Seoul National UniversitySeoul National University Prof. SangProf. Sang--imim, , YooYoo

Lecture ContentsLecture Contents

The thermodynamic description of mixturesThe thermodynamic description of mixtures

5.15.1 Partial molar quantitiesPartial molar quantities5.25.2 The thermodynamics of mixingThe thermodynamics of mixing5.35.3 The chemical potentials of liquidsThe chemical potentials of liquids

5. Simple mixtures5. Simple mixtures

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5.1 Partial molar quantities

Thermodynamic description of mixturesThermodynamic description of mixtures

(a) Partial molar volume

The change in volume per mole of a substance added to a large volume of the mixture

- it varies with composition (see Fig. 5.1)

- definition: for a substance J

where p , T and amount of other components (n’) are constant

- for a binary mixture,

Provided the composition is held constant by the addition ofA and B, the total volume, V = nAVA + nBVB

',, nTpJJ n

VV ⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

=

BnTpB

AnTpA

dnnVdn

nVdV

AB ,,,,⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

=

BBAA dnVdnV +=

Seoul National UniversitySeoul National University Prof. SangProf. Sang--imim, , YooYoo

',, nTpJJ n

VV ⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

=

- Molar volumes are always positive but partial molar volumes can be negative!(see Fig. 5.2 & 5.5)

5.1 Partial molar quantities5.1 Partial molar quantities

Seoul National UniversitySeoul National University Prof. SangProf. Sang--imim, , YooYoo

(b) Partial molar Gibbs energies

For a pure substance, the chemical potential µ = Gm

For a substance in a mixture, (see Fig. 5.4)

For a binary mixture (with constant p, T and n’)

G = nA µA + nB µB

For a multi-component system,

dG = Vdp – SdT + µAdnA + µBdnB + …

Fundamental equation of chemical thermodynamics

At constant p and T,

dG = µAdnA + µBdnB + …

Since dG = dwadd,max , additional (non-expansion) maximumwork can arise from the changing composition of a system.

',, nTpJJ n

G⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

=μ

5.1 Partial molar quantities5.1 Partial molar quantities

Seoul National UniversitySeoul National University Prof. SangProf. Sang--imim, , YooYoo

(c) The wider significance of the chemical potential : Since G = H – TS = U + PV – TS,

dU = - pdV - Vdp + SdT + TdS + dG = - pdV - Vdp + SdT + TdS + (Vdp - SdT + μAdnA + μBdnB + …)

= - pdV + TdS + μAdnA + μ BdnB + …

At constant V and S, dU = μAdnA + μBdnB + …

Hence, In the same way,',, nVSJ

J nU

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

=μ',, nTVJ

J nA⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

=μ',, npSJ

J nH⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

=μ

(d) The Gibbs-Duhem equation:

At constant p and T, for a binary system, nAdµA + nBdµB = 0

since dG = µAdnA + µBdnB + nAdµA + nBdµB, while dG = µAdnA+ µBdnB

In a binary mixture,

: Gibbs-Duhem equation∑ =J

JJ dn 0μ

AB

AB d

nnd μμ −=

5.1 Partial molar quantities5.1 Partial molar quantities

Seoul National UniversitySeoul National University Prof. SangProf. Sang--imim, , YooYoo

5.2 The thermodynamics of mixing5.2 The thermodynamics of mixing(a) The Gibbs energy of mixing of perfect gases

For a mixture of two perfect gases,

where µo is the standard chemical potential

The Gibbs energy of total system is,

Let p/po = p,

After mixing, pA + pB = p

The difference Gf – Gi = ΔmixG ,

Using Dalton’s law (see Section 1.2), pJ/p = xJ,(see Fig. 5.7)

oo

ppRT ln+= μμ

⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛+=+= o

oBBo

oAABBAAi p

pRTnppRTnnnG lnln μμμμ

{ } { }pRTnpRTnG oBB

oAAi lnln +++= μμ

{ } { }Bo

BBAo

AAf pRTnpRTnG lnln +++= μμ

ppRTn

ppRTnG B

BA

Amix lnln +=Δ

( )BBAAmix xxxxnRTG lnln +=Δ

Seoul National UniversitySeoul National University Prof. SangProf. Sang--imim, , YooYoo

(b) Other thermodynamic mixing functions

Entropy of mixing for two perfect gases (see Fig. 5.9)

Enthalpy of mixing

( )BBAAnnp

mixmix xxxxnR

TGS

BA

lnln,,

+−=⎟⎠⎞

⎜⎝⎛

∂Δ∂

−=Δ

0=Δ Hmix since ΔG = ΔH -TΔS

5.2 The thermodynamics of mixing5.2 The thermodynamics of mixing

Seoul National UniversitySeoul National University Prof. SangProf. Sang--imim, , YooYoo

5.3 The chemical potentials of liquids5.3 The chemical potentials of liquids(a) Ideal solutions

Chemical potentials are equal at equilibrium (see Fig. 5.10)

If a solute is also present in the liquid,

By combining two equations,

Raoult’s law for ideal solution

For an ideal solution,

Some solutions seriously deviating from Raoult’s law However, a good approximation for a solvent with a dilute solute!

∗∗ += Ao

AA pRT lnμμ

Ao

AA pRT ln+= μμ

∗∗ +=

A

AAA p

pRT lnμμ

∗= AAA pxp

AAA xRT ln+= ∗μμ

Seoul National UniversitySeoul National University Prof. SangProf. Sang--imim, , YooYoo

5.3 The chemical potentials of liquids5.3 The chemical potentials of liquids

Seoul National UniversitySeoul National University Prof. SangProf. Sang--imim, , YooYoo

(b) Ideal-dilute solutions

For real solutions at low concentrations,proportionality constant ≠ vapor pressure of the pure liquid

Henry’s law for ideal dilute solution (see Fig. 5.15)

BBB Kxp =

- xB : the mole fraction of the solute- KB : empirical constant (with the dimension of pressure) & the slope at xB = 0

Ideal–dilute solutions :Mixtures of the solute obeying Henry’s law with the solvent obeying Raoult’s law

5.3 The chemical potentials of liquids5.3 The chemical potentials of liquids