real gases part 1 - the edelstein center for the … · real gases –part 1 treatments: 1....

Real Gases – Part 1

Treatments:1. Lennard-Jones Potential Energy Function

2. Z Compressibility Factor (מקדם דחיסות) 3. van der Waals Equation of State

4. Redlich-Kwong Eqn.

5. Berthelot Eqn.

6. Virial (Kamerlingh Onnes) Eqn.

7. Beattie-Bridgeman Eqn.

8. Reduced Equations of State

9. Activity Coefficient - based on G (next semester)

10. Many Many Others

1 © Prof. Zvi C. Koren 21.07.10

Intermolecular Forces: Attractions & Repulsions

req

dr

dU- F

U,

r

12n9 ,r

B

r

A F

1n7

12 n9 r

B

r

A- U

n6,

Attraction Repulsion

Lennard-Jones Potential Energy:

SirJohn Edward

Lennard-Jones1894 – 1954

England


RT/VP Factor n)Compressio(or ility Compressib Z

(00C)


Compressibility Factors

N2CH4

Find TB, the Boyle Temperature


(TB)

For any real gas:

dZ/dPZTemp.

> 0> 1> TB

0 1= TB

Low P: < 0< 1

< TB Medium P: > 0

High P: > 0> 1

Variation of Z with T: General Considerations


van der Waals Equation of State & Intermolecular Forces

Johannes Diderik van der Waals1837–1923

Netherlands

Nobel Laureate in Physics, 1910

for his work on

The equation of state for gases and liquids

Definition of “van der Waals forces”:

Any force between neutral molecules!


van der Waals Equation of State (continued)

Rewrite Ideal Gas Eqn. of State as: Pideal ·Voccupied = nRT

Pideal > Preal:Attractive forces reduce the FREQUENCY and FORCE of collisions with the walls

of the container.

The reduction in EACH factor is proportional to the concentration of the gas, n/V:

Preduction (n/V)2

= a(n/V)2 = n2a/V2,

a effective intermolecular force parameter

Voccupied by ideal > Voccupied by real:V occupied by ideal gas = V of entire container

V occupied by real gas = V of entire container – V of molecule themselves

= V – nb,

b effective molar volume parameter

“Corrections” for P & V in Ideal Gas Equation of State

nRT nb-VV

anP

2

2


nRT nb-VV

anP

2

2

van der Waals Equation of State (continued)

a effective intermolecular force parameter

b effective molar volume parameter

a, b f(T)

RT b-VV

aP 2

Units of “a”: ______________

Units of “b”: ______________

V/n V V VolumeMolar m or


baFormulaGas

0.03714.17NH3Ammonia

0.03221.35ArArgon

0.04273.59CO2Carbon dioxide

0.076911.62CS2Carbon disulfide

0.03991.49COCarbon monoxide

0.138320.39CCl4Carbon tetrachloride

0.05626.49C12Chlorine

0.102215.17CHCl3Chloroform

0.06385.49C2H6Ethane

0.05714.47C2H4Ethylene

0.02370.034HeHelium

0.02660.244H2Hydrogen

0.04434.45HBrHydrogen bromide

0.04282.25CH4Methane

0.01710.211NeNeon

0.02791.34NONitric oxide

0.03911.39N2Nitrogen

0.03181.36O2Oxygen

0.05646.71SO2Sulfur dioxide

0.03055.46H2OWater

van der Waals

Constants for

Various Gases(a in atmּL2/mol2;

b in L/mol)

He

Ne

Ar

Kr

Xe


Comparison of Ideal Gas Law and van der Waals Equation

(at 1000C)

Carbon DioxideHydrogen

%P Calc.%P%P Calc.%PObserved

Devia-van derDevia-Calc.Devia-van derDevia-Calc.P

tionWaalstionIdealtionWaalstionIdeal(atm)

-1.049.5+14.057.0+0.450.2-2.648.750

-2.373.3+17.392.3+0.975.7-3.672.375

-4.295.8+33.5133.5+0.8100.8-5.095.0100


Note:

For each observed P there is an observed V and that is plugged into the

equation to calculate a theoretical P.

“Cubic” Equations of State

vdW: The van der Waals eqn. is an example. It is cubic in “V”:

RT b-VV

aP 2

Original form:

Multiply out and by V2 and rearrange:

0 P

VP

VP

PbRTV

23

abaCubic form:

Redlich-Kwong (1949):

RT V

VVTP

1/2

or2

V

a

b-V

RTP

Show the cubic form:


Berthelot Equation of State

Low-Pressure form (P 1 atm):

2

2

T

61

T128

9P1RT VP

cT

P

T

c

c

Tc = critical temperature

Pc = critical pressure

No additional constants


Beattie-Bridgeman Equation of State

432

VVVV

RT P

(5 constants)

Explicit in P:

Explicit in V:

2

32P

RTP

RTRTP

RT V

δγβ

200T

RRT

cABβ

2

000

T

RRT

cBaAbB

2

0

T

R

bcB

Very accurate even for P 100 atm and

T -1500C

(Table on next slide)13 © Prof. Zvi C. Koren 21.07.10

c x 10-4bBoaAoGas

0.00400.014000.059840.0216He

0.10100.020600.21960.2125Ne

5.9900.039310.023281.2907Ar

0.050-0.043590.02096-0.005060.1975H2

4.20-0.006910.050460.026171.3445N2

4.800.0042080.046240.025621.4911O2

4.34-0.011010.046110.019311.3012Air

66.000.072350.104760.071325.0065CO 2

12.83-0.015870.055870.018552.2769CH4

33.330.119540.454460.1242631.278(C2H5)2O

BEATTIE-BRIDGEMAN CONSTANTS FOR SOME GASES*(P in atm, Vm in L/mol)

*J. Am. Chem. Soc., 50, 3136 (1928). See also Maron and Turnbull, Ind. Eng. Chem., 33, 408 (1941).

14 © Prof. Zvi C. Koren 21.07.10Gas Problems: Real Gases: 14-17, 22.

The Kamerlingh Onnes Virial Equation (vires = forces)

Pin a as VP Express :Equation Virial

RTVP :Equation Gas Ideal

esPower Seri

DP CP BP A VP 32

2nd virial coefficient. Units =

3rd virial coefficient. Units =

= RT

Notes:A, B, C, … = f(T)

A > 0

B < 0 (low T), B = 0 (certain T), B > 0 (higher T)

A >> B >> C >>

BP A VP :) 15( P low relativelyAt atm

:0B A VP ,TTat Also B

(טור חזקות)

(האיבר הראשון)

2CP BP A VP :)atm 05( P mediumAt

HeikeKamerlingh

Onnes1853 – 1926,

Holland


E x 1011D x 108C x 105B x 102At (°C)

Nitrogen

4.657-14.47014.980-2.879018.312-50

1.704-6.9108.626-1.051222.4140

0.9687-3.5344.4110.666230.619100

0.7600-2.3792.7751.476338.824200

Carbon Monoxide

6.225-17.91117.900-3.687818.312-50

1.947-7.7219.823-1.482522.4140

0.9235-3.6184.8740.403630.619100

0.7266-2.4493.0521.316338.824200

Hydrogen

1.022-1.7411.1641.202718.312-50

0.7354-1.2060.78511.3638 22.4140

0.1050-0.16190.10031.797463.447500

VIRIAL COEFFICIENTS OF SOME GASES

(P in atm, Vm in L/mol)

0B

A VP ,TTAt

T)(higher 0 B

T),(certain 0 B

T), (low 0 B

0 A

f(T) C, B, A,

C B A

:Notes

B


-4

-2

0

2

-50 0 50 100 150 200

t (0C)B

x 1

02

E x 1011D x 108C x 105B x 102At (°C)

6.225-17.91117.900-3.687818.312-50

1.947-7.7219.823-1.482522.4140

0.9235-3.6184.8740.403630.619100

0.7266-2.4493.0521.316338.824200

VIRIAL COEFFICIENTS OF CO (P in atm, Vm in L/mol)

Determination of TB: Graphical & Regression

:0B A VP ,TTAt B


Z Meets Virial

PRT

D P

RT

C P

RT

B 1 Z 32

RT

VP Z DP CP BP A VP 32

PD' PC' PB' 1 Z 32

RT

D D' ,

RT

C C' ,

RT

B B'

A = RT

We can also express “Z” as an inverse power series in the molar volume

Vδ V γ Vβ 1 Z-3-2-1

These two forms obey the conditions for “Good” Mathematical-Chemical Functions:

V1 )V Z(lim

0P1 Z(P)lim

and

:1הצגה

:2הצגה


Relationships Between the Two Sets of Coefficients

PD' PC' PB' 1 Z /RTVP 32

Vδ V γ Vβ 1 Z /RTVP -3-2-1

) Vδ V γ Vβ VRT( P -4-3-2-1 Solve for P = f(V) from 2nd eqn.:

V γ Vβ 1 Z -2-1

P D' P C' P B' 1 Z 32

) Vδ V γ Vβ V((RT)D'

) Vδ V γ Vβ V((RT) C'

) Vδ V γ Vβ V( RT B' 1 Z

34-3-2-1-3

24-3-2-1-2

-4-3-2-1

V](RT)C' RTβ[B' VRTB' 1 Z -22-1

= B’RT = (B/RT)RT B =

= B’RT + C’(RT)2 = B + CRT = B2 + CRT C = ( –2)/(RT)

Substitute P = f(V) into 1st eqn.:

Compare with 2nd eqn.:

:השוואת מקדמים של חזקות שוות

19 © Prof. Zvi C. Koren 21.07.10Gas Problems: Real Gases: 18, 19, 25.

real gases part 1 - the edelstein center for the … · real gases –part 1 treatments: 1....

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