Real gas

1. molecules not always in motion (condense phase can be formed)

2. molecular size is non-negligible (there is molecular repulsion)

3. Molecules do interact (there is molecular attraction)

- Compression factor (Z)

idealm

m

V

VZ

Vm = molar volume or volume per mole

If Z < 1 molar volume of real gas is less than ideal gas molecular attraction is dominant

Z > 1 molar volume of real gas is larger than ideal gas molecular repulsion is dominant

Substitute P

RTV ideal

m

RT

PV

PRT

VZ mm

The variation of the compression factor, Z, with pressure for several gases at 0 oC

-Equation of state

is equation that represents the state of system

Ideal gas equation is the equation of state for gas

TRnPV

Virial equation in search for equation of state for real gas, one looks at

P dependent of Z

.......1 2 PCPBZ

.......12

mm V

C

V

BZ

or

Since V

P1

.......,, CB .......,, CB

and are 2nd, 3rd, …, virial coefficient respectively

Replacing Z by RT

PVm

.......12

mm

m

V

C

V

B

RT

PV

......1

2mm V

C

V

B

V

nRTP

virial equation of state

virial coefficients vary with gas type

So does the virial equation of state

PV plot for real gas

The experimental isotherms of CO2 at several temperatures.

Critical constants

- Reduced properties

cr

c

mr

cr T

TT

V

VV

P

PP ,,

The compression factors of four gas.

Van der waals equationanother attempt to find equation of state for real gas

To correct what’s wrong in kinetic molecular model, Van der waals proposed

idealmeasured PPP molecular attraction reduces impact force to wall

VVV idealmeasured

volume of gas is the space where gas can travel in a container

•Ideal gas has no molecular volume, volume of gas is volume of container

•real gas has molecular volume (repulsion), space that gas can travel is

less than volume of container

P (rate of collision) x (impact force

V

nX

V

n

Attraction reduces rate of collision and impact force

Reduction in pressure (due to attraction)2

V

na

From ideal gas equation

)1(V

nRTP

Substitute V in (1) with V - nb

)2(2

V

na

nbV

nRTP

then

)3(2

nRTnbVV

naP

“Van der waals ‘s equation”

a, b = van der waals parameter (obtained from experiment)

real gas volume is less than ideal gas

Given b as molecular volume

nbVVreal

Since Van der Waals equation comes from modification of kinetic molecular model (theory) while a, b comes from experiment

Van der waals equation is “semi-empirical”

virial equation is “empirical”

Van der waals parameters of gas.

- Van der waals loop

Isotherms calculated by using the van der waals equation of state.

Comparing ideal and real gas

V = 10 L , n = 1 mol P (atm)

Ideal gas V d w Virial*

273 KN2 2.239 2.234 2.236

CO2 2.239 2.212 2.205

O2 2.239 2.232

Air** 2.239 2.233

600 KN2 4.920 4.926 4.931

CO2 4.920 4.905 4.914

O2 4.920 4.922

Air 4.920 4.925

** AIR = 80%N2 + 20%O2

mV

B

V

nRTP 1*

Example 2.1 Estimate the molar volume of CO2 at 500 K and 100 atm by treating it as a van der waals gas.

From nRTnbVV

naP

2

2

V

na

nbV

nRTP

n

VVm

2mm V

a

bV

RTP

RTbVV

aP m

m

2

0

;

23

2232

2

abaVVRTPbPV

RTVabaVVPbPVxV

RTV

ab

V

aPbPV

mmm

mmmmm

mmm

must solve for roots of cubic equation

023

P

ab

P

aVV

P

RTbV m

mm

a = 3.610 L2 atm mol-2 ; b = 4.29 x 10-2 L mol-1

141.0

100

500082.0

molL

atm

K

Kmol

atmL

P

RTVm

41.00429.0100

610.31

41.00429.0

2

2

2

2

nVnV

nV

VbPV

a

P

RTbVb

PV

aV

VbP

a

P

RTbVV

mm

m

mm

mm

m

mmm

assume 41.00 mV

366.04

366.03

367.02

374.01

m

m

m

m

V

V

V

V

Vm = 0.366 L mol-1