Under all temperature & pressure.
No gas is ideal but they are real gases which obeys ideal gas eq. at low pressure & high temperature.

Ideal gas is characterised by following postulates:

(1) Ideal gas cannot be liquefied.

(2) There is no force of attraction b/w gas molecule.

(3) Volume of ideal gas molecule is negligible as compared to containce



For ideal gas
Compressibility factor z = 1

For real gas z 1.
z > 1 (Volume of gas is relatively more dominant)
It shows +ve deviation. This implies gas is less compressible.
z < 1 (Force of attraction of gas molecule is relatively more dominant)
It shows -ve deviation. This implies gas is more compressible.
z = ................................................ (i)
If gas shows ideal behaviour
Pv_ideal = nRT    v_ideal =
Substituting in (i)
z =

into account, force of attraction among molecule as nell as volume of gaseous molecule.


Derivation:

Correction for volume: Suppose volume occupied by gas molecules is v. When molecules are moving their effective is 4 times factual volume i.e. 4v.
b = 4v (excluded volume)
corrected volume = (v - nb) for n moles

Dumb Question: Why effective volume is 4 times of actual volume ?
Ans:


Excluded volume of 2 molecules is sphere of radius of 2R where R radius of ...

Excluded volume for two molecule
            
Excluded volume for one molecule ?
            
        
Constant 'a' measures force of attraction. Greater value 'a', higher intermolecular force of attraction.

Dumb Question: How unit of 'a' = atm L mol ?
Ans: P =
a = atm²L mol^-2
Units of 'b' = L mol^-1

Different forms of Vander Waal Equation:

(i) At very low pressure: v is very large. Hence, correction term a/v² is negligible correction term 'b' is also negligible. Now Equation reduced to

Pv = nRt

That's why real gas behave like ideal gas at very low pressure.

(ii) At mederate pressure:V decreases. Hence a/v² increases & cannot neglected but b is negligible because volume is still high.
  v = RT (for 1 mole)


(iii) At high pressure: v is so small so b cannot be neglected. Factor a/v² is no doubt large but as P is very high, a/v² can be neglected.


(iv) At high temperature: v is very large (at constant pressure). So, that both correction factors are negligible. (a/v² & b)

PV = RT

So, at high temperature, real gases behave like ideal gas.

*Tip: Gases tend to behave ideally at high temperature and low pressure and non ideally at low temperature and high pressure.

Illustration: Calculate pressure exerted by 110g of CO₂ in vessel of 2L at 27⁰C. Given that Vander Waal's constants are a = 3.59 L² atm mol^-2 & b = 0.0427 L mol^-1.
Ans: According to Vander Waal's Equation

n = = 2.5 mol.
P =
P =
  = (33.61 - 5.61) atm
  = 28 atm

Shows ideal gas behaviour.
Mathematically defined.


= 0
= 0
(v - b) = RT (For 1 mole)
P =
Pv =