chapter 3 real gases

Chemical Thermodynamics : Georg Duesberg Chemical Thermodynamics : Georg Duesberg

Chapter 1 : Slide 1

Chapter 3

Real gases

Chapter11111 1 : Slide 1

Chemical Thermodynamics : Georg Duesberg

Real Gases •  Perfect gas: only contribution to energy is KE of molecules •  Real gases: Molecules interact if they are close enough, have a

potential energy contribution. •  At large separations, attractions predominate (condensation!) •  At contact molecules repel each other (condensed states have volume!)

Ideal (Isotherms) Real (CO2)

F

A

p Thermo meter

Pressure gauge 2



Pressure region

I (very Low) Molecules have large separations -> no interactions -> Ideal Gas Behavior: Z =1

II (moderate) Molecules are close -> attractive forces apply -> The gas occupies less volumes as expected from Boyles law: Z<1

III (high) Molecules compressed highly -> repulsive forces dominate -> hardly further decrease in volume Z>1

Deviations from ideality can be described by the COMPRESSION FACTOR, Z (sometimes called the compressibility). Z = pV/(nRT) = pVm/(RT) For ideal gases Z = 1


Chapter 1 : Slide 4

Microscopic interpretation: Leonard Jones Potential

When p is very high, r is small so short-range repulsions are important. The gas is more difficult to compress than an ideal gas, so Z > 1. When p is very low, r is large and intermolecular forces are negligible, so the gas acts close to ideally and Z ∼ 1. At intermediate pressures attractive forces are important and often Z < 1.


Real Gases: What happens if we press down the piston

A – B perfect gas behavior (isotherm) B – C slight deviation from perfect

behavior – less pressure than expected C – D – E no change in pressure reading

over further compression – but increasing amount of liquid observed

E – F : steep in crease in P, only liquid visible (At contact molecules repel each other condensed states have volume!)

5

( at 20 °C, gas: carbon dioxide)

The line C – D – E is the vapour pressure of a liquid at this tempeature


•  Attractive forces vary with nature of gas •  At High Pressures repelling forces dominate


Z =


•  At Low Temperatures the attractive regime is pronounced •  higher Temperature ->faster motion -> less interaction


Z =

7


Chapter 1 : Slide 8

Boyle Temperatur The temperature at which this occurs is the Boyle temperature, TB, and then the gas behaves ideally over a wider range of p than at other temperatures. Each gas has a characteristic TB, e.g. 23 K for He, 347 K for air, 715 K for CO2.

The compression factor approaches 1 at low pressures, but does so with different slopes. For a perfect gas, the slope is zero, but real gases may have either positive or negative slopes, and the slope may vary with temperature. At the Boyle temperature, the slope is zero and the gas behaves perfectly over a wider range of conditions than at other temperatures.

Virial Equation of State

B = 0 at Boyle temperature

Most fundamental and theoretically sound Polynomial expansion Viris (lat.): force (Kammerling Onnes 1901)

Also allow derivation of exact correspondence between virial coefficients and intermolecular interactions

Virial coefficients: p Vm = RT (1 + B’p + C’p2 + ...)

i.e. p Vm = RT (1 + B/Vm + C/Vm2 + ...)

This is the virial equation of state and B and C are the second and third virial coefficients. The first is 1. B and C are themselves functions of temperature, B(T) and C(T). Usually B/Vm >> C/Vm

2



Johannes Diderik van der Waals got the Noble price in physics in 1910

( )22

or ⎟⎟⎠

⎞⎜⎜⎝

⎛−

−==−⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎠

⎞⎜⎝

⎛+mm Va

bVRTPnRTnbV

VnaP

Real gas – Van der Waals equation.

( )( ) nRTyVxP =−+

nRTgasideal =PV :


Chapter 1 : Slide 11

1.  The molecules occupy a significant fraction of the volume. -> Collisions are more frequent. -> There is less volume available for molecular motion.

Real gas molecules are not point masses (Vid = Vobs - const.) or Vid = Vobs - nb

–  b is a constant for different gases

Real gas – Van der Waals equation: b

Other explanation: What happens if we reduce T to zero. Is volume of the gas, V, going to become zero? We can set P ≠ 0. By the ideal gas law we would have V = 0, which cannot be true. We can correct for it by a term equal to the total volume of the gas molecules, when totally compressed (condensed) nb. Now at T = 0 and P ≠ 0 we have V = nb. nRTnbVP =− )(

Very roughly, b ∼ 4/3 πr3 where r is the molecular radius.


Real gas – Van der Waals equation: a

aVn2

2 a describes attractive force between pairs of molecules. Goes as square of the concentration (n/V)2 .

2) There are attractive forces between real molecules, which reduce the pressure: p ∝ wall collision frequency and

p ∝ change in momentum at each collision. Both factors are proportional to concentration, n/V, and p is reduced by an amount a(n/V)2, where a depends on the type of gas. [Note: a/V2 is called the internal pressure of the gas]. Real gas molecules do attract one another (Pid = Pobs + constant) Pid = Pobs + a (n / V)2 a is also different for different gases


13

Van der Waals equation of state

•  Parameters depend on the gas, but are taken to be independent of T. •  a is large when attractions are large, b scales in proportion to molecular size

(note units)

Substance a/(atm dm6 mol−2) b/(10−2 dm3 mol−1)

Air 1.4 0.039

Ammonia, NH3 4.169 3.71

Argon, Ar 1.338 3.20

Carbon dioxide, CO2 3.610 4.29

Ethane, C2H6 5.507 6.51

Ethene, C2H4 4.552 5.82

Helium, He 0.0341 2.38

Hydrogen, H2 0.2420 2.65

Nitrogen, N2 1.352 3.87

Oxygen, O2 1.364 3.19

Xenon, Xe 4.137 5.16

( )22

or ⎟⎟⎠

⎞⎜⎜⎝

⎛−

−==−⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎠

⎞⎜⎝

⎛+mm Va

bVRTPnRTnbV

VnaP

If 1 mole of nitrogen is confined to 2l and is at P=10atm what is Tideal and TVdW? Tip: R =0.082dm3atmK-1mol-1



CONDENSATION or LIQUEFACTION This demonstrates that there are attractive forces between gas molecules, if they are pushed close enough together. E.G. CO2 liquefies under pressure at room temperature. Above 31 0C no amount of pressure will liquefy CO2: this is the CRITICAL TEMPERATURE, Tc.

Chemical Thermodynamics : Georg Duesberg Chemical Thermodynamics : Georg Duesberg Chapter 1 :

Slide 15

Carbon dioxide: a typical pV diagram for a real gas:

Tc, pc and Vm,c are the critical constants for the gas. The isotherm at Tc has a horizontal inflection at the critical point dp/dV = 0 and d2p/dV2 = 0.

Experimental isotherms of carbon dioxide at several temperatures. The `critical isotherm', the isotherm at the critical temperature, is at 31.04 °C. The critical point is marked with a star.


16

Critical Point

dp/dV = 0 and d2p/dV2 = 0. Consider 1 mol of gas, with molar volume V, at the critical point (Tc, pc, Vc) 0 = dp/dV = -RTc(Vc-b)-2 + 2aVc

-3 0 = d2p/dV2 = 2RTc(Vc-b)-3 - 6aVc

-4 The solution is Vc = 3b, pc = a/(27b2), Tc = 8a/(27Rb).

At the critical temperature the densities of the liquid and gas become equal - the boundary disappears. The material will fill the container so it is like a gas, but may be much denser than a typical gas, and is called a 'supercritical fluid'. The isotherm at Tc has a horizontal inflection at the critical point



Critical Point drying

Applications: TEM sample prep, porous materials, MEMS


CNT


Metal contacts on CNT


Etch


Etch

Freely suspended CNT


Freely suspended CNT

Etch

TEM electron beam


Protective resist

Etch


Suspended on contacted individual CNTs – Platform for combined investigations

Structure and Electronic Properties can be related:

Individual tubes or bundels?

What kinds of CNT

(MWCNT, SWCNT, (n,m), peapods..)


Combined TEM and Raman investigations on individual SWCNTs


31

Maxwell Construction Below Tc calculated vdW isotherms have oscillations that are unphysical. In the Maxwell construction these are replaced with horizontal lines, with equal areas above and below, to generate the isotherms. (The line is the vapour pressure of a liquid at this temperature, or liquid-vapor equilibrium)


32

Features of vdW equation •  Reduces to perfect gas equation at high T

and V •  Liquids and gases coexist when

attractions ≈ repulsions •  Critical constants are related to

coefficients. •  Flat inflexion of curve when T=Tc. •  Can derive (by setting 1st and 2nd

derivatives of equation to zero) expression for critical constants •  Vc = 3b, •  pc = a/27b2, •  Tc =8a/27Rb

•  Can derive expression for the Boyle Temperature •  TB = a/Rb

,


33

•  Can derive (by setting 1st and 2nd derivatives of equation to zero) expression for critical constants

00.,. 2

2

=⎟⎟⎠

⎞⎜⎜⎝

⎛

∂

∂=⎟

⎠

⎞⎜⎝

⎛∂

∂

== cc TTTT vpand

vpei we have,

2va

bvRTP −−

=

( ) 32

2va

bvRT

vp

T

+−

−=⎟

⎠

⎞⎜⎝

⎛∂

∂∴

( ) 432

2 62va

bvRT

vpand

T

−−

=⎟⎟⎠

⎞⎜⎜⎝

⎛

∂

∂

At critical points the above equation reduces to

( )02

32 =+−

−

va

bvRT

( )062

43 =−− v

abvRTand

Features of vdW equation



We also can say 2va

bvRT

P cc −

−=

By dividing those equations and simplifying we get

3cvb =

Substituting for b and solving for ‘a’ from 2nd derivative we get, a = 9RTcvc Substituting these expressions for a and b in equation (P(c) and solving for vc, we get

c

cc p

RTv

83

=c

c

pRT

b8

=∴c

c

pTR

aand22

6427⎟⎠

⎞⎜⎝

⎛=

Note: Usually constants a and b for different gases are given.

Features of vdW equation


Critical constants pc

atm Vc cm3/mol

Tc

K Zc TB

K

Ar 48.0 75.3 150.7 0.292 411.5

CO2 72.9 94.0 304.2 0.274 714.8

He 2.26 57.8 5.2 0.305 22.64

O2 50.14 78.0 154.8 0.308 405.9

The general Van der Waals pVT surface



The principle of corresponding states Gases behave differently at a given pressure and temperature, but they behave very much the same at temperatures and pressures normalized with respect to their critical temperatures and pressures. The ratios of pressure, temperature and specific volume of a real gas to the corresponding critical values are called the reduced properties. Define reduced variables pr = p/pc

Tr = T/Tc Vr = Vm/Vm,c

Van der Waals hoped that different gases confined to the same Vr at the same Tr would have the same pr.


Substitute for the critical values:

22rr

r2

r

9bVa

b)3b27Rb(V8aRT

27bap

−−

=

2rr

rr V

313V

8Tp −−

=

Thus

Thus all gases have the same reduced equation of state (within the Van der Waals approximation).

Also:

Zc =pcVc/RTc = 3/8 =0.375

Proof: rewrite Van der Waals equation for 1 mol of gas, p = RT/(V-b)-a/V2, in terms of reduced variables:

2c

2rcr

crcr VV

abVV

TRTpp −−

=

Principle of Corresponding States

Compression factor plotted using reduced variables. Different curves are different TR

With reduced variables, different gases fall on the same curves -> Degree of generality ( principle of corresponding states) According to this law, there is a

functional relationship for all substances, which may be expressed mathematically as vR = f (PR,TR). From this law it is clear that if any two gases have equal values of reduced pressure and reduced temperature, they will have the same value of reduced volume. This law is most accurate in the vicinity of the critical point.

The compressibility factor of any gas is a function of only two properties, usually temperature and pressure so that Z1 = f (TR, PR) except near the critical point. This is the basis for the generalized compressibility chart. The generalized compressibility chart is plotted with Z versus PR for various values of TR. This is constructed by plotting the known data of one or more gases and can be used for any gas.

It may be seen from the chart that the value of the compressibility factor at the critical state is about 0.25. Note that the value of Z obtained from Van der waals’ equation of state at the critical point,

83

==c

ccc RT

vPZ which is higher than the actual value.

The following observations can be made from the generalized compressibility chart:

Ø  At very low pressures (PR <<1), the gases behave as an ideal gas regardless of temperature.

Ø  At high temperature (TR > 2), ideal gas behaviour can be assumed with good accuracy regardless of pressure except when (PR >> 1).

Ø  The deviation of a gas from ideal gas behaviour is greatest in the vicinity of the critical point.



There are many other equations of state for real gases 1)  The Berthelot Equation. Replace Van der Waals' "a" with a temperature dependent term, a/T:

[p + a/(Vm2T)] [Vm - b] = RT

2)  The Dieterici Equation : [p exp(a/VmRT)] [Vm - b] = RT 3) Redlich-Kwong 4) Peng-Robinson

Redlich-Kwong, Peng-Robinson are quantitative in region where gas liquefies Berthelot,Dieterici and others with more than ten parameters can give good

fits… with four free parameters, you can describe an elephant. With five his tail is

rotating …

)(2/1 BVVTA

BVRTp

mmm +−

−=

)()( βββα

β −++−

−=

mmmm VVVVRTp



Summary: Real gases •  REAL GASES: the COMPRESSION FACTOR and

INTERMOLECULAR FORCES. •  pV diagrams: LIQUEFACTION and the CRITICAL POINT. •  BOYLE TEMPERATURE •  The VAN DER WAALS approximate equation of state p = RT/(Vm-b) -

a/Vm2 is more realistic for REAL GASES. There are other equations of

state which work well e.g The VIRIAL EQUATION REDUCED VARIABLES and the PRINCIPLE OF CORRESPONDING

STATES

Pressure region

I (very Low) Molecules have large separations -> no interactions -> Ideal Gas Behavior: Z =1

II (moderate) Molecules are close -> attractive forces apply -> The gas occupies less volumes as expected from Boyles law: Z<1

III (high) Molecules compressed highly -> repulsive forces dominate -> hardly further decrease in volume Z>1


Real gas – Van der Waals equation.

For nitrogen a=0.14 and b=3.87x10-5. If 1.0 mole of nitrogen is confined to 2.00l and is at P=10atm what is Tideal and TVdW?

Under these conditions the temperature only changes by ~1%. Chapter 1 : Slide 43

( )

( )

( ) 240082.01/0387.0122114.01

/

2440.0822/110 /

2

2

2

=××−⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠

⎞⎜⎝

⎛+

=−⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠

⎞⎜⎝

⎛+

=−⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠

⎞⎜⎝

⎛+

=××==

TnRnbVVnaP

nRTnbVVnaP

TnRPVnRTPV

chapter 3 real gases

Documents