This document summarizes key differences between ideal and real gases. It discusses that real gases interact with each other, taking up physical space and attracting one another, resulting in lower than expected pressures. Real gas properties depart from ideal behavior at low temperatures or high pressures. The partition function approach is used to derive equations of state for both ideal and real gases. For real gases, the virial equation accounts for intermolecular interactions through virial coefficients.

1. DEPARTMENT OF PHYSICS
M.SC. 1ST SEMESTER (2015-2016)
SUB: STATISTICAL PHYSICS
PRESENTED BY: CHITRA JAIN
SUBMITTED TO : Dr H.S. SINGH

2. REAL GAS

3.  REAL GAS:
Real gas is one in which mutual interaction between
molecules can not be neglected.
i.e. Potential energy of interaction is non zero.
IDEAL GAS:
Ideal gas is one in which mutual interaction between
molecules are negligible.
i.e. Potential energy of interaction is zero.

4.  PROPERTIES OF REAL GAS:
1. Real molecules do take up space & do interact with each other.
2. Real gas molecules are not point masses.so,
the actual volume free to move in is less because of particle size.
V’ = V – nb “b” is a constant that differs for each gas.
3. Molecules do attract each other therefore pressure on the container will
be less than ideal.
2
observed )
V
n
(aPP 

5. 4. The FUGACITY represent chemical potential for real gas.
5. Most real gas depart from ideal behaviour at deviation from
- Low temperature
- High Pressure
6. As in real gas Interaction between molecules is not negligible so due
to interaction between molecules potential energy arises.

6. 6. As in real gas Interaction between molecules is not negligible so due to
interaction between molecules potential energy arises.
Acc. to plot
1. At larger distance the atoms
virtually do not interact and 𝑢(𝑟) is
zero.
2. At smaller distance forces of mutual
attraction tend to bring the atoms
closer and 𝑢(𝑟) diminishes.
3. At a distance r0 𝑢(𝑟) is minimum.
4. At 𝑟 < r0 , repulsive force dominant
and 𝑢(𝑟) increases.
𝑢(𝑟) = u0 [(
𝑟 𝑜
𝑟
)12 - 2(
𝑟 𝑜
𝑟
)6]

7.  Statistical mechanics of Ideal & Real Gas
Ideal Gas
Since we know that an ideal gas is one in which mutual interactions b/w molecules
are negligible i.e. potential energy of interaction
U=0
Hence
1.The total energy:
E= K.E.+P.E.
E=
𝑝²
2𝑚
+ U ⟹ 𝑖=1
𝑁 𝑝²
2𝑚

8. 2.The Partition Function:
Partition Function for an Ideal gas:
Z=[(
2𝑚𝜋𝑘𝑇
ℎ²
)
3
2
V] 𝑁
m= mass of molecule;
𝑘= Boltzmann constant;
T = Temperature;
h= Planck constant;
V= Volume of container;
N= Number of molecules;
Z= 𝑒−𝛽𝐸𝓈

9. ln 𝑍 = N [ln 𝑉 +
3
2
ln (
2𝜋𝑚
ℎ2 ) -
3
2
ln𝛽]
3.The pressure P
P=
1
𝛽
𝜕ln 𝑍
𝜕𝑉
P=
𝑁
𝛽𝑉
=
𝑁𝑘𝑇
𝑉
PV= NkT
that is the equation of state of Ideal Gas.

10. Real Gas
1.The total energy:
Since we know that in case of Real gases mutual interactions can not be neglected
so,
The energy of a monatomic gas of N identical atoms, each of mass m is
E= 𝑖=1
𝑁 𝑝²
2𝑚
+ U
Where first term gives the K.E. of atoms & U is the sum of the potential energies
of interaction b/w the pairs of atoms.
U=u12+u13+ ………+ u23+……..=
1
2 𝑖≠𝑗 𝑢ij

11. 2.The Partition Function:
Z=
𝑍 𝑖𝑑
𝑉 𝑁 × 𝑒−𝛽𝑈
𝑑 𝑞1
3
𝑑 𝑞2
3
……..𝑑 𝑞 𝑁
3
this is interacting Partition Function.
& so,
Z= 𝑍𝑖𝑑 ⋅ 𝑍 𝜙
where,
𝑍 𝜙=
1
𝑉 𝑁 𝑒−𝛽𝑈
𝑑 𝑞1
3
𝑑 𝑞2
3
……..𝑑 𝑞 𝑁
3
or
𝑍 𝜙=
1
𝑉 𝑁 𝑖>𝑗 𝑒−𝛽𝑢 𝑖𝑗
𝑖 𝑑 𝑞 𝑖
3
is called “ Configurational Partition Function "or “Configurational Integral”.

12. Evaluation of 𝒁 𝝓:
now introducing
fij= 𝑒−𝑈(𝑞 𝑖,𝑞 𝑗)𝛽
-1 …………………(1)
which has the property that fij is only appreciable when
the particles are close together .
in terms of this parameter the configurational integral
is
𝑍 𝜙=
1
𝑉 𝑁 𝑖<𝑗(1 +fij) 𝑑 𝑞𝑖
3𝑁
……………….. (2)
Where exponentials of the sum has been factored into
product of exponentials.
Expansion of product is as follows
𝑖<𝑗(1 +fij)=1+ 𝑖<𝑗 𝑓𝑖𝑗+ 𝑖<𝑗 𝑘<𝑙 𝑓𝑖𝑗 𝑓𝑘𝑙+…(3)

13. With this expansion it is possible to find terms of different order., in terms of number of particles that
are involved.
The 1st term is single particle term, the 2nd term corresponds to the two particle interaction, the 3rd
to the three particle interaction & so on.
Such expansion is called the CLUSTER EXPANSION(Series expansion to handle inter-particle
interactions)
Each term represent the interaction within clusters of a certain number of particles.
Contribution to third term may be represented as
i,j,k,l,distict i=k i=k & j=l

14. Substituting the 𝑒𝑞 𝑛
( 3 ) in (2)
expansion of
𝑍 𝜙= 1 +
𝑁
𝑉
𝛼1+
𝑵(𝑵−𝟏)
𝟐𝑽 𝟐 𝛼2+………….( 4)
now substituting the 𝑒𝑞 𝑛
for free energy
eq. of state of real gas
PV= NKT (1+
𝑁
𝑉
𝐵2(T) +
𝑁2
𝑉2 𝐵3(T) +……
known as “virial equation” and components 𝐵𝑖(T) are the “virial coefficients.
Now 𝐵2= -
1
2
( 𝑒−𝛽𝑢(𝑟)
-1) 𝑑3
r = -
1
2
( 𝑒−𝛽𝑢(𝑟)
-1) 4𝜋𝑟2
dr
= −2𝜋 ( 𝑒−𝛽𝑢(𝑟)
-1) 𝑟2
dr

15. 𝐵2 = −2𝜋 ( 𝑒−𝛽𝑢(𝑟)
-1) 𝑟2
dr
= −2𝜋[ ( 𝑒−𝛽𝑢 (𝑟)
− 1) 𝑟2
dr
𝑟0
0
+ ( 𝑒−𝛽𝑢 (𝑟)
− 1) 𝑟2
dr
∞
𝑟0
=
2𝜋 𝑟 𝑜
3
3
(1-
𝑢 𝑜
𝐾𝑇
)
Hence,
𝐵2 𝑇 = 𝑏′
−
𝑎′
𝐾𝑇
where b’=
2𝜋𝑟 𝑜
3
3
and a’=
2𝜋𝑟 𝑜
3
3
𝑢 𝑜
Hence,
𝑃
𝐾𝑇
= n + 𝐵2 𝑇 𝑛2
(neglecting higher terms)
𝑠𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑖𝑛𝑔 𝑣𝑎𝑙𝑢𝑒𝑠 n=
𝑁
𝑉
=
𝑁 𝐴
𝑉
where 𝑁𝐴is Avogadro
number and v is molar volume.

16. (𝑃 +
𝑎
𝑉2)(V-𝑁𝐴 𝑏′) = 𝑁𝐴K
where a=a’NA; b=b’NA; R= NAK
(𝑃 +
1
𝑉2)(V-𝑏) = RT
The correction term
1.‘a’ comes from the long range weak attractive
force b/w molecules.
2.‘b’ comes from volume of the molecule.

17. THANK YOU