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Home , Perfect gas, Real gas, Reduced properties, Van der Waals equation

Chapter 3

Real

Chapter11111Chapter 1 :1 Slide: Slide 1 1 Chemical : Georg Duesberg Real Gases • Perfect : only contribution to is KE of • Real gases: Molecules interact if they are close enough, have a potential energy contribution. • At large separations, attractions predominate (!) • At contact molecules repel each other (condensed states have !) Ideal (Isotherms) Real (CO2) F

A

p Thermo meter

Pressure

2 gauge

Chemical Thermodynamics : Georg Duesberg Deviations from ideality can be described by the COMPRESSION FACTOR, Z (sometimes called the ).

Z = pV/(nRT) = pVm/(RT) For ideal gases Z = 1

Pressure region

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

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

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

Chapter33333 1 : Slide 3 : Georg Duesberg 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 attractive forces are important and often Z < 1. Chapter 1 : Slide 4 Chemical Thermodynamics : Georg Duesberg Real Gases: What happens if we press down the piston ( at 20 °C, gas: carbon dioxide) 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 observed E – F : steep in crease in P, only liquid visible (At contact molecules repel each other condensed states have volume!)

The line C – D – E is the vapour pressure of a liquid at this tempeature 5

Chemical Thermodynamics : Georg Duesberg 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 • Attractive forces vary with nature of gas • At High Pressures repelling forces dominate

Z =

Chemical Thermodynamics : Georg Duesberg 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 • At Low the attractive regime is pronounced • higher ->faster motion -> less interaction

Z =

7 Chemical Thermodynamics : Georg Duesberg 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. Chapter 1 : Slide 8 Chemical Thermodynamics : Georg Duesberg Virial Most fundamental and theoretically sound Polynomial expansion Viris (lat.): (Kammerling Onnes 1901) Virial coefficients: 2 p Vm = RT (1 + B’p + C’p + ...) 2 i.e. p Vm = RT (1 + B/Vm + C/Vm + ...) 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). 2 Usually B/Vm >> C/Vm

B = 0 at Boyle temperature Also allow derivation of exact correspondence between virial coefficients and intermolecular interactions – .

ideal gas : PV = nRT (P + x)(V − y) = nRT

2 2 ⎛ ⎛ n ⎞ ⎞ RT ⎛ a ⎞ ⎜ P + a⎜ ⎟ ⎟(V − nb) = nRT or P = − ⎜ ⎟ ⎜ V ⎟ V b ⎜V ⎟ ⎝ ⎝ ⎠ ⎠ m − ⎝ m ⎠

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

Chapter1010101010 1 : Slide 10 Chemical Thermodynamics : Georg Duesberg Real gas – Van der Waals equation: b 1. The molecules occupy a significant fraction of the volume. -> 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 Very roughly, b ∼ 4/3 πr3 where r is the molecular radius. 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 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. P(V − nb) = nRT Chapter 1 : Slide 11 Chemical Thermodynamics : Georg Duesberg Real gas – Van der Waals equation: a 2) There are attractive forces between real molecules, which reduce the pressure: p ∝ wall 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 of the gas].

Real gas molecules do attract one another (Pid = Pobs + constant) 2 Pid = Pobs + a (n / V) a is also different for different gases n2 a describes attractive force between pairs of molecules. Goes as square of the 2 a V concentration (n/V)2 .

Chemical Thermodynamics : Georg Duesberg Van der Waals equation of state 2 2 ⎛ ⎛ n ⎞ ⎞ RT ⎛ a ⎞ ⎜ P + a⎜ ⎟ ⎟(V − nb) = nRT or P = − ⎜ ⎟ ⎜ V ⎟ V b ⎜V ⎟ ⎝ ⎝ ⎠ ⎠ m − ⎝ m ⎠ 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 If 1 of nitrogen is Carbon dioxide, CO2 3.610 4.29

Ethane, C2H6 5.507 6.51 confined to 2l and is at Ethene, C2H4 4.552 5.82 P=10atm what is Tideal Helium, He 0.0341 2.38 and TVdW? Hydrogen, H2 0.2420 2.65 Tip: R =0.082dm3atmK-1mol-1 Nitrogen, N2 1.352 3.87

Oxygen, O2 1.364 3.19 Xenon, Xe 4.137 5.16

• 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 (note13 units) Chemical Thermodynamics : Georg Duesberg 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.

Chapter 1 : Slide 14 Chemical Thermodynamics : Georg Duesberg Carbon dioxide: a typical pV diagram for a real gas:

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.

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.

Chapter 1 : Chemical Thermodynamics : Georg Duesberg Slide 15 Critical Point 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 '. The isotherm at Tc has a horizontal inflection at the critical point dp/dV = 0 and d2p/dV2 = 0. Consider 1 mol of gas, with V, at the critical point (Tc, pc, Vc)

-2 -3 0 = dp/dV = -RTc(Vc-b) + 2aVc

2 2 -3 -4 0 = d p/dV = 2RTc(Vc-b) - 6aVc

The solution is 2 V c = 3b, pc = a/(27b ), 16 T c = 8a/(27Rb). Chemical Thermodynamics : Georg Duesberg Critical Point drying

Applications: TEM sample prep, porous materials, MEMS

Chapter1717171717 1 : Slide 17 Chemical Thermodynamics : Georg Duesberg CNT

Chemical Thermodynamics : Georg Duesberg Metal contacts on CNT

Chemical Thermodynamics : Georg Duesberg Etch

Chemical Thermodynamics : Georg Duesberg Freely suspended CNT

Etch

Chemical Thermodynamics : Georg Duesberg TEM electron beam

Freely suspended CNT

Etch

Chemical Thermodynamics : Georg Duesberg Protective resist

Etch

Chemical Thermodynamics : Georg Duesberg Protective resist

Etch

Chemical Thermodynamics : Georg Duesberg Chemical Thermodynamics : Georg Duesberg Chemical Thermodynamics : Georg Duesberg Chemical Thermodynamics : Georg Duesberg Chemical Thermodynamics : Georg Duesberg 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..)

Chemical Thermodynamics : Georg Duesberg Combined TEM and Raman investigations on individual SWCNTs

Chemical Thermodynamics : Georg Duesberg 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)

31 Chemical Thermodynamics : Georg Duesberg , Features of vdW equation

• Reduces to perfect gas equation at high T and V • 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, 2 • pc = a/27b , • Tc =8a/27Rb • Can derive expression for the Boyle Temperature

• TB = a/Rb

32

Chemical Thermodynamics : Georg Duesberg Features of vdW equation • Can derive (by setting 1st and 2nd derivatives of equation to zero) expression for critical constants 2 RT a ⎛ ∂p ⎞ ⎛ ∂ p ⎞ P i.e., 0and⎜ ⎟ 0 we have, = − 2 ⎜ ⎟ = ⎜ 2 ⎟ = v − b v ∂v T T ∂v ⎝ ⎠ = c ⎝ ⎠T =Tc

2 ⎛ ∂p ⎞ − RT 2a ⎛ ∂ p ⎞ 2RT 6a ∴⎜ ⎟ = + and ⎜ ⎟ = − ∂v v b 2 v3 ⎜ 2 ⎟ 3 4 ⎝ ⎠T ( − ) ⎝ ∂v ⎠T (v − b) v At critical points the above equation reduces to − RT 2a 2RT 6a 2 + 3 = 0 and − = 0 (v − b) v (v − b)3 v 4 33

Chemical Thermodynamics : Georg Duesberg Features of vdW equation By dividing those equations and simplifying we get

vc RTc a b = We also can say Pc = − 3 v − b v 2 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 3RT RT 2 2 c c ⎛ 27 ⎞ R Tc vc = ∴b = and a 8p 8p = ⎜ ⎟ c c ⎝ 64 ⎠ pc

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

Chapter3434343434 1 : Slide 34 Chemical Thermodynamics : Georg Duesberg Critical constants

pc Vc Tc Zc TB atm cm3/mol K 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

Chemical Thermodynamics : Georg Duesberg 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 of a real gas to the corresponding critical values are called the .

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.

Chapter 1 : Slide 36 Chemical Thermodynamics : Georg Duesberg Proof: rewrite Van der Waals equation for 1 mol of gas, p = RT/(V-b)- a/V2, in terms of reduced variables: RT T a p p = r c − r c 2 2 Substitute for the critical values: VrVc − b Vr Vc

pra RTr 8a a 2 = − 2 2 27b 27Rb(Vr 3b − b) Vr 9b Thus

Thus all gases have the same reduced 8Tr 3 pr = − 2 equation of state (within the Van der Waals 3Vr −1 Vr approximation).

Also:

Zc =pcVc/RTc = 3/8 =0.375

Chemical Thermodynamics : Georg Duesberg Principle of Corresponding States 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 Compression factor plotted using reduced the critical point. variables. Different curves are different TR The 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,

Pcvc 3 Zc = = which is higher than the actual value. RTc 8 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: 2 [p + a/(Vm T)] [Vm - b] = RT

2) The Dieterici Equation : [p exp(a/VmRT)] [Vm - b] = RT

3) Redlich-Kwong RT A p = − 1/ 2 Vm − B T Vm (Vm + B) 4) Peng-Robinson RT α p = − Vm − β Vm (Vm + β ) + β (Vm − β )

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 … Chapter 1 : Slide 41 Chemical Thermodynamics : Georg Duesberg

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) - 2 a/Vm is more realistic for REAL GASES. There are other equations of state which 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 Chapter 1 : Slide 42 Chemical Thermodynamics : Georg Duesberg 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?

PV = nRT PV / nR = T 10 × 2/1×0.082 = 244 2 ⎛ ⎛ n ⎞ ⎞ ⎜ P + a ⎟ V − nb = nRT ⎜ ⎜ ⎟ ⎟( ) ⎝ ⎝V ⎠ ⎠ 2 ⎛ ⎛ n ⎞ ⎞ ⎜ P + a ⎟ V − nb / nR = T ⎜ ⎜ ⎟ ⎟( ) ⎝ ⎝V ⎠ ⎠ 2 ⎛ ⎛ 1 ⎞ ⎞ ⎜1 + 0.14 ⎟ 2 −1×0.0387 /1×0.082 = 240 ⎜ ⎜ ⎟ ⎟( ) ⎝ ⎝ 2 ⎠ ⎠

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