L12 Mixture properties

SNU NAOE Y. Lim

L12 Mixture properties

Mixtures

n (mol) 100 400 500

In Mixture, x_A=0.2 x_B=0.8

SNU NAOE Y. Lim

Volume of an ideal mixture

• 𝑣 𝑖𝑑𝑒𝑎𝑙 = 𝑥 𝐴 𝑣 𝐴 + 𝑥 𝐵 𝑣 𝐵 = 1.6

3

0 0.2 1

x_A v_B=1

v_A=4

n (mol) 100 400 500 V (m³) 400 400 800 v(m³/m

ol)

4 1 1.6

Volume of a real mixture

0 0.2 1

v_B=1

v_A=4 Ideal

Real Excess volume v^ex =v_real-v_ideal

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Excess volume for ethanol/water

5

0 0.2 1

x_A v_B=1

v_A=4

Ideal Real Excess volume v^ex =v_real-v_ideal

http://commons.wikimedia.org/wiki/File:Excess_Volume_Mixture_of_Etha nol_and_Water.png

Partial Molar Volume 𝑉 ത

• 𝑉 ത 𝐴 = 𝜕𝑉

𝜕𝑛

_𝐴

𝑇,𝑃,𝑛

_𝐵

• The volume change of a mixture when one mole of A added to the mixture.

• T, P, number of other molecules are

constant. (only number of A molecule is

change)

SNU NAOE Y. Lim

Partial Molar Volume

7

0 0.2 1

x_A

v_A

v_B

𝑉ത_𝐴

𝑉ത_𝐵 𝑣 = 𝑥_𝐴𝑉ത_𝐴 + 𝑥_𝐵𝑉ത_𝐵

𝑉ത_𝐴 = 𝑣 + 1 − 𝑥_𝐴 𝑑𝑣 𝑑𝑥_𝐴 𝑉ത_𝐵 = 𝑣 − 𝑥_𝐴 𝑑𝑣

𝑑𝑥_𝐴

Partial Molar Volume varies with composition

0 0.3 1

x_A

v_A

v_B

𝑉ത_𝐴

𝑉ത_𝐵

𝑣 = 𝑥_𝐴𝑉ത_𝐴 + 𝑥_𝐵𝑉ത_𝐵

𝑉ത = 𝑣 + 1 − 𝑥 𝑑𝑣 𝑉ത = 𝑣 − 𝑥 𝑑𝑣

SNU NAOE Y. Lim

Water/Ethanol Mixture

• 1 mole of water mixed with 9 moles of ethanol

• mole fraction of ethanol=?

• partial molar volume

– for ethanol≈?

– for water≈?

• Molar volume of mixture =?

• an excess volume=?

9

Water/Ethanol Mixture

0 x_A 0.9 1

v_A =58 cm³/mol

v_B =18 cm³/mol

≈ ത𝑉_𝐴

𝑉ത_𝐵 ≈ 15

𝑣 = 𝑥_𝐴𝑉ത_𝐴 + 𝑥_𝐵𝑉ത_𝐵

= 53.7 𝑣_{𝑖𝑑𝑒𝑎𝑙} = 𝑥_𝐴𝑣_𝐴 + 𝑥_𝐵𝑣_𝐵

= 54

𝑉_{𝑖𝑑𝑒𝑎𝑙} = 𝑛_𝐴𝑣_𝐴 + 𝑛_𝐵𝑣_𝐵

= 540

SNU NAOE Y. Lim

Partial Molar Volume

• For a pure substance,

• 𝑣 = 𝑓 𝑇, 𝑃

• 𝑑𝑣 =

^𝜕𝑣

𝜕𝑇 𝑃

𝑑𝑇 +

^𝜕𝑣

𝜕𝑃 𝑇

𝑑𝑃

• For a mixture with m species,

• 𝑉 = 𝑓 𝑇, 𝑃, 𝑛

₁

, 𝑛

₂

, 𝑛

₃

, … , 𝑛

_𝑚

11

𝑑𝑉

= 𝜕𝑉

𝜕𝑇 𝑃,𝑛

𝑑𝑇 + 𝜕𝑉

𝜕𝑃 𝑇,𝑛

𝑑𝑃 + 𝜕𝑉

𝜕𝑛_{1 𝑇,𝑃,𝑛}

2,𝑛₃…𝑛_𝑚

𝑑𝑛₁

+ 𝜕𝑉

𝜕𝑛_{2 𝑇,𝑃,𝑛}

1,𝑛₃…𝑛_𝑚

𝑑𝑛₂ + ⋯ + 𝜕𝑉

𝜕𝑛_{𝑚 𝑇,𝑃,𝑛}

1,…𝑛_𝑚−1

𝑑𝑛_𝑚

= 𝜕𝑉

𝜕𝑇 𝑃,𝑛

𝑑𝑇 + 𝜕𝑉

𝜕𝑃 𝑇,𝑛

𝑑𝑃 + ෍

𝑖=1

𝑚 𝜕𝑉

𝜕𝑛_{𝑖 𝑇,𝑃,𝑛}

𝑗≠𝑖

𝑑𝑛_𝑖

Partial Molar Volume

• 𝑑𝑉 =

^𝜕𝑉

𝜕𝑇 𝑃,𝑛

𝑑𝑇 +

^𝜕𝑉

𝜕𝑃 𝑇,𝑛

𝑑𝑃 + σ

_𝑖=1^{𝑚 𝜕𝑉}

𝜕𝑛_𝑖 𝑇,𝑃,𝑛_𝑗≠𝑖

𝑑𝑛

_𝑖

• If we define partial molar volume 𝑉 ത 𝑖 = 𝜕𝑉

𝜕𝑛

_𝑖

𝑇,𝑃,𝑛

_𝑗≠𝑖

• 𝑑𝑉 = 𝜕𝑉

𝜕𝑇 𝑃,𝑛 𝑑𝑇 + 𝜕𝑉

𝜕𝑃 𝑇,𝑛 𝑑𝑃 + σ 𝑖=1 𝑚 𝑉 ത 𝑖 𝑑𝑛 𝑖

SNU NAOE Y. Lim

Thermodynamic properties of mixture

• At constant T, P,

• 𝑉 = σ ത 𝑉

_𝑖

𝑛

_𝑖

, ത 𝑉

_𝑖

=

^𝜕𝑉

𝜕𝑛_𝑖 𝑇,𝑃,𝑛_𝑗≠𝑖

• 𝑈 = σ ഥ 𝑈

_𝑖

𝑛

_𝑖

, ഥ 𝑈

_𝑖

=

^𝜕𝑈

𝜕𝑛_𝑖 𝑇,𝑃,𝑛_𝑗≠𝑖

• 𝐻 = σ ഥ 𝐻

_𝑖

𝑛

_𝑖

, ഥ 𝐻

_𝑖

=

^𝜕𝐻

𝜕𝑛_𝑖 𝑇,𝑃,𝑛_𝑗≠𝑖

• 𝑆 = σ ҧ 𝑆

_𝑖

𝑛

_𝑖

, ҧ 𝑆

_𝑖

=

^𝜕𝑆

𝜕𝑛_𝑖 𝑇,𝑃,𝑛_𝑗≠𝑖

• 𝐺 = σ ҧ 𝐺

_𝑖

𝑛

_𝑖

, ҧ 𝐺

_𝑖

=

^𝜕𝐺

𝜕𝑛_𝑖 𝑇,𝑃,𝑛_𝑗≠𝑖

13

Ex 6.8

• HYSYS

Δh_mix

NRTL 0

SNU NAOE Y. Lim

Ex 6.11

15

1atm, 25C Δs_mix (HYSYS)

Δs_mix=

−𝑅 ෍ 𝑦_𝑖 ln 𝑦_𝑖

C1:C2 (5:5), PR 5.76 5.762 C1:C4 (5:5), PR 5.73 5.762 Ar:Ne (5:5), PR 5.76 5.762