Thermodynamics ecture Series Reference: Chap 0 Halliday & Resnick Fundamental of Physics 6 th edition Kinetic Theory of Gases Microscopic Thermodynamics Applied Sciences Education Research Group (ASERG) Faculty of Applied Sciences Universiti Teknologi MARA email: drjjlanita@hotmail.com http://www.uitm.edu.my/faculties/fsg/drjj1.html
Review Steam Power Plant Working fluid: Water High T Res., T H Furnace q in q H Boiler ω in Pump Turbine ω out Condenser q in -q out ω out - ω in q out q ow T Res., T Water from river q in -q out ω net,out A Schematic diagram for a Steam Power Plant Copyright DRJJ, FSG, UiTM
Review - Steam Power Plant Working fluid: Water High T Res., T H Furnace q in q H Purpose: Produce work, W out, ω out Steam Power Plant ω net,out q out q ow T Res., T Water from river An Energy-Flow diagram for a SPP Copyright DRJJ, FSG, UiTM 3
Review - Steam Power Plant Thermal Efficiency for steam power plants η desired output η required input ω η net q rev in,out 1 q q T T in H q in out ω 1 net q q in q,out out in 1 For real engines, need to find q and q H. Copyright DRJJ, FSG, UiTM 4 q q H
Review - Entropy Balance Entropy Balance Steady-flow device Heat echanger Case 1 blue border Case red border 1 Cold water Inlet 3, Hot water inlet Q in Out 4 Copyright DRJJ, FSG, UiTM 5
Review - Entropy Balance Entropy Balance Steady-flow device 3 Heat echanger: energy balance; m m1 where 3 m 4 m Q in 1 Assume ke mass 0, pe mass 0 Case 1 Q in Q out + in W out W 4 ( mϑ ) eit ( mϑ ) inlet, kw 0 m 4 h4 m 3 h3 + m h m1 h1, m 4 h 4 h 3 m h1 h, ( ) ( ) kw kw Copyright DRJJ, FSG, UiTM 6
Review - Entropy Balance Entropy Balance Steady-flow device 3 Heat echanger: energy balance; where m 4 m 3 m m1 Assume ke mass 0, pe mass 0 m 4 h 4 h 3 m h1 h Q in Q out m θ m 1 θ 1 Q in 0 m h h1, Case 1 ( ) ( ), kw Case ( ) kw 1 kw Copyright DRJJ, FSG, UiTM 7, 4 Q in
Review - Entropy Balance Entropy Balance Steady-flow device Heat echanger: Entropy Balance where Case 1 m m m 4 m 3 1 1 S gen 0 0 + m 4 s4 m 3 s3 + m s m1 s1, S m ( s s ) m ( s s ) gen 4 4 3 + 1, kw Copyright DRJJ, FSG, UiTM 8 3 4 kw K Q in K
Review - Entropy Balance Entropy Balance Steady-flow device Heat echanger: Entropy Balance where Case m m m 4 m 3 1 Q Q out in S gen + m s m1 s1, Tout Tin Q in S gen 0 + m ( s s 1 ) T in 1 kw Copyright DRJJ, FSG, UiTM 9, K kw K 3 4 Q in
Introduction - Objectives Objectives: 1. State terminologies and their relations among each other for ideal gases.. Write the ideal gas equation in terms of the universal gas constant and in terms the Boltzmann constant. 3. Derive and obtain the relationship between pressure and root mean square speed of molecules. 4. Obtain the relationship of rms speed and gas temperature
Microscopic Variables Classical Thermodynamics Properties are macroscopic measurables: P,V,T,U No inclusion of atomic behaviour Did not discuss about the origin of P,T or eplain V. T 30 C P 4.46 kpa H O: Sat. liquid Copyright DRJJ, FSG, UiTM 11
Microscopic Variables-Molecular Approach Kinetic Theory of Gases Pressure eerted by gas related to molecules colliding with walls T and U related to kinetic energies of molecules V filled by gas relate to freedom of motion of molecules. Must look at same number of molecules when measure size of samples High density Copyright DRJJ, FSG, UiTM 1
Microscopic Variables-Molecular Approach Kinetic Theory of Gases: Sizes Mole: the number of atoms contained in 1 g sample of carbon-1 Avogadro s number: N A 6.0 10 3 atoms/mol Number of moles is n is the ratio of number of molecules with respect to N A n High density N N Copyright DRJJ, FSG, UiTM 13 A
Microscopic Variables-Molecular Approach Kinetic Theory of Gases: Sizes Number of moles is n is the ratio of sample mass to the molar mass, M or molecular mass m N M sample n N M A M sample mn Where the molar mass is related to the molecular mass by Avogadro number High density Copyright DRJJ, FSG, UiTM 14 A M mn A
Ideal Gases ow density (mass in 1 m 3 ) gases. Molecules are further apart High density Real gases satisfying condition P gas << P crit ; T gas >> T crit, have low density and can be treated as ideal gases ow density Molecules far apart Copyright DRJJ, FSG, UiTM 15
Ideal Gases Equation Equation of State -P-ν-T behaviour PνRTRT (energy contained by 1 kg mass) where ν is the specific volume in m 3 /kg, R is gas constant, kj/kg K, T is absolute temp in Kelvin. High density ow density Molecules far apart Copyright DRJJ, FSG, UiTM 16
Ideal Gases Equation Equation of State -P-ν-T behaviour PνRT, since ν V/M sam then, P(V/ M sam )RT. So, PVM sam RT, in kpa m 3 kj. Total energy of a system. High density ow density Copyright DRJJ, FSG, UiTM 17
Ideal Gases Equation Equation of State -P-ν-T behaviour PV M sam RT nmrtn(mr)t But R u MR. Hence, can also write PV nr u T where n is no of kilomoles, kmol, M is molar mass in kg/kmole, R is a gas constant and R u is universal gas constant; R u MR 8.314 kj/kmol kmol K High density ow density Copyright DRJJ, FSG, UiTM 18
Ideal Gases Equation Equation of State -P-ν-T behaviour PV nr u T nkn A T(N/N A )(kn A )T. Hence, can also write PV nkt where n is no of kilomoles, kmol, N is no of molecules, k is Boltzmann constant; nr u Nk. R u 8.314 kj/kmol kmol K k R u / N A 1.38 10-3 J/K High density ow density Copyright DRJJ, FSG, UiTM 19
Pressure, Temperature and Root Mean Square Speed How is the pressure P that an ideal gas of n moles confined to a cubical bo of volume V and held at temperature T, related to the speeds of the molecules?? z m y v r Normal To wall Copyright DRJJ, FSG, UiTM 0 Before collision
Pressure, Temperature and Root Mean Square Speed Assume elastic collision, then after collide with right wall, only component of velocity will change. Then momentum change is: p p f p mv mv mv i z m y v r Normal To wall Copyright DRJJ, FSG, UiTM 1 After collision
Pressure, Temperature and Root Mean Square Speed So momentum change received by the wall is: p + t mv The time to hit the right wall again is v z m y v r Normal To wall Copyright DRJJ, FSG, UiTM After collision
Pressure, Temperature and Root Mean Square Speed So average rate of momentum transfer received by the wall due to 1 molecule is: p t p t + mv / v mv F z m y v r Normal To wall Copyright DRJJ, FSG, UiTM 3 After collision
Pressure, Temperature and Root Mean Square Speed So average rate of momentum transfer received by the wall due to N molecules is: N m y v r 1 z Copyright DRJJ, FSG, UiTM 4
Pressure, Temperature and Root Mean Square Speed The total force along is the sum due to collision by all N molecules with different speeds. The pressure on the wall is the force eerted for each unit area and is then: P F mv 1 / + mv / +.. + m P + ( ) v + v + v 3 1.. N mv N / Copyright DRJJ, FSG, UiTM 5
Pressure, Temperature and Root Mean Square Speed The total force along is the sum due to collision by all N molecules with different speeds. The pressure on the wall is then: m P + ( ) v + v + v 3 1.. N But there are N velocities representing N molecules and so we can represent the different speeds by an average speed. Note also that n N/N A. So, N nn A. Then the pressure on the wall is: Copyright DRJJ, FSG, UiTM 6
Pressure, Temperature and Root Mean Square Speed But there are N velocities representing N molecules and so we can represent the different speeds by and average speed. Note also that n N/N A. So, N nn A. Then the pressure on the wall is: But mn mnn ( ) A is the molar mass, A M of the gas mass of 1 mol P v 3 avg and 3 is the volume of the bo. So, P nm V ( ) v avg Copyright DRJJ, FSG, UiTM 7
Pressure, Temperature and Root Mean Square Speed Then the pressure is: P P mnn nm V A 3 ( ) v avg ( ) v avg But mn A is the molar mass, M of the gas mass of 1 mol and 3 is the volume of the bo. So, In the 3D bo each molecule has speed along,y and z direction. v v + v y + v z Copyright DRJJ, FSG, UiTM 8
Pressure, Temperature and Root Mean Square Speed Since there are many molecules in the bo each moving with different velocities and in random directions, the average square of velocity components are equal. y v v v z Then, v v + v + v Hence v v 3 Finally, P nm 3V ( v ) avg Copyright DRJJ, FSG, UiTM 9
Pressure, Temperature and Root Mean Square Speed The square root of the average of the square of the velocity is called root-mean mean-square speed of the molecules. It means square each speed, find the mean, then take its square root. v rms v ( ) avg Hence, the pressure is: So, ( ) avg So, v rms v P nmv 3V rms Copyright DRJJ, FSG, UiTM 30
Pressure, Temperature and Root Mean Square Speed The square root of the average of the square of the velocity is called root-mean mean-square speed of the molecules. It means square each speed, find the mean, then take its square root. v rms v ( ) avg Hence, the pressure is: P nmv 3V rms So, ( ) avg So, v rms v The rms speed can be determined If P,T is known. Using PVnR u T nr T u nmv 3 Copyright DRJJ, FSG, UiTM 31 rms
Pressure, Temperature and Root Mean Square Speed Since the square of the root mean square of the velocity is: v rms 3R u M T The root mean square is then: v rms 3RuT M Copyright DRJJ, FSG, UiTM 3
Pressure, Temperature and Root Mean Square Speed Gas (Values taken at T300K) Hydrogen (H ) Helium (He) Water vapor (H O) Nitrogen (N ) Oygen(O ) Carbon dioide (CO ) Sulphur Dioide (SO ) Molar mass, M (10-3 kg/kmol).0 4.0 18.0 8.0 3.0 44.0 64.1 ν rms, (m/s) 190 1370 645 517 483 41 34 Copyright DRJJ, FSG, UiTM 33
Temperature-Translational kinetic Energy Consider a molecule in the bo which are colliding with other molecules and changes speed after collision. It moves with translational kinetic energy at any instant KE mv But the average translational kinetic energy is over a period of time is: mv m KE avg avg avg m ( ) v v rms Copyright DRJJ, FSG, UiTM 34
Temperature-Translational kinetic Energy Substitute the rms speed in terms of T, then: KE avg m3r u M T m/ 3RuT mn / A 3R Copyright DRJJ, FSG, UiTM 35 u N Note that the molar mass MmN A. Note also that R u kn A. Hence the average translational kinetic energy is: Regardless of mass, all ideal 3RuT 3 KE avg kt gas molecules at temperature N A T have the same avg. translational KE. T A