This document discusses the concept of fugacity, which is a property that accounts for the non-ideal behavior of real gases. It introduces fugacity and explains that fugacity can be used to represent real gases in thermodynamic equations in place of pressure. The document then discusses methods for determining fugacity, including calculating fugacity at low pressures. It also covers the fugacity of gas mixtures and the physical significance of fugacity. Finally, it defines excess functions which quantify the non-ideal behavior of solutions and provides expressions for excess chemical potential, Gibbs free energy, entropy and enthalpy.
1. Shri Shivaji Science College, Nagpur
Seminar Topic :
Concept of fugacity, Determination of fugacity,excess
function of non-ideal solution
By
Bhagyashree S. bokde
M.Sc Chemistry (sem 2)
2. C0NTENT
➢ Concept of fugacity.
➢ Fugacity at low pressure.
➢ Determination of fugacity of gas.
➢ Calculation of fugacity at low pressure.
➢ Physical significance of fugacity.
➢ Fugacity of gas in gaseous mixture.
➢ Excess function of non-ideal solution.
3. Concept of fugacity
The great American chemist G. N. Lewis (1875-1946)
introduced the concept of Fugacity for representing the actual behavior of real gases
which is distinctly different from the behavior of ideal gases.
Variation of free energy with pressure at constat temperature is given by,
1
𝛿𝐺
𝛿𝑝 𝑇
This equation is applicable to all gases whether ideal or non ideal.
If one mole of gas is under consideration,then v-refer to molar
volume. For an ideal gas. The above equation may be written as
ⅆ𝐺 𝑇 = 2𝑇
ⅆ𝑃
𝑝
…………………...
and for n moles as,
ⅆ𝐺 𝑇 = 𝑛𝑅𝑇
ⅆ𝑃
𝑝
2
= nRT d(lnp)……… 3
Integration of this equation is,
G= 𝐺∗ + nRT lnp ..........................4
=Y .....................
4. 𝐺∗
be the integration constant ,which is the free energy of n moles of the ideal
gas at temperature T, then pressure p is unity .
Integration of eq. 2 is between pressure p1 and p2 at constant temp. is,
G = 𝑝1
𝑝2
𝑛𝑅𝑇
𝑑𝑃
𝑃
= nRT . ln
𝑃2
𝑃1
…………..
The corresponding equation for 1 mole of the gas would be,
∆𝐺 = 𝑅𝑇 𝑙𝑛
𝑃2
𝑃1
……………….
Equation 4 and 6 are not valid for real gases , since v is not exactly equal to
𝑅𝑇
𝑃
• In order to make them simple equation applicable to real gases , lewis
Introduced a new fuction F called fugacity function . It takes the plane of in
equation which for ideal gases may be expressed as
(𝑑𝐺)𝑇= nRT d (lnp) …………………
And equation may be represented as,
G = 𝐺∗
+ nRT lnf ………………
Where, 𝐺∗
is the free energy of n moles of a real gas when its fugacity happens
to be 1.
5
6
7
8
5. • Thus, fugacity is a sort of ‘frictious pressure’ which is used in order to retain for
real gases simple from of equations which are applicable to ideal gases only.
• Eq. 8 eventully gives the free energy of a real gas at temperature T and pressure P
at which its fugacity can be taken as f.
• Eq. 7 an integration between fugacities f1 and f2 at constant temp. T yields,
∆𝐺 = 𝑛𝑅𝑇 𝑖𝑛
𝑓2
𝑓1
………………
The corresponding equation for 1mole of the gas would be
∆𝐺 = 𝑅𝑇 𝑖𝑛 (
𝑓2
𝑓1
) ……………..
As discussed above, equation 9 and 10 are applicable to real gases.
❖Fugacity at low pressure :-
The ratio f and p , where p is the actual pressure approaches unity where p
approaches zero . Since in that cost a real gas approximates to ideal behavior. The
fugacity function therefore may be defined as,
limit
𝑓
𝑝
= 1
9
10
p→0
6. • Evidently, at low pressure, fugacity is equal to pressure whrere two terms
differ materially only at high pressure.
❖ Determination of Fugacity of a gas :-
from equation 8 for 1 mole of a gas may be put as,
G = 𝐺∗
+ RT ln f ……………………. 12
• Determination of eq. 12 with respect to pressure at constant temperature and
constatnt no of moles of the various constituents , i.e. in closed system gives,
𝛿𝐺
𝛿𝑝 𝑇
= RT 𝜕(ln 𝑓)
𝜕𝑃
… … … . . 13
Since
𝛿𝐺
𝛿𝑝 𝑇
=v
It, follows that
𝛿 ln 𝑓
𝜕𝑃 𝑇
=
𝑣
𝑅𝑇
……………..14
• Thus, at definite temperature equation 14 may written as,
RT d ( ln f ) =v dp ………………15
Since, one mole of the gas is under consideration. V is the molar
volume of the gas.
7. • Knowing that for an ideal gas ,
v =
𝑅𝑇
𝑃
, the quantity d, defined as departure from ideal behavior at a given
temperature is given by,
α =
𝑅𝑇
𝑃
- V ……………..16
Multiplying by dp throughout we get,
α = RT
𝑑𝑝
𝑝
- vdp ………….17
Combining equation 15 and 17 we have,
RTd (lnf ) = RT
𝑑𝑝
𝑝
- α dp
Or, d (lnf ) = d ( lnp ) – α dp (RT) ……………18
Integrating equation 18 between pressure 0 and p we have,
ln
𝑓
𝑝
=
−1
𝑅𝑇
0
𝑃
∝ (𝑑𝑝) …………..19
8. ❖ Calculation of fugacity at low pressure :-
• It has been found that the experiment value of α at low pressure assumes almost a
constant value under such conditions , therefore eq. 19 gives,
ln
𝑓
𝑝
= -α
𝑝
𝑅𝑇
…………………..20
now, at low pressure since gases tend to be ideal f = p 𝐹
𝑝
≈ 1 ……….21
9. • Making use of the fact that ln x is approximately equal to -1 , when x
approaches unity, we have
ln
𝑓
𝑃
=
𝑓
𝑃
- p
Hence,
𝑓
𝑝
= 1 + ln
𝑓
𝑃
………………..22
= 1 – α
𝑃
𝑅𝑇
=
𝑃𝑉
𝑅𝑇
f =
𝑃2𝑉
𝑅𝑇
…………………….23
This equation is useful in calculating fugacity at moderately low pressure.
❖Fugacity of gas in gaseous mixture :-
• Remembering that for one mole of a pure substance,the free energy (G) is identical with
chemical potential. In eq. 7 for one mole of any gaseous component i of a gaseous
mixture may be written as
d𝑢𝑖 = RTd (ln fi ) ………….24
equation 8 may be written as,
10. 𝑢𝑖 = 𝑢𝑖
∗
+ RT ln 𝑓𝑖 ………………25
Where, 𝑢𝑖
∗
is the chemical potential of the gaseous component i as its unit fugacity.
❖ Physical significance of fugacity :-
In order to understand the physical significance of the term Fugacity,
• A system consisting of liquid water in contact with its vapour.
• Water molecules in the liquid phase will have a tendancy to escap into the vapour
phase by evaporation.
• While those one the vapour phase will have a tendancy to escap into the liquid
phase by condensation.
• At equilibrium the two escaping tendancies will be equal.
• It is now accepted that each substance in a given state has a tendancy to escap
from that state.
• This escaping tendancy was term by Lewis as Fugacity.
11. ❖ Excess function of non – ideal solution :-
• The deviation from ideal behavior can be expressed in terms of excess
thermodynamic functions which gives more quantitative idea about the nature of
molecular interaction.
• The difference between thermodynamic function of mixing for a non – ideal system
and the corresponding value for an ideal system at same temperatrure and pressure
is called ‘thermodynamic excess function’.
• It is denoted by subscript E. This quantity represents the excess ( positive or
negative) of a given thermodynamic property of the solution over that in the ideal
solution.
𝑌𝐸
= ∆𝑌𝑚𝑖𝑥𝑖𝑛𝑔𝑟𝑒𝑎𝑙
- ∆Y𝑚𝑖𝑥𝑖𝑛𝑔𝑖𝑑𝑒𝑎𝑙
= ∆Y𝑚𝑖𝑥𝑖𝑛𝑔(𝑛𝑜𝑛−𝑖𝑑𝑒𝑎𝑙)
- ∆𝑌𝑚𝑖𝑥𝑖𝑛𝑔𝑖𝑑𝑒𝑎𝑙
• Where, 𝑌 can be any thermodynamic function.
• The excess volume,internal energy and enthalpy are identical to the corresponding
mixing properties.
𝑉𝐸
= ∆Vmix
𝐻𝐸
= ∆𝐻mix
𝑈𝐸
= ∆Umix
12. ❖ Expression for excess thermodynamic function:-
1. Excess chemical potential 𝜇𝐸
𝜇𝑖
𝐸
= 𝜇𝑟𝑒𝑎𝑙 - 𝜇𝑖𝑑𝑒𝑎𝑙
= ( 𝜇𝑖 + RTln𝐺𝑖) - ( 𝜇𝑖 + RTln𝑋𝑖)
= ( 𝜇𝑖 + RTln𝑋𝑖- γ𝑖) - ( 𝜇𝑖 + RTln𝑋𝑖)
= 𝜇𝑖 + RTln𝑋𝑖 + RTlnγ𝑖 - 𝜇𝑖 - RTln𝑋𝑖
𝜇𝑖
𝐸
= RTln𝛾𝑖 …………………①
The standard state of the component in the ideal and non-ideal state have been
taken the same and,
μ𝑖 (𝑟𝑒𝑎𝑙)
°
= μ𝑖 (𝑖𝑑𝑒𝑎𝑙)
°
2. Excess gibbs free energy 𝐺𝐸
𝐺𝐸
= ∆𝐺𝑟𝑒𝑎𝑙
𝑀
- ∆𝐺𝑖𝑑𝑒𝑎𝑙
𝑀
= RT σ 𝑛𝑖ln𝑎𝑖 - RT σ 𝑛𝑖ln𝑋𝑖
= RT [σ 𝑛𝑖ln𝑋𝑖 + σ 𝑛𝑖lnγ𝑖 - σ 𝑛𝑖ln𝑋𝑖]
= RTσ 𝑛𝑖lnγ𝑖
And for binary mixture, 𝐺𝐸
= RT (𝑛1ln𝛾1 + 𝑛2ln𝛾2) …………..②
