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1 Chapter 3 Real gases Chapter : 1 : Slide 1 1

2 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 (CO ) F A p Thermo meter Pressure gauge

3 Deviations from ideality can be described by the COMPRESSION FACTOR, Z (sometimes called the compressibility). Z pv/(nrt) pv m /(RT) For ideal gases Z 1 Pressure region I (very Low) Molecules have large separations -> no interactions -> Ideal Gas Behavior: Z 1 II (moderate) III (high) Molecules are close -> attractive forces apply -> The gas occupies less volumes as expected from Boyles law: Z<1 Molecules compressed highly -> repulsive forces dominate -> hardly further decrease in volume Z>1 Chapter : Slide 3

4 Microscopic interpretation: Leonard Jones Potential When p is very high, r is small so shortrange 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. Chapter 1 : Slide 4

5 Real Gases: What happens if we press down the piston ( at 0 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 liquid observed E F : steep in crease in P, only liquid visible (At contact molecules repel each other condensed states have volume!) 5 The line C D E is the vapour pressure of a liquid at this tempeature

6 Deviations from ideality can be described by the COMPRESSION FACTOR, Z (sometimes called the compressibility). Z pv/(nrt) pv m /(RT) For ideal gases Z 1 Attractive forces vary with nature of gas At High Pressures repelling forces dominate Z

7 Deviations from ideality can be described by the COMPRESSION FACTOR, Z (sometimes called the compressibility). Z pv/(nrt) pv m /(RT) For ideal gases Z 1 At Low Temperatures the attractive regime is pronounced higher Temperature ->faster motion -> less interaction Z 7

8 Boyle Temperatur The temperature at which this occurs is the Boyle temperature, T B, and then the gas behaves ideally over a wider range of p than at other temperatures. Each gas has a characteristic T B, e.g. 3 K for He, 347 K for air, 715 K for CO. 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

9 B 0 at Boyle temperature Virial Equation of State Most fundamental and theoretically sound Polynomial expansion Viris (lat.): force (Kammerling Onnes 1901) Virial coefficients: p V m RT (1 + B p + C p +...) i.e. p V m RT (1 + B/V m + C/V m +...) 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/V m >> C/V m Also allow derivation of exact correspondence between virial coefficients and intermolecular interactions

10 Real gas Van der Waals equation. ideal gas : PV nrt ( P + x)( V y) nrt P + a n V ( V nb) nrt or P RT V b a m V m Johannes Diderik van der Waals got the Noble price in physics in 1910 Chapter : Slide 10

11 Real gas Van der Waals equation: b 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 (V id V obs - const.) or V id V obs - nb b is a constant for different gases Very roughly, b 4/3 πr 3 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 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. P ( V nb) nrt Chapter 1 : Slide 11

12 Real gas Van der Waals equation: a ) 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), where a depends on the type of gas. [Note: a/v is called the internal pressure of the gas]. Real gas molecules do attract one another (P id P obs + constant) P id P obs + a (n / V) a is also different for different gases n V a describes attractive force between pairs a of molecules. Goes as square of the concentration (n/v).

13 P + a n V 13 (note units) Van der Waals equation of state ( V nb) Substance a/(atm dm 6 mol ) b/(10 dm 3 mol 1 ) Air Ammonia, NH Argon, Ar Carbon dioxide, CO Ethane, C H Ethene, C H Helium, He Hydrogen, H Nitrogen, N Oxygen, O Xenon, Xe nrt 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 or P RT V b a m V m If 1 mole of nitrogen is confined to l and is at P10atm what is T ideal and T VdW? Tip: R 0.08dm 3 atmk -1 mol -1

14 CONDENSATION or LIQUEFACTION This demonstrates that there are attractive forces between gas molecules, if they are pushed close enough together. E.G. CO liquefies under pressure at room temperature. Above 31 0 C no amount of pressure will liquefy CO : this is the CRITICAL TEMPERATURE, T c. Chapter 1 : Slide 14

15 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 C. The critical point is marked with a star. T c, p c and V m,c are the critical constants for the gas. The isotherm at T c has a horizontal inflection at the critical point dp/dv 0 and d p/dv 0. Chapter 1 :

16 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 fluid'. The isotherm at T c has a horizontal inflection at the critical point dp/dv 0 and d p/dv 0. Consider 1 mol of gas, with molar volume V, at the critical point (T c, p c, V c ) 0 dp/dv -RT c (V c -b) - + av c -3 0 d p/dv RT c (V c -b) -3-6aV c -4 The solution is V c 3b, p c a/(7b ), T c 8a/(7Rb). 16

17 Critical Point drying Applications: TEM sample prep, porous materials, MEMS Chapter : Slide 17

18 CNT

19 Metal contacts on CNT

20 Etch

21 Freely suspended CNT Etch

22 TEM electron beam Freely suspended CNT Etch

23 Protective resist Etch

24 Protective resist Etch

25

26

27

28

29 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..)

30 Combined TEM and Raman investigations on individual SWCNTs

31 Maxwell Construction Below T c 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

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 TT c. Can derive (by setting 1 st and nd derivatives of equation to zero) expression for critical constants V c 3b, p c a/7b, T c 8a/7Rb Can derive expression for the Boyle Temperature T B a/rb 3

33 33 Can derive (by setting 1 st and nd derivatives of equation to zero) expression for critical constants 0 0.,. c T c T T T v p and v p e i we have, v a b v RT P ( ) 3 v a b v RT v p T + ( ) v a b v RT v p and T At critical points the above equation reduces to ( ) v a b v RT ( ) v a b v RT and Features of vdw equation

34 By dividing those equations and simplifying we get b v c 3 RTc a We also can say Pc v b v Substituting for b and solving for a from nd derivative we get, a 9RT c v c Substituting these expressions for a and b in equation (P(c) and solving for v c, we get v c 3RT 8p c c Features of vdw equation b RT 8p c c and a 7 64 Note: Usually constants a and b for different gases are given. R p T c c Chapter : Slide 34

35 Critical constants p c atm V c cm 3 /mol T c K Z c T B K Ar CO He O The general Van der Waals pvt surface

36 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 p r p/p c T r T/T c V r V m /V m,c Van der Waals hoped that different gases confined to the same V r at the same T r would have the same p r. Chapter 1 : Slide 36

37 p Proof: rewrite Van der Waals equation for 1 mol of gas, p RT/(V-b)- a/v, in terms of reduced variables: r p c RTrTc V V b r c a V V r c Substitute for the critical values: pra 7b RTr 8a 7Rb(V 3b r b) V r a 9b Thus p r 8Tr 3V 1 r 3 V r Thus all gases have the same reduced equation of state (within the Van der Waals approximation). Also: Z c p c V c /RT c 3/

38 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 v R f (P R,T R ). 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. Compression factor plotted using reduced variables. Different curves are different T R

39 The compressibility factor of any gas is a function of only two properties, usually temperature and pressure so that Z 1 f (T R, P R ) except near the critical point. This is the basis for the generalized compressibility chart. The generalized compressibility chart is plotted with Z versus P R for various values of T R. This is constructed by plotting the known data of one or more gases and can be used for any gas.

40 It may be seen from the chart that the value of the compressibility factor at the critical state is about 0.5. Note that the value of Z obtained from Van der waals equation of state at the critical point, Z c P v c RT c c 3 8 which is higher than the actual value. The following observations can be made from the generalized compressibility chart: Ø At very low pressures (P R <<1), the gases behave as an ideal gas regardless of temperature. Ø At high temperature (T R > ), ideal gas behaviour can be assumed with good accuracy regardless of pressure except when (P R >> 1). Ø The deviation of a gas from ideal gas behaviour is greatest in the vicinity of the critical point.

41 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/(v m T)] [V m - b] RT ) The Dieterici Equation : [p exp(a/v m RT)] [V m - b] RT 3) Redlich-Kwong 4) Peng-Robinson p p RT V RT V β V m A ( V 1/ m B T Vm m + m B) α ( V + β ) + β ( V β ) m m 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

42 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/(V m -b) - a/v m 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) III (high) Molecules are close -> attractive forces apply -> The gas occupies less volumes as expected from Boyles law: Z<1 Molecules compressed highly -> repulsive forces dominate -> hardly further decrease in volume Z>1 Chapter 1 : Slide 4

43 Real gas Van der Waals equation. For nitrogen a0.14 and b3.87x10-5. If 1.0 mole of nitrogen is confined to.00l and is at P10atm what is T ideal and T VdW? PV P P 1 + a + a nrt n V n V PV / nr T 1 ( V nb) ( V nb) 10 / nrt / nr T 44 ( ) / Under these conditions the temperature only changes by ~1%. Chapter 1 : Slide 43

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