Summary Extended Formulae sheet for Chemical Thermodynamics

CHEN10192 – Chemical Thermodynamics
Formulae Sheet



Pressure: 1 bar = 105 Pa

Temperature: T (K) = T (°C) + 273.15

Ideal gas constant: R = 8.314 J mol–1 K–1

Thermodynamic potentials:

𝜕𝑈
𝑑𝑈 = 𝑇𝑑𝑆 − 𝑃𝑑𝑉 + ∑ ( ) 𝑑𝑛𝑖
𝜕𝑛𝑖 𝑆,𝑉,𝑛
𝑖 𝑗
𝜕𝐻
𝑑𝐻 = 𝑇𝑑𝑆 + 𝑉𝑑𝑃 + ∑ ( ) 𝑑𝑛𝑖
𝜕𝑛𝑖 𝑆,𝑃,𝑛
𝑖 𝑗
𝜕𝐴
𝑑𝐴 = −𝑆𝑑𝑇 − 𝑃𝑑𝑉 + ∑ ( ) 𝑑𝑛𝑖
𝜕𝑛𝑖 𝑇,𝑉,𝑛
𝑖 𝑗
𝜕𝐺
𝑑𝐺 = −𝑆𝑑𝑇 + 𝑉𝑑𝑃 + ∑ ( ) 𝑑𝑛𝑖
𝜕𝑛𝑖 𝑇,𝑃,𝑛
𝑖 𝑗
-
Maxwell relations: ↳ Chemical
potential (Mil
𝜕𝑇 𝜕𝑃
( ) = −( )
𝜕𝑉 𝑆,𝑛𝑖 𝜕𝑆 𝑉,𝑛𝑖
𝜕𝑇 𝜕𝑉
( ) =( )
𝜕𝑃 𝑆,𝑛𝑖 𝜕𝑆 𝑃,𝑛𝑖
𝜕𝑃 𝜕𝑆
( ) =( )
𝜕𝑇 𝑉,𝑛𝑖 𝜕𝑉 𝑇,𝑛𝑖
𝜕𝑉 𝜕𝑆
( ) = −( )
𝜕𝑇 𝑃,𝑛𝑖 𝜕𝑃 𝑇,𝑛𝑖

Ideal gas:
𝑇
ℎ𝐼𝐺 (𝑇) = ℎ𝐼𝐺 (𝑇0 ) + ∫ 𝑐𝑃𝐼𝐺 𝑑𝑇 gF0 = hE0 .
TsF =
gFP(T ,
P) +
RTInG
𝑇0
𝑇 𝐼𝐺
𝑐𝑃 𝑃 PuEr
git gFP (T
F6
P) -
RT
𝑠 𝐼𝐺 (𝑇, 𝑃) = 𝑠 𝐼𝐺 (𝑇0 , 𝑃0 ) + ∫
:
a
𝑑𝑇 − 𝑅 𝑙𝑛
.
,



𝑇 𝑃0
𝑀,𝐼𝐷
𝑇0
d
ℎ (𝑇, 𝑥⃗) = 0
= RTCE -
1)
𝑠 𝑀,𝐼𝐷 (𝑇, 𝑃, 𝑥⃗) = −𝑅 ∑ 𝑥𝑖 𝑙𝑛𝑥𝑖
𝑖
𝑀,𝐼𝐷 (𝑇,
𝑔 𝑃, 𝑥⃗) = −𝑊𝑚𝑖𝑛 = 𝑅𝑇 ∑ 𝑥𝑖 𝑙𝑛𝑥𝑖
𝑖


Compressibility Factor (2) =
behavior
↳ represents deviation of real
gas
from ideal

↳
Ideal : 1
gas z =

, B(T)

↓


ia = /P :
Virial EoS: -z = 1 +
Bp

𝐵𝑀𝐼𝑋 = ∑ ∑ 𝑦𝑖 𝑦𝑗 𝐵𝑖𝑗 =>
,B B
y , Ba
y 2y
+ +
,
, y ,
, z


𝑖 𝑗 -




(1 − 𝛿𝑖𝑗 ) S
𝐵𝑖𝑖 + 𝐵𝑗𝑗
𝐵𝑖𝑗 = 𝐵𝑗𝑖 =
2


pl-togeth
𝑃

- ( +a
𝜕𝑣


Residual i s
𝛥ℎ𝑅 (𝑇, 𝑃) = ∫ [𝑣 − 𝑇 ( ) ] 𝑑𝑃 + =


0 𝜕𝑇 𝑃
𝑃
𝜕𝑣 𝑅
𝛥𝑠 𝑅 (𝑇, 𝑃) = − ∫ [( ) − ] 𝑑𝑃
𝑃
0 𝜕𝑇 𝑃 𝑃 :
Fugi - 𝑙𝑛 𝜑 = ∫ (𝑅𝑇 − 𝑃) 𝑑𝑃 -
𝑣 1
0 B B Brix &
- Zy 2y Zy Baz 2y B Brix
+ -




/lim Flim
+
,
,
, 2
, -
𝑃 ,

𝑣̄ 𝑖 1
, ,




𝑙𝑛𝜑𝑖 = ∫ ( − ) 𝑑𝑃
0 𝑅𝑇 𝑃 · ziBij-Brix)
Underwa SoreRedlich-Eng (1
Peng-Ros a




=b
a(T)
z
= Cubic EoS: - P
(v + (b)(v + db)
=> ,
d = - m)
a f (Tc f(T
P,
=- Energy pramater w)
=
-
> a =
Tc Pc
𝑎𝑀𝐼𝑋 = ∑ ∑ 𝑦𝑖 𝑦𝑗 𝑎𝑖𝑗 ya
,
=>
yea 2y
, ,
+ 9 + ,

y , 2 ,


Substituting b f (Tc
&
,
,
P ) Cordur
f(Tc
=

b
-
=
𝑖 𝑗 , . >
- =
0. )
,


these into the
𝑎𝑖𝑗 = 𝑎𝑗𝑖 = √𝑎𝑖𝑖 𝑎𝑗𝑗 (1 − 𝑘𝑖𝑗 ) accentric factor
equation for
Mixing
elsa 𝑏𝑀𝐼𝑋 = ∑ 𝑦𝑖 𝑏𝑖
-



𝑖
= >
y ,
b
,
+
y ,
b
~ =

-log ,

(
𝑣
+
𝜕𝑃
𝛥ℎ𝑅 (𝑇, 𝑃) = 𝑅𝑇(𝑧 − 1) + ∫ [𝑇 ( ) − 𝑃] 𝑑𝑣 RT(z-1) =
d =
RT(t -
1) -




2
∞ 𝜕𝑇 𝑣

em()
𝑣
𝜕𝑃 𝑅
𝛥𝑠 𝑅 (𝑇, 𝑃) = 𝑅 𝑙𝑛 𝑧 + ∫ [( ) − ] 𝑑𝑣
∞ 𝜕𝑇 𝑣 𝑣
rInz -Edu ·
+ :
Rit +


𝑣
1 𝑃
𝑙𝑛 𝜑 = (𝑧 − 1) − 𝑙𝑛 𝑧 + ∫ ( − ) 𝑑𝑣
∞ 𝑣 𝑅𝑇

~
∞
1 𝜕𝑃 1
𝑙𝑛𝜑𝑖 = − 𝑙𝑛𝑧𝑀𝐼𝑋 + ∫ [ ( ) − ] 𝑑𝑉
𝑅𝑇 𝜕𝑛𝑖 𝑉
Intrix In ) =
- +


𝑉




Partial Molar
(U) Activity coefficients:

𝑙𝑛𝛾𝑖 = (
𝜕(𝐺 𝐸 /𝑅𝑇)
)
-chri i
,
o
P (v" -

v") : a " av -




&
specific Helmholtz
V
energy

--For-a
↑
~
-

Properties 𝜕𝑛𝑖 𝑇,𝑃,𝑛 𝑗 reference
State
Asymmetric 𝛾𝑖
Activity
> 𝛾𝑖∗ =
𝛾𝑖∞ >
-
symmetric activity crefficient at infinite dilution
Coefficient

ideal
Vapour-liquid equilibrium: vapor liquid ideal
:


At low Pressure :
↓ d
Component 𝑦𝑖 Ideal mixture :
y
P =
xi P Real mixture : P ViciP
𝐾𝑖 = y:
:

>
- :


Volatility 𝑥𝑖




i
𝑉 U
. > 1 ,
repulsive forces prevail Ceasier to bril
𝑉𝑎𝑝𝑜𝑢𝑟 𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛 𝛼 =
𝐹 Vi < 1 attractive furces prevail (harder to boil
𝑧𝑖 𝐾𝑖 ,




&
𝑦𝑖 =
1 + 𝛼(𝐾𝑖 − 1) Vi
: k
+ P


- v. s : (1 )

F =D
y x
-




, ,

= d =



(1 y )x ,
F
-




, zi ,




T 2x ,

y =

, 1 + (a 1)x ,
-




I
↳ ,: