lecture 2_introduction to statistical thermo.jnt (1)

Advanced Thermodynamics

Yaguo Wang

Assistant Professor

Mechanical Engineering

Lecture 2

Review Table _ Posted on Canvas

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

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

Specific Heats for Monoatomic Ideal Gases

Constant Specific Heat for Monoatomic Ideal Gas

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

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

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 =

Exp. A_Quasi-static Isothermal Compression

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 =

Exp. B_Quasi-static Adiabatic Compression

Introduction to Statistical Thermo.

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?


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?


Problem 2: What if without regard to order within the boxes?


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.


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?


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.

Quantum States_Bohr Hydrogen Atom

Main Idea:

Electron energy

States are discrete

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

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?


() =

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?


() =

The number of microstates in a given Macrostate


Principle of Equal Priori Probabilities

All microstates of motion

occur with equal frequency.

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:

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?

Molecular Distributions

lecture 2_introduction to statistical thermo.jnt (1)

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