT/PV is 1).
• A gas having critical pressure, critical temperature and critical volume is said to be in  critical state.

Boylesinglequotes Temperature:

The temperature at which real gases confirm to Boylesinglequotes law is called Boylesinglequotes temperature T_b.

The numerical value of Boylesinglequotes temperature is T_b = a / Rb where , a and b are van der waal constants. At boylesinglequotes temperature gases follow boylesinglequotes law over a wide range of temperature.

Inversion temperature: Inversion temperature is studied in the Joule-Thompson in thermodynamics. The numerical value of this inversion temperature (T_i) is related to the the boylesinglequotes temperature by the following equaiton:

                                     T_i = 2T_b

                                                      = 2a / Rb

Collsion frequency and mean free path:

As the name suggsests the collision frequency denotes the rate or frequency of collision of miolecules.It is defined as the number of collisions taking place per unit time per unit volume and is denoted by z

Mathematically,                        z = πn²σ²C_av/?2

Mean Free path:  The average of distances travelled by the molecules between two successive collisions is called mean free path (λ) 

Mathematically,                  λ = 1/ ?2πnσ²

 where,

n = number of molecules per unit molar volume , molar volume = 0.0224 m³

n = 6.023 X 10²³/ 0.0224  m^-3

Cav = average velocity,

σ = collision diameter ( the minimum distance between the centre of the molecules at the point of collision) .

Based on the kinetic theory of gases , it is observed that the mean free path,

                                     λ    T/P .

Thus, from the above two relations on the mean free path , it can observed that,

• Mean free path is inversely proportional to the size of the molecules
  
• Mean free path is inversely proportional to the number of of molecules per unit volume
  
• Mean free is directly proportional to temperature
  
• Mean free path is inversely proportional to pressure.

 [SolvedExample]

Example9:  The value of the van der waals constant b for Neon is 17.10 x 10^-6 . Calculate its molecular diameter.

Solution: We know, b = 4 X volume occupied by the molecule in one mole of the gas

 ?                                  = 4 X N₀ X (4/3πr³)

 ?    17.10 x 10^-6           = 4 X 6.02 X 10^-23 X 4/3 X 3.14 X r³

?                   r₃             = (4 X 6.02 X 4 X 3.14 X 10^-17_)/ (3 X 17.10)

                                      = 5.89 X 10^-17

?                   r             =  3.89 X 10^-6 m

[/SolvedExample]