Rudolf Žitný, Ústav procesní a zpracovatelské techniky ČVUT FS 2010

1. This course is approximately at this level CHEMISTRYE182019 CH4 State variables and state equations Some pictures and texts were copied from www.wikipedia.com Rudolf Žitný, Ústav procesní a zpracovatelské techniky ČVUT FS 2010

2. Intermolecular forces CH4 Van der Waals forcesact between individual molecules or between molecule and atoms. VdW forces are much weaker than covalent bonds (typically 400 times less) • Keesom's electrostatic attraction of polar molecules (permanent dipole-dipole) • Debye’s forces between permanent dipole and induced dipole • London dispersion forces between nonpolar molecules (induced temporary polarisation) Hydrogen bondsact between individual molecules or between different parts of a long macromolecule (for example macromolecule of proteins – hydrogen bonds determine spatial arrangement of macromolecule in this case).

3. State variables CH4 State variable is a characteristic of system, that is independent of history (independent of previous chemical reactions, mechanical actions, heat transfer, etc). Some state variables can be directly measured (e.g. pressure, density, temperature), other must be identified from measurable properties and basic physical principles (e.g. conservation of energy).

4. p,v,T – directly measurable state variables CH4 Temperature is a measure of inner kinetic energy of random molecular motion. In case of solids the kinetic energy is the energy of atom vibration, in liquids and gases the kinetic energy includes vibrational, rotational and translational motion. Thermodynamic temperature is expressed in Kelvins [K]. Measurement by Thermocouples (different metals electrically connected generate voltage), RTD (Resistance Temperature Detectors – temperature dependent electrical resistance) – thermistors (semiconductors), Infrared thermometers. Pressure is normal force acting on unit surface, that causes volumetric changes of material. Units are Pascals. Measurement by U-tube manometers, strain-gauge or piezoelectric pressure transducers. Specific volume (reciprocal value is density). Measurement by weighting (electronic balances), pycnometers.

5. p 2 mm (!) Silicon membrane with integrated strain-gauges (pressure transducer Kulite Semiconductors). p,v,T – instruments CH4 Temperature THERMOCOUPLE Pressure PIEZORESISTIVE TRANSDUCERS Specific volume

6. Gibbs phase rule CH4 State of system is characterized by state variables T,p,v,u,h,s (temperature, pressure, specific, volume, inner energy, enthalpy, entropy) and in case of mixtures also by concentrations c1, c2,… Not all state variables are independent. Number of independent variables (DOF, Degree Of Freedom) is given by Gibbs rule NDOF = Ncomponents – Nphases + 2 • 1 component, 1 phase (e.g.gaseous oxygen) NDOF=2 . In this case only two state variables can be selected arbitrarily, e.g. p,v, or p,T or v,T. • 1 component, 2 phases (e.g. equilibrium mixture of water and steam at the state of evaporation/condensation). In this case only one state variable can be selected, e.g. pressure (boiling point temperature is determined by p)

7. p,v,T for ideal gases CH4 State equation of ideal gas (Boyle’s law) Universal gas constant Specific volume Molecular mass Check units p (Pa=N/m2 =J/m3), v (m3/kg), M (g/mol=kg/kmol), T (K) What to substitute for R numerically: 8.314 or 8314? Example: density of air at 300K and atmospheric pressure p=105 Pa.

8. p,v,T for ideal gases CH4 Equivalent formulations Molar volume No.of moles Volume Dalton’s law and partial pressures

9. p,v,T for ideal gases CH4 Partial pressure and molar fraction Molar fraction Example: partial pressure of oxygen in air (21%) at atmospheric pressure:

10. p,v,T for real gases VdW CH4 Van der Waals equation Volume of 1 mol Cohesion pressure It is possible to express pressure explicitly as a function of temperature a volume Volume evaluation (given p,T) requires solution of cubic equation.

11. p,v,T for real gases VdW CH4 Van der Waals equation isotherms Critical point, solution of these two equations give a,b parameters as a function of critical temperature and critical pressure

12. pc,vc,Tc critical point VdW CH4 Solution of a,b from state variables at critical point It is usually easier to measure critical temperature and pressure and therefore Memorize critical temperatures for Water 374 0C Air -141 0C

13. p,v,T real gas Redlich Kwong CH4 Little bit more accurate state equation (modified pressure term)

14. pvT diagram H2O CH4

15. pvT real gases - parameters CH4

16. Treated product Wine feed Wine Water 400MPa Pressure multiplier pvT liquids/solids CH4 Liquids and solids, unlike gases, are characterised by the close packing of molecules - both the phases are called condensed matter. Intermolecular van der Waals forces are very significant, and generally speaking are of an electrical nature: Keesom's electrostatic attraction of polar molecules (dipole-dipole, water), attraction of polar molecules and ions (hydrates), London dispersion forces between nonpolar molecules (induced temporary polarisation) enabling for example liquefaction or solidification of nonpolar noble gases /helium/ at sufficiently low temperatures). Tait’s equation of state Tait’s equation is suitable for description of liquids at very high pressures (of the order of 100 MPa), when even liquids change their volume considerably. E.g. in the technology of high pressure treatment of food products (sterilisation of juices, fermentation of vine).

17. Dmi B Dmi B C D B C D D B C Amaj RELAX CH4 Still under pressure is workoholic working, volume of work expanding like a gas no matter how the temperature is growing he parses in vain thisRidiculous Text.

18. H H H H-C-C-C-H H H H Pv=RT tutorial CH4 Example: Pressure vessel of the volume V=50 m3 contains gaseous propane C3H8 at T=200C and overpressure (gauge pressure) p=0.5 MPa. Barometric pressure 770 torr. Calculate mass of propane. MC3H8=3x12+8=44 kg/kmol Barometric pressure pa=101.3x770/760=102.66 kPa V=50 m3 Absolute pressure p=102.66+500=602.66 kPa Thermodynamic temperature T=30+273.15=293.15 K pV=nRT n=pV/(RT)=602.7x50/(8.314x293.15)=12.36 kmol m = n x MC3H8 =12.36x44= 545 kg

19. Pv=RT VdW tutorial CH4 Example: Methane CH4 at T=250C and density =60 kg/m3. Calculate pressure. MCH4=12+4=16 kg/kmol Specific volume v=1/=1/60 Molar volume = v.MCH4 = 0.267 m3/kmol =60 kg/m3 State equation ideal gas p=8.314x298.15/0.267 =9.284 MPa H H-C-H H Van der Waals a=228.79 kN.m4/kmol2 b=0.04278 m3/kmol = 7.846 MPa

20. Critical Tc VdW tutorial CH4 Example: Calculate critical molar volume and critical temperature for methane CH4 Van der Waals a=228.79 kN.m4/kmol2 b=0.04278 m3/kmol 192 K Critical parameters according to Wikipedia Tc = 190.4 K

21. Pv=RT tutorial Baloon CH4 Example: Calculate load capacity of a baloon filled by hot air. D=20m, T=600C, Te=200C, p=105 Pa. M=29 (air) D m = 599 kg

22. x D V Pv=RT tutorial Syringe CH4 Record time change of temperature of air compressed in syringe. Thermocouple P-pressure transducer Kulite XTM 140 Example: V2/V1=0.5 =cp/cv=1.4 T1=300 K T2=396 K temperature increase 96 K!!