lecture 2_introduction to statistical thermo.jnt (1)
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statistical thermo class notesTRANSCRIPT
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Advanced Thermodynamics
Yaguo Wang
Assistant Professor
Mechanical Engineering
Lecture 2
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Review Table _ Posted on Canvas
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Ideal Gas _ Heat Capacities and Specific Heats
Approximately,
Heat capacity (CV) and specific heat capacity (cV) for constant volume process
Heat capacity (CP) and specific heat capacity (cP) for constant pressure process
EnergyC
T
v vv v
U uC and c
T T
P PP P
H hC and c
T T
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Specific Heats for Ideal Gases
For an ideal gas, h = h(T)
Similarly, for ideal gas u = u(T)
2
1
T
PP
PT
h dhc
T dTh c dT Always
2
1
T
VV
VT
u duc
T dTu c dT Always
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Specific Heats for Monoatomic Ideal Gases
Constant Specific Heat for Monoatomic Ideal Gas
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Isentropic Process for Monoatomic Ideal Gases
Define k = cP/cV
k 1
2 1
1 2s s
T v
T v
k 1
k2 2
1 1s s
T P
T P
If cP and cV independent of temperature
p v p v
1R=c -c =c (1- )=c (k-1)
k
v
v v
dT dvds [c ]
T v
dT dv =c +c (k-1)
T v
=0
R
dT dPds [c ]
T
dT 1 dP =c c (1- )
T
=0
p
p p
RP
k P
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Quasi-static Processes Assume massless, frictionless and well-insulated piston
Slowly apply increasing force (FP)
to piston causing piston to move
down in compression process,
PCM = (FP/AP + PATM) at all times
Process approximatelyreversible by slowly removing each infinitesimal mass
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Exp. A_Quasi-static Isothermal Compression
Uninsulated piston cylinder sitting in a constant-temperature water bath, a force applied on top of piston compress the monoatomic gases very slowly (quasi-static) from position 1 to 2.
1. Calculate the work in this process with 1st and 2nd
law
2. Verify that same result is reached with =
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Exp. A_Quasi-static Isothermal Compression
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Exp. A_Quasi-static Isothermal Compression
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Exp. A_Quasi-static Isothermal Compression
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Exp. B_Quasi-static Adiabatic Compression
Insulated piston cylinder, a force applied on top of piston compress the monoatomic gases very slowly (quasi-static) from position 1 to 2.
1. Calculate the work in this process with 1st and 2nd
law
2. Verify that same result is reached with =
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Exp. B_Quasi-static Adiabatic Compression
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Exp. B_Quasi-static Adiabatic Compression
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Exp. B_Quasi-static Adiabatic Compression
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Introduction to Statistical Thermo.
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Ways of Arranging Objectives
Problem 1: How many ways can we arrange N
Distinguishable objects? For example, we wish to arrange N books in various ways on a shelf?
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Ways of Arranging Objectives
Problem 2: How many ways can we arrange N
distinguishable objects into r distinguishable boxes? Such that there are N1 objects in the first box, N2 in the second, , and Nr in the rth box?
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Ways of Arranging Objectives
Problem 2: What if without regard to order within the boxes?
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Ways of Arranging Objectives
Problem 3: How many ways can we select N
distinguishable objects from a set of g distinguishable objects? e.g. putting N books on one of two shelves and g-N books on the other.
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Ways of Arranging Objectives
Problem 4: How many ways we can put N indistinguishable objects into g distinguishable boxes? There is no limit on the number of objects in any box?
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Ways of Arranging Objectives
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Ways of Arranging Objectives
Problem 5: How many ways we can put N distinguishable objects into g distinguishable boxes? Each of N different books can be put on any of g shelves.
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Quantum States_Bohr Hydrogen Atom
Main Idea:
Electron energy
States are discrete
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The issue in Thermo is manner in which a fixed number of particles are distributed among the available quantum states (microstates).
Many quantum states have same energy __ number of quantum states at each energy level is called degeneracy.
Thermodynamic Probability
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Microstate: description of a system which relies on the states of each element of the system.
Q1: how many coordinates do we need to describe a system containing ONE particle?
Q2: how many coordinates do we need to describe a system containing N=100 particles?
Thermodynamic Probability
() =
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Macrostate: description of a system which relies on some macroscopic properties.
Think about the 100 particles, each is labeled as 1, 2, 100. Put these 100 particles into five boxes, each with N1, N2, N3, N4, N5 particles.
If we dont care about the label of individual particle,
but only the total number in each box.
Q: how many coordinates do we need to describe this system?
Thermodynamic Probability
() =
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The number of microstates in a given Macrostate
Thermodynamic Probability
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Principle of Equal Priori Probabilities
All microstates of motion
occur with equal frequency.
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Specification of Molecular Microstates
An array of N particles has a total energy U. The energies of the individual particles are assumed to take on the discrete values, , , , , . The number of particles with energy is .
Thus:
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Example:
An array of 2 particles has a total energy of 2, each energy level has a degeneracy of 2. The energies of the individual particles are assumed to take on the discrete values, , , .The number of particles with energy is .How many possible macrostates for this system?
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Molecular Distributions