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Thermodynamics Lecture Series Applied Sciences Education Research Group (ASERG) Faculty of Applied Sciences Universiti Teknologi MARA Kinetic Theory of Gases – Microscopic Thermodynamics Reference: Chap 20 Halliday & Resnick Fundamental of Physics 6 th edition

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Copyrights DR JJ, ASERG, FSG, UiTM Shah Alam, Review – Steam Power Plant Pump Boiler Turbin e Condenser High T Res., T H Furnace q in = q H in out Low T Res., T L Water from river A Schematic diagram for a Steam Power Plant q out = q L Working fluid: Water q in - q out = out - in q in - q out = net,out

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Copyrights DR JJ, ASERG, FSG, UiTM Shah Alam, Review - Steam Power Plant Steam Power Plant High T Res., T H Furnace q in = q H net,out Low T Res., T L Water from river An Energy-Flow diagram for a SPP q out = q L Working fluid: Water Purpose: Produce work, W out, out

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Copyrights DR JJ, ASERG, FSG, UiTM Shah Alam, Review - Steam Power Plant Thermal Efficiency for steam power plants For real engines, need to find q L and q H.

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Copyrights DR JJ, ASERG, FSG, UiTM Shah Alam, Entropy Balance –Steady-flow device Review - Entropy Balance Heat exchanger Q in , Hot water inlet Cold water Inlet Out Case 1 – blue border Case 2 – red border

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Copyrights DR JJ, ASERG, FSG, UiTM Shah Alam, Entropy Balance –Steady-flow device Review - Entropy Balance Heat exchanger: energy balance; Assume ke mass = 0, pe mass = 0 where Q in Case 1

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Copyrights DR JJ, ASERG, FSG, UiTM Shah Alam, Entropy Balance –Steady-flow device Review - Entropy Balance Heat exchanger: energy balance; Assume ke mass = 0, pe mass = 0 where Q in Case 1 Case 2

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Copyrights DR JJ, ASERG, FSG, UiTM Shah Alam, Entropy Balance –Steady-flow device Review - Entropy Balance Heat exchanger: Entropy Balance where Q in Case 1

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Copyrights DR JJ, ASERG, FSG, UiTM Shah Alam, Entropy Balance –Steady-flow device Review - Entropy Balance Heat exchanger: Entropy Balance where Q in Case 2

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Introduction - Objectives 1.State terminologies and their relations among each other for ideal gases. 2.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 Objectives:

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Copyrights DR JJ, ASERG, FSG, UiTM Shah Alam, Microscopic Variables New Way of Looking at Gases 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 explain V. T = 30 C P = kPa T = 30 C P = kPa H 2 O: Sat. liquid

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Copyrights DR JJ, ASERG, FSG, UiTM Shah Alam, Microscopic Variables-Molecular Approach New Way of Looking at Gases Kinetic Theory of Gases Pressure exerted 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

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Copyrights DR JJ, ASERG, FSG, UiTM Shah Alam, Microscopic Variables-Molecular Approach New Way of Looking at Gases Kinetic Theory of Gases: Sizes Mole: the number of atoms contained in 12 g sample of carbon-12 Avogadro’s number: N A =6.02 x atoms/mol Number of moles is N is the ratio of number of molecules with respect to N A High density

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Copyrights DR JJ, ASERG, FSG, UiTM Shah Alam, Microscopic Variables-Molecular Approach New Way of Looking at Gases Kinetic Theory of Gases: Sizes Number of moles is N is the ratio of sample mass to the molar mass, M (kg/kmol) or molecular mass m (kg/atoms) High density Where the molar mass is related to the molecular mass by Avogadro number

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Copyrights DR JJ, ASERG, FSG, UiTM Shah Alam, Ideal Gases New Way of Looking at Gases Low density (mass in 1 m 3 ) gases. Molecules are further apart P gas > T crit Real gases satisfying condition P gas > T crit, have low density and can be treated as ideal gases High density Low density Molecules far apart

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Copyrights DR JJ, ASERG, FSG, UiTM Shah Alam, Ideal Gases New Way of Looking at Gases Equation of State Equation of State - P- -T behaviour P =RT RT P =RT (energy contained by 1 kg mass) where is the specific volume in m 3 /kg, R is gas constant, kJ/kg K, T is absolute temp in Kelvin. High density Low density Molecules far apart

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Copyrights DR JJ, ASERG, FSG, UiTM Shah Alam, Ideal Gases New Way of Looking at Gases Equation of State Equation of State - P- -T behaviour P =RT PV=M sam RT P =RT, since = V/M sam then, P(V/ M sam )=RT. So, PV=M sam RT, in kPa m 3 =kJ. Total energy of a system. Low density High density

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Copyrights DR JJ, ASERG, FSG, UiTM Shah Alam, Ideal Gases New Way of Looking at Gases Equation of State Equation of State - P- -T behaviour PV =M sam RT R u =MRPV = NR u T PV =M sam RT =NMRT=N(MR)T But R u =MR. Hence, can also write PV = NR u T where N N is no of kilomoles, kmol, M M is molar mass in kg/kmole, R R is a gas constant and R u R u =MR= kJ/kmol K R u is universal gas constant; R u =MR= kJ/kmol K Low density High density

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Copyrights DR JJ, ASERG, FSG, UiTM Shah Alam, Ideal Gases New Way of Looking at Gases Equation of State Equation of State - P- -T behaviour PV =NR u T PV = nkT PV =NR u T =NkN A T=(n/N A )(kN A )T. Hence, can also write PV = nkT where N N is no of kilomoles, kmol, n n is no of molecules, k k is Boltzmann constant; R u = kJ/kmol K = R u = kJ/kmol K = kN A k = R u / N A = 1.38 x J/K Low density High density

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Copyrights DR JJ, ASERG, FSG, UiTM Shah Alam, Pressure, Temperature and Root Mean Square Speed New Way of Looking at Gases How is the pressure P that an ideal gas of N moles confined to a cubical box of volume V and held at temperature T, related to the speeds of the molecules?? y m L L L z x Normal To wall Before collision

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Copyrights DR JJ, ASERG, FSG, UiTM Shah Alam, Pressure, Temperature and Root Mean Square Speed New Way of Looking at Gases Assume elastic collision, then after collide with right wall, only x component of velocity will change. Then momentum change is: y MsMs L L L z x Normal To wall After collision

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Copyrights DR JJ, ASERG, FSG, UiTM Shah Alam, Pressure, Temperature and Root Mean Square Speed New Way of Looking at Gases So momentum change received by the wall is: Normal To wall y m L L L z x After collision The time to hit the right wall again is

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Copyrights DR JJ, ASERG, FSG, UiTM Shah Alam, Pressure, Temperature and Root Mean Square Speed New Way of Looking at Gases So average rate of momentum transfer received by the wall due to 1 molecule is: Normal To wall y m L L L z x After collision

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Copyrights DR JJ, ASERG, FSG, UiTM Shah Alam, Pressure, Temperature and Root Mean Square Speed New Way of Looking at Gases The total force along x is the sum due to collision by all N molecules with different speeds. The pressure on the wall is the force exerted for each unit area and is then:

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Copyrights DR JJ, ASERG, FSG, UiTM Shah Alam, Pressure, Temperature and Root Mean Square Speed New Way of Looking at Gases The total force along x is the sum due to collision by all n molecules with different speeds. The pressure on the wall is then: 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:

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Copyrights DR JJ, ASERG, FSG, UiTM Shah Alam, Pressure, Temperature and Root Mean Square Speed New Way of Looking at Gases 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 A is the molar mass, M of the gas mass of 1 mol and L 3 is the volume of the box. So,

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Copyrights DR JJ, ASERG, FSG, UiTM Shah Alam, Pressure, Temperature and Root Mean Square Speed New Way of Looking at Gases But mN A is the molar mass, M of the gas mass of 1 mol and L 3 is the volume of the box. So, Then the pressure is: In the 3D box each molecule has speed along x,y and z direction.

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Copyrights DR JJ, ASERG, FSG, UiTM Shah Alam, Pressure, Temperature and Root Mean Square Speed New Way of Looking at Gases Since there are many molecules in the box each moving with different velocities and in random directions, the average square of velocity components are equal. Finally, Then, Hence

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Copyrights DR JJ, ASERG, FSG, UiTM Shah Alam, Pressure, Temperature and Root Mean Square Speed New Way of Looking at Gases The square root of the average of the square of the velocity is called root-mean-square speed of the molecules. It means square each speed, find the mean, then take its square root. Hence, the pressure is: So,

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Copyrights DR JJ, ASERG, FSG, UiTM Shah Alam, Pressure, Temperature and Root Mean Square Speed New Way of Looking at Gases The square root of the average of the square of the velocity is called root-mean-square speed of the molecules. It means square each speed, find the mean, then take its square root. Hence, the pressure is: So, The rms speed can be determined If P,T is known. Using PV=NR u T

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Copyrights DR JJ, ASERG, FSG, UiTM Shah Alam, Pressure, Temperature and Root Mean Square Speed New Way of Looking at Gases The root mean square is then: Since the square of the root mean square of the velocity is:

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Copyrights DR JJ, ASERG, FSG, UiTM Shah Alam, Pressure, Temperature and Root Mean Square Speed New Way of Looking at Gases Gas (Values taken at T=300K) Molar mass, M (10 -3 kg/kmol) rms, (m/s) Hydrogen (H 2 ) Helium (He) Water vapor (H 2 O) Nitrogen (N 2 ) Oxygen(O 2 ) Carbon dioxide (CO 2 ) Sulphur Dioxide (SO 2 )

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Copyrights DR JJ, ASERG, FSG, UiTM Shah Alam, Temperature-Translational kinetic Energy New Way of Looking at Gases Consider a molecule in the box which are colliding with other molecules and changes speed after collision. It moves with translational kinetic energy at any instant But the average translational kinetic energy is over a period of time is:

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Copyrights DR JJ, ASERG, FSG, UiTM Shah Alam, Temperature-Translational kinetic Energy New Way of Looking at Gases Substitute the rms speed in terms of T, then: Note that the molar mass M=mN A. Note also that R u = kN A. Hence the average translational kinetic energy is: Regardless of mass, all ideal gas molecules at temperature T have the same avg. translational KE.

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