Chapter 3 Real gases Chapter11111Chapter 1 :1 Slide: Slide 1 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 2 gauge 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 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 Chapter33333 1 : Slide 3 Chemical Thermodynamics : 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 pressures 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 liquid 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 Temperatures the attractive regime is pronounced • higher Temperature ->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 Equation of State Most fundamental and theoretically sound Polynomial expansion Viris (lat.): force (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 Real gas – Van der Waals equation. 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. -> 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 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 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 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 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) 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 mole 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 fluid'. 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 molar volume 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 • 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