Physical Chemistry with Formulas and Examples

Physical Chemistry with Formulas and Examples Ihsan Basaran July 4, 2014 Abstract It is a document includes most common formulas of Physical Chemistry

1

Thermodynamic

1.1 1.1.1

First Law General expressin of Internal Energy ΔU = q + w

(1)

Where U is internal energy, q is heat, and w is work 1.1.2

General expression of Work W = Pex .ΔV

(2)

Where W is work, Pex is external pressure, ΔV is change in the volume 1.1.3

Work at reversible processes

V2 (3) V1 Where V2 is final volume and V1 is the volume at the begining of process W = −RT ln

1.1.4

Heat capacity at constant volume : Cv Cv =

 δU  δT V

(4)

ΔU = n.Cv .ΔT

(5)

Qv = n.Cv .ΔT

(6)

Also Qv = Δ U 1.1.5

Heat capacity at constant pressure : Cp Cp =

 δU  δT p

(7)

ΔU = n.Cp .ΔT

(8)

Qp = n.Cp .ΔT

(9)

Cp = Cv + R

(10)

1

• for an unimolecular atom ; Cv = 3/2 R and Cp = 5/2 R • for an bimolecular atom ; Cv = 5/2 R and Cp = 7/2 R • for an multimolecular atom ; Cv = 7/2 R and Cp = 9/2 R

2

Phase Diagrams and Chemical Potential

2.1

General formula for chemical potential dG = (μ1 − μ2 )dn

(11)

μ1 : chemical potential at point 1 μ2 : chemical potential at point 2 dG : Gibbs Energy

2.2 2.2.1

The dependence of stability on conditions Temperature     δμ δG = −Sm −→ = −Sm δT p δT p

(12)

−Sm : molar entropy 2.2.2

Pressure



δμ δp

 T

= Vm −→ Δμ = Vm P

(13)

Vm : molar volume

2.3

Effect of pressure on vapour pressure ln p Vm × ΔP = ∗ p RT

2.4

(14)

Clepeyron Equations for Phase Boundaries

Slope of the diagrams are determined via Clepeyron equations Δtrs S dp = dT Δtrs V

(15)

Δtrs S : entropy change for transferred substance Δtrs V : Volume of transferred substance 2.4.1

Solid-Liqud Phase Boundary dp Δm H = dT T Δm V

Δm H : enthalpy change for melting Δm V : Volume change while melting

2

(16)

2.4.2

Liquid-Gas Boundary Δvap H dp = dT T Δvap V

(17)

Δvap H dp = dT T (RT /P )

(18)

ΔV g ≫ ΔV liq → ΔV vap = ΔV g V = RT/P →

2.4.3

Finding the vapour pressure of a liquid at desired temperature −

p = p∗e and γ=

3

γ

1 1 ΔH vap × ( ) − ( ∗) R T T

(19) (20)

Simple Mixtures

3.1 3.1.1

Partial molar quantities Partial molar volume

For a substance J, molar volume in a solution is Vj =

 δV  nj p,t,n

(21)

In a mixture including liquid a,b V = Va na + Vb nb 3.1.2

Partial Molar Gibbs energy G = n a μA + n b μB

3.1.3

(22)

(23)

Other important equations for chemical potential

U = −pV +T S+G −→ dU = −pdV +T ds+dG = −pdV +T ds+μdna +μdnb +.. (24) When the volume and entropy are constant dU = μdna + μdnb + ....... 3.1.4

(25)

Gibbs-Duhem equation

In a two componnent mixture, the chemical potential of one substance can’t change independently without the other substance’s chemical potemtial. dμB = −

3.2

nA dμA nB

(26)

Termodynamics of mixtures

  δGmix = −nRT xa lnxA + xB lnxB

ΔS mix =

   δGmix  = −nR xa lnxA + xB lnxB δT

3

(27) (28)

3.3

Chemical potentials of Liquids μA = μA∗ − RT ln

pA pA ∗

(29)

μA∗ : chemical potential of pure liquid A pA ∗ : vapour pressure of pure liquid A

3.4 3.4.1

Ideal solutions Rault’s law pA = xA × pA ∗

(30)

μA = μA ∗ + RT lnxa

(31)

pA ∗ : vapour pressure of pure liquid A xA : mole fraction of liquid A in the solution μA ∗ : chemical potential of pure liquid A 3.4.2

Ideal dilute sulutions and Henry’s law pB = xB KB

(32)

KB : extrapolated Henry pressure

4

Properties of Solutions

4.1 4.1.1

Liquid mixtures Ideal solutions

for an Ideal solution ;

4.1.2

ΔH mix = ΔGmix + T ΔS mix = 0

(33)

ΔV mix = 0, ΔHmix = 0

(34)

Regular solutions

in regular solutions ; ΔH = 0 and ΔS = 0 H E = nβRT xA xb H

(35)

E

:exceeded entalphy β :is a function for molecular interactions ΔGmix = nRT (xA ln xA + xB ln xB + βxA xB )

4.2

(36)

Collugative preperties

Chemical potential change when a solute introduced to a liquid μ∗ = μ∗ + RT ln xA

4

(37)

4.2.1

Increasing of boiling point xB =

ΔH vap  1 1 − ∗ R T T

T ∗ : normal boiling point

5

(38)