CN2121 Formula List - Summary Chemical Engineering Thermodynamics PDF

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Summary

Summary of formulas used...

Description

For Ideal gas:

𝑑𝑊 = −𝑃𝑒𝑥𝑡𝑑𝑉

Irreversible process

𝑑𝑊 = −𝑃𝑑𝑉

Reversible process

𝑑𝑈 = 𝑑𝑄 + 𝑑𝑊 = 𝑑𝑄 – 𝑃𝑑𝑉

First Law

𝐻 = 𝑈 + 𝑃𝑉 𝑑𝐻 = 𝑑𝑈 + 𝑃𝑑𝑉 + 𝑉𝑑𝑃

Enthalpy

𝐶𝑉 =

Heat Capacities

𝑑𝑆 =

Entropy

𝑉

𝑑(𝑚𝑈)𝐶𝑉 +∆ 𝑑𝑡

𝑃

1 𝐻 + 𝑢 2 + 𝑔𝑧 𝑚 = 𝑄 + 𝑊𝑠 2 𝑄𝑗 = 𝑆𝐺 ≥ 0 𝑇𝜎,𝑗

𝑗

𝑃𝑉 = 𝑅𝑇

𝐵𝑃 𝑃𝑉 = 1+ 𝑅𝑇 𝑅𝑇 𝐵𝑃𝑟 = 1+ 𝑇𝑟

𝑍 =

Virial EOS 𝐵=

Generalised Correlations

𝜕𝐻 𝜕𝑇

−𝑄 𝑑𝑄𝑟𝑒𝑣 𝑎𝑛𝑑 ∆𝑆𝑠𝑢𝑟𝑟 = 𝑇𝜎 𝑇𝑠𝑢𝑟𝑟

𝑑(𝑚𝑆)𝐶𝑉 + ∆ 𝑚𝑆 − 𝑑𝑡

Ideal Gas EOS

van der Waals EOS

𝑎𝑛𝑑 𝐶𝑃 =

𝑑𝑚𝐶𝑉 = 𝑚𝑖𝑛 − 𝑚𝑜𝑢𝑡 𝑑𝑡

Mass Balance

Entropy Balance

𝜕𝑈 𝜕𝑇

𝐵𝑃𝑐 = 𝐵0 + 𝜔𝐵1 𝑅𝑇𝑐

𝐵0 = 0.083 − 𝐵1 = 0.139 −

Gas: 𝑍 =

𝑍0

+

𝜔𝑍1

Liquid: 𝑉 𝑠𝑎𝑡 = 𝑉𝑐 𝑍𝑐

For Multiple Component: 𝑀 = 1 − Where M represents V,U,H,S.

𝑥𝑣

𝑀𝑙

+

1−𝑇𝑟

𝑥 𝑣𝑀𝑣

2

𝑇2 𝑇1

∆𝐻 =

𝐶𝑉 𝑑𝑇

𝐶𝑉 𝑉2 𝑑𝑇 + 𝑅 ln 𝑉1 𝑇

𝑑𝑄 = 𝐶𝑉 𝑑𝑇 +

∆𝑆 =

𝑑𝑄 = 𝐶𝑝 𝑑𝑇 −

𝑅𝑇 𝑑𝑉 𝑉

𝑇2 𝑇1

𝑇1

For Residual Properties: 𝐶𝑃 𝑑𝑇

𝐶𝑃 𝑃2 𝑑𝑇 − 𝑅 ln 𝑃1 𝑇

Isochoric Process: dV = 0

2 1

2

∆𝐻 =

∆𝑈 =

𝑊 = −

Adiabatic Process: dQ = 0

7

Polytropic Process

1

2

1

𝐶𝑉 𝑑𝑇

𝐶𝑃 𝑑𝑇

𝑉2 𝑑𝑉 = 𝑅𝑇𝑙𝑛 𝑊= − 𝑉 𝑉1 1 𝑉2 𝑃2 𝑄 = −𝑊 = 𝑅𝑇𝑙𝑛 = −𝑅𝑇𝑙𝑛 𝑉1 𝑃1 ∆𝐻 =

Isobaric Process: dP = 0

𝑃𝑑𝑉 = 0

𝑄 = ∆𝑈 =

2 𝑅𝑇

Isothermal Process: dT = 0

𝑅𝑇 𝑑𝑉 𝑉

𝑅𝑇 𝑑𝑊 = −𝑅𝑑𝑇 + 𝑑𝑃 𝑃 𝑊= −

0.172 𝑇𝑟 4.2

𝑑𝑊 = −

𝑅𝑇 𝑑𝑃 𝑃

0.422 𝑇𝑟 1.6

27𝑅 2𝑇𝑐 2 𝑎= 64𝑃𝑐 𝑅𝑇𝑐 𝑏= 8𝑃𝑐 𝑇 𝑃 𝑇𝑟 = , 𝑃𝑟 = 𝑇𝑐 𝑃𝑐

𝑎 𝑅𝑇 − 𝑃 = 𝑉 − 𝑏 𝑉2

∆𝑆 =

𝑇1

𝑇2

For Ideal Gas and Reversible Process:

∆𝑆𝑢𝑛𝑖𝑣 = ∆𝑆𝑠𝑦𝑠𝑡𝑒𝑚 + ∆𝑆𝑠𝑢𝑟𝑟 ≥ 0

Second Law

Energy Balance

∆𝑈 =

𝑇2

2 1

2

1

2

1

𝐶𝑉 𝑑𝑇 = 0

2

1

𝑄 = ∆𝐻 = 𝑊 = ∆𝑈 = ∆𝐻 =

2

1

𝑊= −

𝐶𝑉 𝑑𝑇

𝐶𝑝 𝑑𝑇 2

1

2 1

𝑃𝑑𝑉

𝑄 = ∆𝑈 − 𝑊

𝜕𝑇 𝜕𝑉

𝑆

𝜕𝑃 𝜕𝑇

𝜕𝑃 = − 𝜕𝑆

𝑉

𝜕𝑆 = 𝜕𝑉

𝜕𝑃 𝑑𝑈 = 𝐶𝑉 𝑑𝑇 + 𝑇 𝜕𝑇

𝑑𝑆 =

∆𝐻 =

𝑇2

𝜕𝑉 𝜕𝑇

𝑉

− 𝑃 𝑑𝑉

𝑉

𝑇1

𝐶𝑃 𝑑𝑇 + ∆𝐻𝑅 𝜕𝑃 𝜕𝑇

𝐶𝑝 = 𝐶𝑉 + 𝑇 𝑉𝑅 =

𝑉

𝜕𝑉 𝜕𝑇

𝑅𝑇 (𝑍 − 1) 𝑃

𝐻𝑅 = −𝑇 𝑅𝑇

𝑃 0

𝜕𝑍 𝜕𝑇

𝑑𝑆 =

𝜕𝑆 = − 𝜕𝑃

𝑃

𝑃

∆𝑆 = 𝑑

𝑃

𝐶𝑃 𝜕𝑉 𝑑𝑇 − 𝜕𝑇 𝑇

𝑆 = 𝑆𝑖𝑔 + 𝑆 𝑅

𝑇2 𝐶 𝑖𝑔 𝑃

𝑆𝑅 = −𝑇 𝑅

𝑑𝑃 𝑃

𝑃 0

𝑑𝑃 − 𝑃

𝜕𝑍 𝜕𝑇

𝐺𝑅 = 𝑅𝑇

𝑃

𝑃 𝑃 0

(𝑍 − 1)

𝑑𝐵

𝑃 0

𝑑𝑇

𝑑𝑃 𝑃

=

1 𝜌𝑎 − 1 − 𝜌𝑏 𝑅𝑇 𝐻𝑅 𝜌𝑏 2𝜌𝑎 = − 𝑅𝑇 1 − 𝜌𝑏 𝑅𝑇

𝜌𝑎 𝜌 2 𝑎𝑏 𝑆𝑅 = ln 1 − + 𝑅𝑇 𝑅 𝑅𝑇 𝜌𝑏 2𝑎𝜌 𝜌𝑎 𝜌 2 𝑎𝑏 = − − ln 1 − + 𝑅𝑇 𝑅𝑇 1 − 𝜌𝑏 𝑅𝑇 𝑅𝑇 𝐺𝑅

𝐻𝑅 𝐻𝑅 𝐻𝑅 0 +𝜔 = 𝑅𝑇𝑐 𝑅𝑇𝑐 𝑅𝑇𝑐 𝑅

=

𝑑𝑃 𝑃

𝑅 𝑑𝐵 𝑃𝑐 𝑑𝑇𝑟 𝑑𝐵 𝑑𝐵0 𝑑𝐵1 +𝜔 = 𝑑𝑇𝑟 𝑑𝑇𝑟 𝑑𝑇𝑟 𝑑𝐵0 0.675 = 𝑑𝑇𝑟 𝑇𝑟 2.6 0.722 𝑑𝐵1 = 𝑑𝑇𝑟 𝑇𝑟 5.2

𝑃𝑉 𝐵𝑃 = 1+ 𝑅𝑇 𝑅𝑇 𝑃 𝐵 𝑑𝐵 = − 𝑅𝑇 𝑅 𝑇 𝑑𝑇 𝑆𝑅 𝑃 𝑑𝐵 =− 𝑅 𝑅 𝑑𝑇 𝐵𝑃 𝐺𝑅 = 𝑅𝑇 𝑅𝑇

𝑆𝑅

𝑑𝑃

(𝑍 − 1)

𝑍 =

Generalized Correlations for Gases

𝑃

𝑉𝑅 𝐻𝑅 = 𝑑𝑃 − 𝑑𝑇 𝑅𝑇 𝑅𝑇2

𝐺𝑅 𝑅𝑇

𝐻𝑅

van der Waals EOS

𝑇

𝑃2 𝑑𝑇 − 𝑅 ln + ∆𝑆𝑅 𝑃1

𝑇

𝑇1

𝑍=

Virial EOS

𝜕𝑉 𝜕𝑆

=

𝑆

𝜕𝑉 𝑑𝐻 = 𝐶𝑝 𝑑𝑇 + 𝑉 − 𝑇 𝑑𝑃 𝜕𝑇 𝑃

𝑑𝑉

𝑖𝑔

𝐶𝑝 𝑑𝑇

𝑃𝑉 𝛿 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑇𝑉 𝛿−1 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑇𝑃(1−𝛿)/𝛿 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡

𝜕𝑇 𝜕𝑃

𝑉

𝐻 = 𝐻𝑖𝑔 + 𝐻𝑅

2

𝐶 𝛾 = 𝑝 𝐶 𝑣 𝑃𝑉 𝛾 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑇𝑉 𝛾−1 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑇𝑃(1−𝛾)/𝛾 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡

𝑑𝐺 = −𝑆𝑑𝑇 + 𝑉𝑑𝑃

𝑇

𝐶𝑉 𝜕𝑃 𝑑𝑇 + 𝜕𝑇 𝑇

𝐶𝑉 𝑑𝑇 1

𝑑𝐻 = 𝑇𝑑𝑆 + 𝑉𝑑𝑃

𝑑𝐴 = −𝑆𝑑𝑇 − 𝑃𝑑𝑉

𝐶𝑃 𝑑𝑇 = 0

𝑃𝑑𝑉 = −𝑃 𝑉2 − 𝑉1 = −𝑅 𝑇2 − 𝑇1 ∆𝑈 =

𝑑𝑈 = 𝑇𝑑𝑆 − 𝑃𝑑𝑉

𝑆𝑅 𝑅

0

+𝜔

𝑆𝑅 𝑅

1

1

Step 12

Carnot Engine

Rankine Engine

VC Fridge

Boiler: 𝐿𝑠𝑎𝑡 → 𝑉𝑠𝑎𝑡 ∆𝑃 𝑎𝑛𝑑 ∆𝑇 = 0 𝑄𝐻 = ∆𝐻

Boiler: 𝐿𝑠𝑢𝑏𝑐𝑜𝑜𝑙 → 𝑉 𝑠𝑢𝑝𝑕𝑒𝑎𝑡 ∆𝑃 = 0 𝑄𝐻 = ∆𝐻

Evaporator: 𝐿 + 𝑉 → 𝑉 𝑠𝑎𝑡 ∆𝑃 𝑎𝑛𝑑 ∆𝑇 = 0 𝑄𝐶 = ∆𝐻

Turbine: 𝑉 𝑠𝑎𝑡 → 𝐿 + 𝑉 ∆𝑆 = 0 −𝑊𝑠 = ∆𝐻

Turbine: 𝑉 𝑠𝑢𝑝𝑕𝑒𝑎𝑡 → 𝐿 + 𝑉 ∆𝑆 = 0 −𝑊𝑠 = ∆𝐻

Compressor: 𝑉 𝑠𝑎𝑡 → 𝑉𝑠𝑢𝑝𝑕𝑒𝑎𝑡 ∆𝑆 = 0 𝑊𝑠 = ∆𝐻

Condenser: 𝐿3 + 𝑉3 → 𝐿4 + 𝑉4 ∆𝑃 𝑎𝑛𝑑 ∆𝑇 = 0 −𝑄𝐶 = ∆𝐻

Condenser: 𝐿3 + 𝑉3 → 𝐿𝑠𝑎𝑡 ∆𝑃 𝑎𝑛𝑑 ∆𝑇 = 0 −𝑄𝐶 = ∆𝐻

Condenser: 𝑉 𝑠𝑢𝑝𝑕𝑒𝑎𝑡 → 𝐿𝑠𝑎𝑡 ∆𝑃 = 0 −𝑄𝐻 = ∆𝐻

Pump: 𝐿 + 𝑉 → 𝐿𝑠𝑎𝑡 ∆𝑆 = 0 𝑊𝑠 = ∆𝐻

Pump: 𝐿𝑠𝑎𝑡 → 𝐿𝑠𝑢𝑝𝑐𝑜𝑜𝑙 ∆𝑆 = 0 𝑊𝑠 = ∆𝐻

Throttler: 𝐿𝑠𝑎𝑡 → 𝐿 + 𝑉 ∆𝐻 = 0 𝑊𝑠 = 0

Step 23

Step 34

Step 41

𝑇𝐶 𝑊 = 1− η = 𝑄𝐻 𝑇𝐻

Efficiency

𝑊 = 𝑊𝑖𝑠𝑒𝑛 𝑊𝑖𝑠𝑒𝑛 = = 𝑊

η𝑡𝑢𝑟𝑏 = η𝑝𝑢𝑚𝑝

∆𝐻 ∆𝐻 𝑆 (∆𝐻)𝑆 ∆𝐻

𝑄𝐶 𝐻2 − 𝐻1 𝜔 = = 𝐻3 − 𝐻2 𝑊

For all cycles: ∆𝑈 = ∆𝐻 = ∆𝑆 = 0 𝜕𝑇 𝜕𝑃 𝐻

=

−1 𝜕𝐻 𝐶𝑝 𝜕𝑃

𝑇

When µ > 0, T ↓ as P ↓ and for µ < 0, T ↑ as P ↓

Φ =

𝑓 𝜕(𝑛𝐺) 𝑎𝑛𝑑 𝜇 = 𝑃 𝜕𝑛

𝐺 𝑅 = 𝑅𝑇 ln

Experimental Data Virial Coefficients Generalized Coefficients

−1

=𝐶 𝑉−𝑇 𝑝

𝜕𝑉 𝜕𝑇 𝑃

𝑓 = 𝑅𝑇 ln Φ 𝑃

ln Φ =

0

𝑑 ln Φ = 𝑍 − 1

(𝑉

− 𝑉 𝑖𝑔 ) 𝑅𝑇

𝑑𝑃, 𝑓𝑜𝑟 𝑕𝑖𝑔𝑕 𝑃.

𝑑𝑃 𝑃𝑟 , ln Φ = 𝐵 , 𝑓𝑜𝑟 𝑚𝑜𝑑𝑒𝑟𝑎𝑡𝑒 𝑃 𝑇𝑟 𝑃

ln Φ = ln Φ0 + 𝜔 ln Φ1 𝑃𝑟 𝑃𝑟 𝑑𝑃𝑟 𝑑𝑃𝑟 (𝑍0 − 1) (𝑍1 ) , ln Φ1 = ln Φ0 = 𝑃𝑟 𝑃𝑟 0 0 𝑓𝐿 = 𝑓𝑆 =

𝑉𝐿 𝑠𝑎𝑡 Φ𝑠𝑎𝑡 𝑃 𝑠𝑎𝑡 𝑒 𝑅𝑇(𝑃−𝑃 )

𝑉𝑆 𝑠𝑎𝑡 Φ𝑠𝑎𝑡 𝑃 𝑠𝑎𝑡 𝑒𝑅𝑇 𝑃−𝑃

= Φ 𝑠𝑎𝑡𝑃 𝑠𝑎𝑡

𝑀𝑖𝑔 𝑇, 𝑃 + 𝑀𝑅 𝑇, 𝑃 = 𝑀(𝑇, 𝑃) For Ideal Gas Mixture: 𝑀𝑖

𝑁

𝑁

𝑖=1

Partial Molar Property:

𝜕(𝑛𝑀) 𝑀𝑖 = 𝜕𝑛𝑖

𝑃,𝑇,𝑛𝑗≠𝑖

𝑉 𝑖𝑔

Enthalpy

𝐻𝑖𝑔

Entropy

𝑆𝑖 𝑖𝑔 − 𝑅𝑙𝑛𝑦𝑖

Gibbs Energy

𝐺𝑖 𝑖𝑔 + 𝑅𝑇𝑙𝑛𝑦𝑖

𝑛𝑀 =

𝑛𝑖 𝑀𝑖

𝑖

Gibbs-Duhem Relation: 𝜕𝑀

𝜕𝑀

𝑑𝑃 +

𝑇,𝑥

𝜕𝑃

𝑃,𝑥

lim 𝑀𝑖 =

𝑥𝑖 →0

𝑥𝑖 →1 ∞ 𝑀𝑖

𝑑𝑇 −

𝑖 𝑥𝑖 𝑑𝑀𝑖

= 0

𝑀𝑖

(infinite dilution) 𝑖𝑑

Lewis Randall Rule: 𝑓𝑖 = 𝑥𝑖 𝑓𝑖 Raoult’s Law: 𝑦𝑖 𝑃 = 𝑥𝑖 𝑃 𝑠𝑎𝑡 Activity Coefficient: 𝛾𝑖 =

𝑑

𝑑

𝑛𝐺 𝑅𝑇 𝑛𝐺 𝑅 𝑅𝑇

𝑛𝐺 𝐸 𝑑 𝑅𝑇

=

𝑛𝑉 𝑛𝐻 𝑑𝑃 − 𝑑𝑇 + 𝑅𝑇 𝑅𝑇 2

=

𝑛𝑉 𝑅 𝑅𝑇

𝑁 𝑖

𝑛𝐻𝑅

𝑑𝑃 − 2 𝑑𝑇 + 𝑅𝑇

𝑛𝑉𝐸 𝑛𝐻 𝐸 = 𝑑𝑃 − 2 𝑑𝑇 + 𝑅𝑇 𝑅𝑇

𝑓𝑖 𝑓𝑖 𝑜𝑟 𝑦𝑖 𝑃 𝑥𝑖 𝑃 𝐺𝑖 𝑑𝑛 𝑅𝑇 𝑖 𝑁 𝑖 𝑁 𝑖

𝑉𝑖

Enthalpy

𝐻𝑖

Entropy

𝑆𝑖 − 𝑅𝑙𝑛𝑥𝑖

Gibbs Energy

𝐺𝑖 + 𝑅𝑇𝑙𝑛𝑥𝑖

𝑖

𝑖

𝑦𝑖 𝑆𝑖

−𝑅

𝑖

𝑦𝑖 𝐺𝑖 𝑖𝑔 + 𝑅𝑇

𝑦𝑖 ln 𝑦𝑖

−𝑅

𝑦𝑖 ln 𝑦𝑖

𝑅𝑇

𝑖

𝑖 𝑖

𝑦𝑖 ln 𝑦𝑖 𝑦𝑖 ln 𝑦𝑖

𝑀 𝑖𝑑 𝑖 𝑖 𝑖 𝑖

𝜕(𝑛𝐺) 𝜕𝑛𝑖

𝑃,𝑇,𝑛 𝑗

𝑛𝐺𝐸 𝜕( 𝑅𝑇 ) ln 𝑖 = 𝜕𝑛𝑖 𝑃,𝑇,𝑛

𝐸

𝑛𝐺𝐸 𝜕( 𝑅𝑇 ) ln 𝛾𝑖 = 𝜕𝑛𝑖 𝑃,𝑇,𝑛

𝑗

𝑖

𝑥𝑖 𝑉𝑖

0

𝑥𝑖 𝐻𝑖

0

𝑥𝑖 𝑆𝑖 − 𝑅

𝑥𝑖 𝐺𝑖 + 𝑅𝑇

𝐺 𝐸 /𝑥1 𝑥2𝑅𝑇

𝑗

∆𝑀𝑚𝑖𝑥 = 𝑀 −

∆𝑀𝑚𝑖𝑥 𝑖𝑑

𝑖

𝑥𝑖 ln 𝑥𝑖

−𝑅

𝑥𝑖 ln 𝑥𝑖

𝑅𝑇

𝑖

𝑥𝑖 𝑀𝑖

𝑖 𝑖

𝑥𝑖 ln 𝑥𝑖 𝑥𝑖 ln 𝑥𝑖

𝑛𝐺𝐸

𝜕( 𝑅𝑇 ) 𝜕𝑛𝑖

𝑃,𝑇,𝑛𝑗

𝛾𝑖 > 1: 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑑𝑒 𝑣𝑖𝑎𝑡𝑖𝑜𝑛 𝑐𝑟𝑜𝑠𝑠 𝑖𝑛𝑡𝑒𝑟𝑎𝑐𝑡𝑖𝑜𝑛𝑠 < 𝑝𝑢𝑟𝑒 𝑠𝑝𝑒𝑐𝑖𝑒𝑠 𝑖𝑛𝑡𝑒𝑟𝑎𝑐𝑡𝑖𝑜𝑛𝑠 𝛾𝑖 < 1: 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 𝑐𝑟 𝑜𝑠𝑠 𝑖𝑛𝑡𝑒𝑟𝑎𝑐𝑡𝑖𝑜𝑛𝑠 > 𝑝𝑢𝑟𝑒 𝑠𝑝𝑒𝑐𝑖𝑒𝑠 𝑖𝑛𝑡𝑒𝑟𝑎𝑐𝑡𝑖𝑜𝑛𝑠 For species dominant in solution (𝑥𝑖 → 1): 𝛾𝑖 = 1, 𝑓𝑖 = 𝑓𝑖 For species infinitely diluted in solution (𝑥𝑖 → 0): 𝛾𝑖 = 𝛾𝑖 ∞ ,𝑖𝑓= 𝑥𝑖 𝐻𝑖 (Henry’s Law) For low pressure VLE, can use Modified Raoult’s Law to determine 𝛾𝑖 and 𝑀𝐸 𝐺𝐸 = 𝑥𝑖 ln 𝛾𝑖 𝑎𝑛𝑑 𝑥𝑖 𝑑 ln 𝛾𝑖 = 0 𝑅𝑇 𝑖 𝑖 ln 𝛾𝑖

ln 𝛾𝑖 ∞

One Constant

A

ln 𝛾1 = 𝐴𝑥2 2 𝑎𝑛𝑑 ln 𝛾2 = 𝐴𝑥1 2

ln 𝛾1 ∞ = ln 𝛾2 ∞ = 𝐴

Two Constant

𝐴21𝑥1 + 𝐴12𝑥2

ln 𝛾1 = 𝑥2 2[𝐴12 + 2 𝐴21 − 𝐴12 𝑥1 ] ln 𝛾2 = 𝑥1 2[𝐴21 + 2(𝐴12 − 𝐴 21)𝑥2 ]

ln 𝛾1 ∞ = 𝐴12 ln 𝛾2 ∞ = 𝐴21

Van Laar

𝐴′12𝐴′ 21 𝐴′12𝑥1 + 𝐴′ 21𝑥2

𝑥 𝑖→1

𝑦𝑖 𝑀𝑖

0

𝐸

For non ideal solution: 𝑀𝐸 = 𝑥𝑖 𝑀 𝑖 − 𝑥𝑖 𝑀 𝑖 𝑖𝑑 lim𝑀𝐸 = 0 ∆𝑀𝑚𝑖𝑥 = 𝑀 −

𝑖

𝑖𝑔

Activity Coefficient: 𝐺𝑖 = 𝑅𝑇 ln 𝛾𝑖 and ln 𝛾𝑖 =

𝑥𝑖 𝑃 𝑠𝑎𝑡

𝑅

𝐺𝑖 𝑑𝑛𝑖 𝑅𝑇

𝑦𝑖 𝐻𝑖 𝑖𝑔

𝑖

𝑓𝑖

𝜇𝑖 =

𝐺𝑖 𝑑𝑛𝑖 𝑅𝑇

0

𝑖𝑑

Molar Volume

𝑥 𝑖 𝑓𝑖 𝑦𝑖 𝑃

Modified Raoult’s Law: 𝛾𝑖 = Φ𝑖 =

𝑅𝑇/𝑃

Ideal Solution Mixture Model: 𝑀𝑖𝑑 𝑇, 𝑃 + 𝑀𝐸 𝑇, 𝑃 = 𝑀 𝑇, 𝑃 For Ideal Solution:

Binary Mixture: 𝑥1 + 𝑥2 = 1, 𝑑𝑥1 = −𝑑𝑥2 𝑑𝑀 𝑑𝑀 𝑀 1 = 𝑀 + 𝑥2 𝑎𝑛𝑑 𝑀2 = 𝑀 − 𝑥1 𝑑𝑥1 𝑑𝑥1 𝑥1𝑑𝑀1 + 𝑥2𝑑𝑀2 = 0 lim𝑀 = lim 𝑀𝑖 = 𝑀𝑖 (purity) 𝑥𝑖 →1

∆𝑀𝑚𝑖𝑥 𝑖𝑔

𝐶𝑕𝑒𝑚𝑖𝑐𝑎𝑙 𝑃𝑜𝑡𝑒𝑛𝑡𝑖𝑎𝑙 𝑜𝑓 𝑆𝑝𝑒𝑐𝑖𝑒𝑠: 𝜇𝑖 𝑖𝑔 = 𝐺𝑖 = 𝐺𝑖 𝑖𝑔 + 𝑅𝑇𝑙𝑛𝑦𝑖 𝑖𝑔 𝑖𝑔 𝐹𝑢𝑔𝑎𝑐𝑖𝑡𝑦 𝑜𝑓 𝑆𝑝𝑒𝑐𝑖𝑒𝑠: 𝑓 𝑖 = 𝑝𝑖 and 𝛷𝑖 = 1

𝑥𝑖 𝑀 𝑖

𝑖

𝑀 𝑖𝑔

𝑖𝑔

Summability Relation: 𝑀 =

𝑖𝑔

Molar Volume

𝑖=1

𝑑𝐺 = −𝑆𝑑𝑇 + 𝑉𝑑𝑃 + 𝜇𝑖 𝑑𝑥𝑖 → 𝐺 = 𝐺(𝑃, 𝑇, 𝑥𝑖 )

𝑅𝑇𝑑 ln Φ = 𝑉 − 𝑉 𝑖𝑔 𝑑𝑃 𝑃

Ideal Gas Mixture Model:

𝑑 𝑛𝐺 = − 𝑛𝑆 𝑑𝑇 + 𝑛𝑉𝑑𝑃 + 𝜇𝑖 𝑑𝑛𝑖

𝐴𝑡 𝑉𝐿𝐸, 𝐺 𝑉 = 𝐺𝐿 , 𝜇 𝑉 = 𝜇 𝐿 𝑃 = 𝑃 𝑠𝑎𝑡, 𝑓 𝑉 = 𝑓 𝐿 = 𝑓 𝑠𝑎𝑡

𝑇,𝑃

Liquids Solids

Multi-Component System:

𝜕𝑃

T-S Diagram

For throttling: Joule/Thomson Coefficient: 𝜇 =

𝑑𝑃 𝑠𝑎𝑡 𝑅𝑇 ∆𝑍𝑙𝑣 ∆𝐻𝑙𝑣 𝑎𝑛𝑑 ∆𝑉 𝑙𝑣 =𝑃 𝑠𝑎𝑡 𝑑 ln 𝑃 𝑠𝑎𝑡 = 𝑙𝑣 𝑙𝑣 −∆𝐻𝑙𝑣 𝑇∆𝑉 𝑠𝑎𝑡 𝑑𝑇 𝑑 ln 𝑃𝑇) ∆𝐻𝑙𝑣 𝑑(1 = 𝑅∆𝑍 = 2 𝑙𝑣 𝑎𝑛𝑑 𝑅𝑇 ∆𝑍 𝑑𝑇

𝐴′12𝑥1 𝐴′ 21𝑥2 𝐴′ 𝑥2 1 + ′ 21 𝐴 12𝑥1

ln 𝛾1 = 𝐴′12 1 + ln 𝛾2 =

𝐴′

21

−2 −2

ln 𝛾1 ∞ = 𝐴′12 ln 𝛾2 ∞ = 𝐴′ 21...