generalization of ideal gas behavior_2

8/3/2019 Generalization of Ideal Gas Behavior_2

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Generalization of Ideal Gas Behavior

Boyles Law

The volume of any definite quantity of

gas at constant temperature varied

inversely as the pressure on the gas.

P1V1 = P2V2

Charles Law

The volume of a definite quantity of

gas at constant pressure is directly

proportional to the absolute temperature.

Gay-Lussacs Law

Combined Gas Law

Avogadros Law

Ideal Gas Equation

PV = nRT

Values of R:

0.08205 L-atm/mol-K

82.05 cc-atm/mol-K

8.314 x 107 ergs/mol-K

8.314 J/mol-K

1.987 cal/mol-K

62360 mmHg/mol-K

Daltons Law of Partial pressure

At constant temperature, the total

pressure exerted by a mixture of gases in a

definite volume is equal to the sum of the

individual pressure without each gases

would exert if it occupied the same total

volume alone.

PT = P1 + P2 + P3 + +Pn

Partial pressures:

P1 = P2 =

P3 =


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Daltons Law:

PT =

PT = nT

Partial Pressure of a gas to the total

pressure of the mixture:

P1 =

P2 =

P3 =

PA/PT = nA/nT = NA (mole fraction)

PB/PT = nB/nT = NB (mole fraction)

PC/PT = nC/nT = NC (mole fraction)

NA + NB + NC = 1

Amagats Law of Partial Pressure

In any gas mixture the total volume

may be considered to be the sum of the

partial volumes of the constituents of the

mixture.

VT = V1 + V2 + V3 + + Vn

V = (n1 + n2 + n3)

V = nT

VA/V = NA/RT = NA

VB/V = NB/RT = NB

VC/V = N

C/R

T= N

C

Graham's Law of Diffusion

At constant temperature and pressure,

the rate of diffusion of various gases

varies inversely as the square roots of

their densities or molecular weights.

=

=

Where:

u or r = rate of diffusion

p1 or p2 = densities

Limitation of Ideal Gas

Low to moderate pressurePV = znRT

Where:

z = compressibility factor

If: z > 1 = dominant adhesive force

z < 1 = dominant cohesive force


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Van Der Walls Equation of State

Van Der Walls deduced that the

magnitude of the pressure correction P' for

n moles of gas present in volume V is given

by:

P' = where a is a constant

Thus:

(P +

(V-nb) = nRT

Note: valid only for wide pressure range

a =

b =

Kamerlingh Onnes Equation of State

PV seriesPVm = A + BP + BP2 + BP3 +

Where:

P = pressure

Vm = molar volume

A,B,C = viral coefficients

Rate: PVM = A

PV2 = nRT

Berthelot Equation

PV = nRT (1 +

Where: T = actual temperature

Tc = critical temperature

P = actual pressure

Pc = critical pressure

Beattie - Bridgeman Equation of State

P =

Vm =

Where:

= RTB0 - A0 -

= -RTB0b + Aoa -

=

T = absolute temperature

R = gas constant

Ao, Bo, a, b, c = constant

characteristic of each gas


4/27

Molecular Weights of Gases

p = p =

MW = M =

Some Methods for Determination of Molecular

Weights

Regnult's Method

> employed to determine the molecular

weights of substances which are gaseous at

room temperature.

Dumas' Method

> used to determine molecular weights

in the vapor phase of readily volatile

liquids.

Wvapor = W(bulb + vapor) - W(bulb + air) + Wair

M =

[1 +

(1- )] (Berthelot

Equation)

Determination of ExactMolecular Weights

Method of limiting densities

> based upon the fact that as zero

pressure is approached, the ideal gas laws

become the exact for all gases

PMW = pRT

p/P = MW/RT = constant

(p/P)P=0 = MW/RT

MW = RT P=0

Plotting:

P (atm)

Kinetic Theory of Ideal Gases

The Kinetic Theory is based on the

following fundamental postulates:

1. Gases are considered to be composed of

minute discrete particles called molecules.


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For any one gas all molecules are thought

to be of the same mass and size but to

differ in these from gas to gas.

2. The molecules within a container are

believed to be in ceaseless chaotic motion

during which they collide with each other

and with the walls of the container.

3. The bombardment of the contaiuner walls

by the molecules gives rise to thephenomenon we call pressure (force/unit

area).

4. In as much as the pressure of the gas

within a container does not vary with time

at any given pressure and temperature. The

molecular collisions must involve no energy

loss due to friction. In other words, allmolecular collisions are elastic.

5. The absolute temperature is a quantity

proportional to the average kinetic energy

of all the molecules in a system.

6. At relative low pressures, the average

distances between molecules are largecompared with the molecular diameters, and

hence the attractive forces between

molecules, which depend on the distance of

molecular separation, may be considered

negligible.

7. Finally, since the molecules are small

compared with the distances between them,

their volume may be considered to benegligible compared with the total volume

of the gas.

Root-Mean-Square Velocity

U =

= Where:

U = root-mean-square velocity

Ux, Uy, Uz = component velocities along

x, y, or z

Momentum = mass x velocity (mu)

Change in momentum

m = m1 - m2

= mux - (-mux) = 2mux

Change in Momentum per Second per Moleccule

= (2mux) =

= 2

Total change in momentum/molecule/second =

2

+ 2

+ 2


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= (U

2x + U2y + U2z)

= U2

For n' molecules in a cube

Total change in momentum per second =

P = =

P = =

=

PV = mn'

Where:

A cube = 6l2

F =

V = l3

Deductions from Kinetic Theory of Gases

Boyle's Law

1/2 mn'U2 = K1T

1/2 PV = 1/3 K1T

PV = 2/3 K1T

PV = C

V/T = C'

P/T = C"

Charle's Law

V = T

V = K2T

Avogadro's Principle

> equal volumes of all gases at the same

pressure and temperature contain equal

numbers of molecules

For two gases

P1V1 = P2V2

1/3 n'1m1= 1/3 n'2m2

m1 = m2

per mole

1/3 n'1m1= 1/3 n'2m2

n'1 = n'2

n' = n x avogadros number

where:

n = moles

n' = molecular wt


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N'1 = N'2 (no of molecule will be the

same)

Avogadro's No (N) = 6.023x1023 molecules/gram-

mole

Graham's Law of Diffusion

PV = C

P1V1 = P2V2

1/3 n'1m1= 1/3 n'2m2

U21/ U22 =

U1/ U2 = =

U1/ U2 = = =

PV = nRT

PV = 1/3 n'mu2 = nRT

PV = = nRT

Where:

n' = nM

n' = no of molecules

n = avogadro's number

Further Deduction from the Kinetic Theory

Velocity of Gas Molecules

> 1/3 nMWU2 = nRT

U2 =

U =

= Kinetic Energy of Translation

Ek =(

nMu

2)

=nRT

Where:

nRT =

MWU2 = 3RT

On division by Avogadro's no.

Ek/N =

= kT

Where:

K = is called the Boltzman constant

and is equal to 1.3805 x 10-16 erg per

degree


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Distribution of Molecular Velocities

Maxwell-Boltzman Distribution Law

dnc/n' = 4(

)3/2 e-MC2/2RT c2dc

dividing by dc:

P =

= 4(

)

3/2 e-MC2/2RT c2dc

W

here:

P = probability of finding molecules

with the velocity

dnc = no of molecules out of a total

n' having velocities between c and c + dc

M = molecular wt

T = temeperature of gas

Dnc/n' = fraction of the total number of

molecules having the same velocities

Most Probable Velocity ()

= Average Velocity (v)

= v = 0.912u

v = 0.921 where:

:v:u = 1:1.128:1.224

Frequency of Collisions and Mean Free Path

> the number of molecules which a single

gas molecule will collide per second is:

v2(n*)2

Where:

v = average velocity in cm per second

= molecular diameter in cm

n* = n x (6.023 x 1023)

Z = v2(n*)2

Z = total no of colliding molecules per cc

per second

Nc = no of molecular collisions occurring

in each cc per second

Nc =

=

v

2(n*)2


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Mean Free Path (l)

> the average distance a molecule traverse

before colliding

l =

Heat Capacity of Gases

Cp = at constant pressure

Cv = at constant volume

Ek = RT

E = Ek2 - Ek1 =R(T2 -T1)

Cp - Cv =

Cv = R

Cp = R

H = E + PV

Specific Heat Ratio

For monoatomic gases:

= =

= = 1.667

For diatomic gases: (H2, N2, O2 I2 Cl2)

Cp - Cv = R Cp = 7/2 R

Cv = 5/2 R = 1.4

For triatomic gases: (CO2, SO2, H2S N2O)

= 1.3

Theory of Non-Ideal Gases

Interaction energy (E')

E' = - +

Force interaction (f')

f' = -

where:

A and B are constant molecule

r = distance of separation

n = constant whose value may range from

9 12


10/27

LIQUIDS

Critical Phenomena in Liquids

> properties of liquids and vapor become

identical

> critical points are:

Critical Temperature, Tc

Critical Pressure, Pc

Critical Volume, Vc

For H2O critical constants are:

Tc = 374.4C

Pc = 219.5 atm

Vc = 58.7 cc per mole

Liquefaction of Gases

Process involve: cooling

Cooling + compressor

For permanent gases:

Adiabatic expansion, Q = 0

Allowing gas to cool itself by performing

work in adiabatic expansion against piston.

Viscosity

> resistance to flow and produce by

shearing effect of moving one layer of

fluid past, another and quite distinct from

intermolecular attraction.

Fluidity > opposite of viscosity

=

Where:

=

= viscosity on absolute

dynamic

Kinematic Viscosity

Viscosity of Gases

= vlp

where:

p = density

l = mean free path

l =


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Poiseville or Stokes Equation

=

1/2 =

Where:

P = pressure

t = time of flow

r = radius of tube

V = volume of liquid

Stoke's Law

> applicable to the fall of spherical

bodies in all types of fluid media provided

the radius of the falling body is large

compared with the distance between the

molecules of the fluid.

W = m g/gc

W = PV g/gc

F1 = 4/3 r3(p - pm)g/gc

F2 = 6rv

Equating F1 and F2:

4/3 r3(p - pm)g/gc = 6rv

=

Fluidity ()

=

1/2 =

> the viscosity of liquid decreases withincreasing temperature

Fg = F1

FR = F2


12/27

First Law of Thermodynamics

Thermodynamics = study of energy conversion

Kinds of Energy

> Kinematic Energy

The energy a system possesses by

virtue of its motion

> Potential Energy

The energy a system possesses by

virtue of its position

Mechanical energy > erg (cgs)

Electrical energy > joule (cgs)

Thermal energy > calorie (cgs)

Thermodynamic System

System

> anything under study or consideration

> as any portion of the universe isolated

on or in an inert container, which maybe

real or imaginary, for purposes of study of

the effect of various variables upon the

contents of the system.

A system can be:

Open System

> flow of energy, flow of mass (pump)

> a system which can exchange both matter

and energy with its surroundings

Closed System

> no mass flow, flow of energy (elec bulb)

> one which no transfer of matter to or

from the surroundings is possible, but that

of energy is

Isolated System

> no mass flow(transfer): no energy flow

> one which cannot exchanged both matter

and energy with its surroundings

It can be:

Homogeneous System = contains only one

phase

Heterogeneous System = two or more

than a single phase may involved


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Phase

> defined as homogeneous, physically

distinct, and mechanically separable

portion of a system.

Steady State

> does not change in time

True Solution

> physically homogeneous mixture of two or

more substances

Thermodynamics SystemProperties:

Extensive property

> dependent on the mass

Intensive Property

> mass independent

First Law of Thermodynamics

Energy can neither be created nor

destroyed but it can only transform from

one form to another.

PE1 + KE1 + E1 + P1V1 + Q = W + P2V2 + KE2 +E2 +

P2V2

First Law Equation:

E = q - W

dW = fdl = pAdl

W =

Wmax is attained at reversible process

W = (+) work is done by the system (V2 > V1)

expansion

W = (-) work is done on the system (V2 < V1)

compression

q = (+) absorbed

q = (-) evolved

E = q - p(V2 - V1)

a. P is constant

E = q - p(V2 - V1)

b. Volume is constant

E = q

c. free expansion

P = 0; W = 0

E = q


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d. P is variable, T is constant

W = nRT ln

W = nRT ln

Enthalpy of a System

H = E + PV

H = H2 - H1

= (E2 + P2V2) - (E1 + P1V1)

= (E2 - E1) + (P2V2 - P1V1)

= E + P(V2 - V1)

H = E + PV

Where:

E = internal energy

If Cp = f(T)

Cp = a + bT + CT2 + dT3

Cp = a + b/T2 + C/T2

H = n

At constant volume:

C =

Cv = (

v

=

E =

E = mCvT

H = E

H = (E2 - E1) + (P2V2 - P1V1)

H = E + Pv

H = q - PV + PV

H = q

H > E if w = 0, w > 0

H = E if w = 0

H = (+) absorption of heat

H = (-) evolution of heat

0


15/27

Heat Capacity (Cp, Cv)

Cp = heat capacity at constant temperature

W = rate of change of the internal energy

with temperature at constant volume.

Cp = (

v

At constant pressure:

Cp = (

P

If Cp is constant:

H = nCpT

=

H = when n = 1

H = when n = 1

H = E + PV

Differentiating it with respect to

temperature at constant pressure:

Cp - Cv = ()P + P(

) - (

)v

Difference of state of state function on

variables:

*E as a function of (T, V) then:

E = f(T,V)

*H = f(T,P)

dH = CpdT + [V - T()P]dP

Thermodynamics Behavior of Ideal Gas

PV = nRT

For ideal gas:

Cp - Cv = nR

Cp - Cv = R

For monoatomic gases:

Cp = 5/2 R

Cv = 3/2 R

For diatoimic gases:

Cp = 7/2 R

Cv = 5/2 R

Note:

=


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For monoatomic = 1.6

For diatomic = 1.4

For triatomic = 1.3

Isothermal andAdiabatic Process

Isothermal Processes

> any process conducted in a manner such

that the temperature remains constant

during the entire operation.

E = q - w

0 = q - w

q = w

Isothermal process in ideal gas

W = nRT ln V2/V1

W = nRT ln P1/P2

If P is constant:

W = P(V2 - V1)

Adiabatic process in ideal gas

q = 0 (no heat evolved nor absorbed)

E = - w

w = - E

Where:

E =

* Gas laws are not applicable in adiabatic

process.

Derived formula:

P1V1 = P2V2

= ( )

= (

= (

w =

w =

Joule - Thompson Effect

Adiabatic:E = - w

E2 - E1 = (P2V2 - P1V1)

E2 + P2V2 = E1 + P1V1

H2 = H1

H = 0


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Joule - Thomson Coefficient:

=()H

Cp = - ()T

Cp = T ()P - V

Where:

Cp = ()P

= 0 at inversion point (gas

neither heat or cools)= (+) cooling

= (-) heating

The Carnot Cycle

> in itial and final states are equal

> composed of several steps/processes

Carnot Cycle Steps

> 2 isothermal (T = C) process (expansion,

compression)

> 2 adiabatic process (q = 0) (expansion,

compression)

Analysis of the cycle

1. AB --- isothermal expansion

E = q2 - w1

E = 0

q2 = w1 = nRT2 lnV1/V2 = nRT lnP2/P1

w = q2 - q1

2. BC --- adiabatic expansion

q = 0

E = - w2

E =

3. CD --- isothermal compression

E3 = 0

E3 = -q1 - w3

q1 = -w3 = -nRT lnV4/V3

4. DA --- adiabatic compression

q = 0

E4 = -w4

E4 =

E cycle = 0

w max = q2 - q1

P

V

Isothermal

Expansion

AdiabaticExpansion

Isothermal

Compression

Adiabatic

Compression P1V1

P2V2

P3V3

P4V4

A

B

C

D


18/27

Thermodynamics Efficiency

Eff =

x 100Eff =

=

=

= x 100

=

x 100

ThermoChemistry

> branch of physical chemistry which deals

with the thermal changes accompanying

chemical and physical transformation.

Heat lost = heat gained

qH = mCpT

Heat of ReaCTION AT Constant Volume or

Pressure

E = q - w

q = E +

at constant volume: V = 0q = E

at constant pressure:

q = H

where:

H + E = (+) heat is absorbed

H + E = (-) heat is evolved

Calculation of E fromH or vice versa

H = E + PV

= E + (PVP - PVR)

= E + (npRT -nRRT)

= E + (NP -NR) RT

H = E + ngRT

Where: ng = nP - nR

Hess' Law of Heat Summation

> the heat of reaction must be independent

of the particular manner in which the

reaction takes place

Heat of Combustion

> for combustible material

> amount of heat liberated per mole of

substance burned

Hrxn =

Heat of Formation

> heat involved in the formation of 1 mole

of substance from the element.

Kirchhoff's Equation

Cp = CpP - CpR

Cp = [

]P

E2 =


19/27

The Second and Third Law of Thermodynamics

> The flow of heat is unidirectional,always from higher temperature region to

lower temperature region.

> All naturally occurring process always

tend to change spontaneous in a direction

which will land to equilibrium.

> Heat cannot be converted to work which

leaving permanent changes either in systems

involved or in their surroundings.

Haber Process

Entropy

> degree of disorder

> state of a function> depends only on initial and final state

of the system

S = S2 - S1

dS =

where dq = infinite quantity of heat

At constant temperature:

=

S =

Entropy Change in Isolated Systems

Reversible expansion:

Ss =

Sr =

V1 - V2 System = Ss + Sr

= +

System = 0

Isolated System Under Non-Isothermal

Condition Founds:

> for any reversible process or cycle S =

0

> for any irreversible process or cycle

S = ; S > 0

The Second Law of Thermodynamics

> All process in nature tend to occur only

with an increase in entropy and that the

directions of change is always such as to

the entropy increase.

dsi > 0

T2

q2

q1

W

q2 = q1 + W


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Entropy Change for Systems Only

ds = ds + dsr > 0system + reservoir

dsr =

ds + dsr > 0 E = q - w

ds - > 0 dE = dq - dw

Tds - dq > 0 dq = dE + dw

Tds - dE - pdV -dw' > 0

For irreversible process

Tds - dE - PdV - dw' > 0

PV = nRT P = P1 dw' = dw'mTds - dE - PdV - dw;mx = 0

Tds - dE - PdV = 0

Entropy Change in Ideal Gases

Isothermal

dE = 0Tds = PdV

ds =

ds =

=

In Isolated System

S = nRln

S = nRln

S = -nRln

Dependence on Temperature andVolume

ds = (

)v dT + (

)T dV

= dT +

)vdV

S = nCv ln

+ nR ln

S = f (T,P)

S = ncP ln

+ nR ln

Entropy of Mixing for Ideal Gas

Sm = n1R ln

+ n2R ln

Sm = n1R ln

+ n2R ln

If

=

= PT (special cases)

Sm = - (n1RlnN1 + n2RlnN2)


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Entropy Change in Physical Transformations

S = S2 - S1

S = =

where: Hv = latent heat

q = Hv = Hf = Hs

Hs = Hv + Hf

L - G S =

(evaporation)

G - L S = (cooling)S - L S =

(melting)

S - G S = =

(sublimation)

q = sensible heat

q = mCpT

Change Entropy in Chemical Reactions

A + bB + = cC + dD +

S = -

S = ;

=

S2 = S1 + nCpln

if cP is constant

Third Law of Thermodynamics

-

ST =

ST =

Free Energy and Equilibrium

> capability of the rxn to do work

Helholtz Free Energy (A)

A = E - TS


A = A2 -A1

= (E2 -TS2) - (E1 - TS1)

= E - T(S2 - S1)

A = E - T(S)

If T is not constant:

A = (E2 -TS2) - (E1 - TS1)

= E - (T2S2 - T1S1)

A = E - TS

S =

qr = TS


A = E - TS

= E - qr

= - wWmax = -A

Where: in first law of thermodynamics

E = q - w

E - q = - w

At constant temperature; ideal gas

W = nRT ln

= nRT ln


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A for Reactions

A = E - TS

dA = dE - (TdS + SdT)

A = f (T,V)

a) dA = ()vdT + (

)dV

b) dA = dE - TdS - SdT

= dE - SdT - dqr

Where:

dqr = dE + dw

dqr = dE + pfV

dA = dE - SdT - (dE + pdV)

dA = - SdT - pdV

Where:

-S = (

)v A = E - TS

-P = ()T

Therefore:

()v =

=

(

)v = -

()v =

dT

Where:

E = f(T)

E =

()v =

dT

The Gibbs Free Energy (G)

G = H - TS

G = G2 -G1

G = H - TS if T is constant

G = H - qr

= E + PV - qr

= E - qr + PV

= A + PV

G = A + PVG = - (Wm - PV)

Where:

A = f (T,V)

G = f (T,P)

G1 = (H1 - T1S1) = G2

G = A + PV

S =

A = - w

E = q - w

E - q = A

G = f (T, P)

dG = VdP - SdT

[

]P =

=

=

(

) =


23/27

Where:

-S = ()P

V = ()T

G = H - TS

H = G + TS

( )P = - S

(

) =

Properties and Significance of G

AT constant T, P:

G = 0 @ equilibrium

G = - spontaneous

G = + non - spontaneous

G = H - TS

G = H - qr

qr = H - G

qr (+) = H >G (absorbed)

qr (-) = G >H (evolved)

Calculations of Free Energy Changes


G = VP if V is constant

If V = f (T) ; ideal gas

G = nRT ln

G = nRT ln

Fugacity andActivity Concepts

G = RT lnf + B

At standard state:

G0 = RTlnf0 + B

G = G0 + nRT ln f/f0

G = G0 + 1

G = G0

Per Mole:

G = nRT ln

Where:

f/f0 = a

a = 1

Standard State for Gases

Where:

f0 = 1 f = P as P = 0

a = f/f0 lim

= 1

a = f P-0

G = G0 + RT lnf

= (activity coefficient)

ideal gas

Determination of Activity Coefficients of

Gases

dG = VdP ; G = VP

G = G0 + RT lnf


24/27

dG = RTdlnf

dlnf =

dlnf =

= ( + )

= +

= dlnP +

Dlnf - dlnP =

W

here: = V -

V = +

=

dln

= dln =

=

lny = (P - 0)lny =

if T and is constant

Standard States for Solid and Liquids

> As the standard of pure solid or liquidis taken the solid or liquid at 1 atm

pressure at each temperatue.

a = 1

The Reaction Isotherm

A + bB cC + dD

GA = + RT ln aA

GB = + RT ln aB

GC = + RT ln aC

GD = + RT ln aD

Grxn =

-

= (cGC + dGP) - (GA + bGB)G = G0 + RT ln (ln /)

Criteria of Equilibrium

Where:

G = 0

H = E + PV

G = H - TS

TdS = dE - PdV = 0

If:

V = C ; E = C

dV = 0 ; dE = 0

Then:

TdS - 0 - 0 = 0

TdS = 0

dS = 0

(S)E,V = 0 at equilibrium


25/27

If:

T and V are constants

dA + 0 + 0 = 0(dA)T,V = 0

(A)T,V = 0

Then:

A = E - TS

dA = dE - TdS - SdT

TdS - dE = -dA - SdT

-dA - SdT -PdV = 0

dA - SdT - PdV = 0

3 Criteria of Equilibrium

(S)E,V = 0

(G)T,P = 0

(A)T,V = 0

AT constant T and P:dG + 0 - 0 = 0

(dG)T,P = 0

(G)T,P = 0

Then:

G = H - TS

= E + PV - TS

G = A + PV

dG = dA + PdV + VdP

dA + PdV = dG - VdP

-SdT = dG - VdP

dG + SdT - VdP = 0

where: A = E - TS

Physical equilibrium Involving Pure

Substances

At equilibrium:

dG = 0 dA = 0

dT = dP = 0 A = 0

G = 0 A2 - A1 = 0

G2 - G1 = 0 A2 = A1

G2 = G1

dG = -SdT + VdP = dG1 = dG2

-S2dT + V2dP = -S1dT + V1dP

(V2 - V1)dP = (S2 - S1)dT

dP/dT = (S2 - S1)/(V2 - V1) =

Where:

S = q/T ; q = H

S = H/T

Therefore: =

--- clayperon equation

Uses of Clayperon Equation

=

=

(

)

=

P2 - P1 = ln


26/27

The Boiling Point of Liquids

Liquid to Vapor (Evaporation/Condensation)

H = Hv =

=

Where:

V = Vg - VL

V = VgPV = nRT

Vg = nRT/P

By substitution: integration:

ln=

(

)

Solid to Liquid(Fusion)

H = Hf where: V = Vl - Vs =

By substitution: integration:

P2 - P1 = ln

Troutons Rule

= c = 21

Sublimation (Solid to Gas)

Hs = Hv + Hf

V = Vg - Vs =

=

By substitution and integration:

ln

=

(

)

where:

Vg >>> Vs

V = Vg

PV = nRT

V = nRT/P


27/27

PhysicalChemistry

I

Physical

Principles

Alfredo Samsona Jr.

generalization of ideal gas behavior_2

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