1. Thermodynamics Lecture Series
3/31/2017
Thermodynamics Lecture Series
Reference: Chap 20 Halliday & Resnick Fundamental of Physics 6th edition
Assoc. Prof. Dr. J.J.
Kinetic Theory of Gases – Microscopic Thermodynamics
Applied Sciences Education Research Group (ASERG)
Faculty of Applied Sciences
Universiti Teknologi MARA
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2. Review – Steam Power Plant
High T Res., TH
Furnace
Working fluid:
Water
qin = qH
Boiler
Turbine
Pump
out
in
Condenser
qout = qL
qin - qout = out - in
Low T Res., TL
Water from river
qin - qout = net,out
A Schematic diagram for a Steam Power Plant
Copyrights DR JJ, ASERG, FSG, UiTM Shah Alam, 2005

3. Review - Steam Power Plant
High T Res., TH
Furnace
Working fluid:
Water
Purpose:
Produce work,
Wout, out
qin = qH
Steam Power Plant
net,out
qout = qL
Low T Res., TL
Water from river
An Energy-Flow diagram for a SPP
Copyrights DR JJ, ASERG, FSG, UiTM Shah Alam, 2005

4. Review - Steam Power Plant
Thermal Efficiency for steam power plants
For real engines, need to find qL and qH.
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5. Review - Entropy Balance
Entropy Balance –Steady-flow device
Heat exchanger 1 2 4
3, Hot water inlet
Cold water Inlet
Out
Case 1 – blue border
Qin
Case 2 – red border
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6. Review - Entropy Balance
Entropy Balance –Steady-flow device
Qin
Heat exchanger: energy balance;
where
Assume kemass = 0, pemass = 0
Case 1
Copyrights DR JJ, ASERG, FSG, UiTM Shah Alam, 2005

7. Review - Entropy Balance
Entropy Balance –Steady-flow device
Qin
Heat exchanger: energy balance;
where
Assume kemass = 0, pemass = 0
Case 1
Case 2
Copyrights DR JJ, ASERG, FSG, UiTM Shah Alam, 2005

8. Review - Entropy Balance
Entropy Balance –Steady-flow device
Qin
Heat exchanger:
Entropy Balance
where
Case 1
Copyrights DR JJ, ASERG, FSG, UiTM Shah Alam, 2005

9. Review - Entropy Balance
Entropy Balance –Steady-flow device
Qin
Heat exchanger:
Entropy Balance
where
Case 2
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10. Introduction - Objectives
3/31/2017
Introduction - Objectives
Objectives:
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.
Derive and obtain the relationship between pressure and root mean square speed of molecules.
Obtain the relationship of rms speed and gas temperature
Copyrights DR JJ, ASERG, FSG, UiTM Shah Alam, 2005

11. New Way of Looking at Gases
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 explain V.
T = 30 C
P = kPa
H2O:
Sat. liquid
Copyrights DR JJ, ASERG, FSG, UiTM Shah Alam, 2005

12. New Way of Looking at Gases
Microscopic Variables-Molecular Approach
High density
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
Copyrights DR JJ, ASERG, FSG, UiTM Shah Alam, 2005

13. New Way of Looking at Gases
Microscopic Variables-Molecular Approach
High density
Kinetic Theory of Gases: Sizes
Mole: the number of atoms contained in 12 g sample of carbon-12
Avogadro’s number:
NA =6.02 x 1023 atoms/mol
Number of moles is
N is the ratio of number of molecules with respect to NA
Copyrights DR JJ, ASERG, FSG, UiTM Shah Alam, 2005

14. New Way of Looking at Gases
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 (kg/kmol) or molecular mass m (kg/atoms)
High density
Where the molar mass is related to the molecular mass by Avogadro number
Copyrights DR JJ, ASERG, FSG, UiTM Shah Alam, 2005

15. New Way of Looking at Gases
High density
Low density Molecules far apart
Ideal Gases
Low density (mass in 1 m3) gases.
Molecules are further apart
Real gases satisfying condition Pgas << Pcrit; Tgas >> Tcrit , have low density and can be treated as ideal gases
Copyrights DR JJ, ASERG, FSG, UiTM Shah Alam, 2005

16. New Way of Looking at Gases
High density
Low density
Molecules far apart
Ideal Gases
Equation of State - P--T behaviour
P=RT (energy contained by 1 kg mass) where  is the specific volume in m3/kg, R is gas constant, kJ/kgK, T is absolute temp in Kelvin.
Copyrights DR JJ, ASERG, FSG, UiTM Shah Alam, 2005

17. New Way of Looking at Gases
Ideal Gases
Low density
High density
Equation of State - P--T behaviour
P=RT , since = V/Msam then, P(V/ Msam)=RT. So, PV=MsamRT, in kPam3=kJ.
Total energy of a system.
Copyrights DR JJ, ASERG, FSG, UiTM Shah Alam, 2005

18. New Way of Looking at Gases
Ideal Gases
Low density
High density
Equation of State - P--T behaviour
PV =MsamRT =NMRT=N(MR)T But Ru=MR. Hence, can also write PV = NRuT where
N is no of kilomoles, kmol,
M is molar mass in kg/kmole ,
R is a gas constant and
Ru is universal gas constant; Ru=MR= kJ/kmolK
Copyrights DR JJ, ASERG, FSG, UiTM Shah Alam, 2005

19. New Way of Looking at Gases
Ideal Gases
Low density
High density
Equation of State - P--T behaviour
PV =NRuT =NkNAT=(n/NA)(kNA)T. Hence, can also write PV = nkT where
N is no of kilomoles, kmol,
n is no of molecules,
k is Boltzmann constant;
Ru = kJ/kmolK = kNA
k = Ru / NA = 1.38 x J/K
Copyrights DR JJ, ASERG, FSG, UiTM Shah Alam, 2005

20. New Way of Looking at Gases
Pressure, Temperature and Root Mean Square Speed y m L z x
Normal
To wall
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??
Before collision
Copyrights DR JJ, ASERG, FSG, UiTM Shah Alam, 2005

21. New Way of Looking at Gases
Pressure, Temperature and Root Mean Square Speed y Ms L z x
Normal
To wall
After collision
Assume elastic collision, then after collide with right wall, only x component of velocity will change. Then momentum change is:
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22. New Way of Looking at Gases
Pressure, Temperature and Root Mean Square Speed y m L z x
So momentum change received by the wall is:
Normal
To wall
The time to hit the right wall again is
After collision
Copyrights DR JJ, ASERG, FSG, UiTM Shah Alam, 2005

23. New Way of Looking at Gases
Pressure, Temperature and Root Mean Square Speed y m L z x
So average rate of momentum transfer received by the wall due to 1 molecule is:
Normal
To wall
After collision
Copyrights DR JJ, ASERG, FSG, UiTM Shah Alam, 2005

24. New Way of Looking at Gases
Pressure, Temperature and Root Mean Square Speed
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:
Copyrights DR JJ, ASERG, FSG, UiTM Shah Alam, 2005

25. New Way of Looking at Gases
Pressure, Temperature and Root Mean Square Speed
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/NA. So, n =NNA. Then the pressure on the wall is:
Copyrights DR JJ, ASERG, FSG, UiTM Shah Alam, 2005

26. New Way of Looking at Gases
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/NA. So, n =NNA. Then the pressure on the wall is:
But mNA is the molar mass, M of the gas mass of 1 mol and L3 is the volume of the box. So,
Copyrights DR JJ, ASERG, FSG, UiTM Shah Alam, 2005

27. New Way of Looking at Gases
Pressure, Temperature and Root Mean Square Speed
Then the pressure is:
But mNA is the molar mass, M of the gas mass of 1 mol and L3 is the volume of the box. So,
In the 3D box each molecule has speed along x,y and z direction.
Copyrights DR JJ, ASERG, FSG, UiTM Shah Alam, 2005

28. New Way of Looking at Gases
Pressure, Temperature and Root Mean Square Speed
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.
Then,
Hence
Finally,
Copyrights DR JJ, ASERG, FSG, UiTM Shah Alam, 2005

29. New Way of Looking at Gases
Pressure, Temperature and Root Mean Square Speed
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.
So,
Hence, the pressure is:
Copyrights DR JJ, ASERG, FSG, UiTM Shah Alam, 2005

30. New Way of Looking at Gases
Pressure, Temperature and Root Mean Square Speed
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.
So,
The rms speed can be determined
If P,T is known. Using PV=NRuT
Hence, the pressure is:
Copyrights DR JJ, ASERG, FSG, UiTM Shah Alam, 2005

31. New Way of Looking at Gases
Pressure, Temperature and Root Mean Square Speed
Since the square of the root mean square of the velocity is:
The root mean square is then:
Copyrights DR JJ, ASERG, FSG, UiTM Shah Alam, 2005

32. New Way of Looking at Gases
Pressure, Temperature and Root Mean Square Speed
Gas
(Values taken at T=300K)
Molar mass, M
(10-3 kg/kmol)
rms,
(m/s)
Hydrogen (H2)
2.02
1920
Helium (He)
4.0
1370
Water vapor (H2O)
18.0
645
Nitrogen (N2)
28.0
517
Oxygen(O2)
32.0
483
Carbon dioxide (CO2)
44.0
412
Sulphur Dioxide (SO2)
64.1
342
Copyrights DR JJ, ASERG, FSG, UiTM Shah Alam, 2005

33. New Way of Looking at Gases
Temperature-Translational kinetic Energy
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:
Copyrights DR JJ, ASERG, FSG, UiTM Shah Alam, 2005

34. New Way of Looking at Gases
3/31/2017
New Way of Looking at Gases
Temperature-Translational kinetic Energy
Substitute the rms speed in terms of T, then:
Note that the molar mass M=mNA. Note also that Ru = kNA. Hence the average translational kinetic energy is:
Regardless of mass , all ideal gas molecules at temperature T have the same avg. translational KE.
Copyrights DR JJ, ASERG, FSG, UiTM Shah Alam, 2005
Copyrights DR JJ, ASERG, FSG, UiTM Shah Alam, 2005