Thermodynamics Lecture 26

8/19/2019 Thermodynamics Lecture 26

1/13

4/11/20

BITS PilaniPilani Campus

–


2/13

4/11/20

Principle of entropy increase

Consider a closed system in adiabatic enclosure undergoing

. b –Sa?

• Return the system to state a by a reversible process, withany heat exchange happening with a single reservoir at Tr .Clearly Qb→a ≠ 0

• or e cyc e, b→a = b→a + b→a• Can Qb→a be positive? No, but why not?

• Because Kelvin-Planck statement will be violated

• Therefore Qb→a < 0, ie., Sa – Sb < 0

• ., a.enclosure cannot decrease. Isolated system a particular case

• (∆S)isolated ≥ 0

BITSPilani, Pilani Campus


3/13

4/11/20

Entropy creation and transfer

• Let a closed system undergo an infinitesimal process

ur ng w c ere s a ea rans er w a reservo r aTr.

• dSsystem + dSreservoir ≥0. But dSr = - δQ/Tr

• dSs + dSr = δSgen ≥ 0, defining the entropy generation

• s r gen

• Entropy transfer accompanies heat transfer

Process Entropy

changeEntropy

transferEntropycreation

Work Lost

work


1 R ln 2 R ln 2 0 RT ln 2 0

2 R ln 2 0 R ln 2 0 ?

3 R ln 2 0.5 R 0.19 R 0.5 RT ?


4/13

4/11/20

Clausius inequality

s r gen, gen ,

• dSs ≥ δQ/Tr , the Clausius inequality (meaning?)

• For a cycle ∫ δQ/Tr ≤ 0



5/13

4/11/20

What is entropy?

Reversible work versus reversible heat transfer

Consider an ideal gas (non-interacting point particles)

What is the effect of a reversible adiabatic decrease in the

volume?

What is the effect of a reversible transfer of heat at constant

volume?

What happens in the course of the free expansion process

that we have discussed in some detail?

What is entropy?



6/13

4/11/20

Spontaneity and Equilibrium

Isolated system spontaneously evolves to states of greater

entropy

What therefore is the criterion that characterizes the

equilibrium state of an isolated system?

• Maximum Entropy



7/13

4/11/20

Irreversibility and entropy

• Due to nature of matter at microscopic level

• Large scale natural processes eventually some kind of mixing –

of particles or over available space

• In some cases the mixing or sharing is of energy, as in the

equa za on o e empera ure o oc s

• In friction, kinetic energy of body as a whole into random energy

of component molecules

• Chemical reaction – at e uilibrium the available ener is

distributed over the available quantum states in the most random

possible manner



8/13

4/11/20

Entropy and Mixing - Spreading

Two distinct kinds

• Spreading of particles over space

• Spreading of available energy of system among available

articles and their states

The two effects may oppose one another – under adiabatic

isolation, equilibrium determined by maximum in the overall‘randomness’ or s readin – confi urational + thermal

Illustrate with discussion of diffusion



9/13

4/11/20

Natural processes and Microstates

• Thermodynamic state or macrostate

• Quantum state or microstate

• A macrostate comprises an astronomical number of

microstates in a t ical macrosco ic s stem

• Natural process is one in which there is an increase in Ω,

the number of microstates

• ,

microstate the system is in



10/13

4/11/20

Entropy andΩ

• Adiabatic mixing of A and B

• Increase in Ω, which in turn is a measure of the decrease

in information about the actual state

• Increase in entro S

• Ω = Ω(U,V,N) and S = S(U,V,N), both are functions of

state. How are they related to one another?

• = –



11/13

4/11/20

Entropy andΩ

• S = k ln Ω. In previous example, assuming no energetic

distinctions, ∆S = k ln(70)

• S as a measure of thermodynamic probability, also

measured by Ω

• Mixed-up-ness, order and disorder – case of adiabatic

change of super cooled liquid water to ice – greater thermaldisorder, greater configurational order

•Increase of S – system spread over larger number of

possible quantum states – loss of information



12/13

4/11/20

Ideal monatomic gas

• Non-interacting structure-less point particles

• Translational states, given by particle in box model, very

closely spaced, virtually a continuum

• Translational state ener ies de end on the size ie.

volume of the box – work versus heat

• Accessible states for a particle, z = (2πmkT/h2

)3/2

V

• = N ,

of thermally accessible quantum states or microstates



13/13

4/11/20

Ideal monatomic gas

S = Nk ln V + 3/2 Nk ln T + constant

• If T1 = T2, ∆S = Nk ln(V2/V1)

Comparing to the result we had earlier, for 1 mole, can find k.

Entro increase due to more accessible levels at lower

energy

• If V1 = V2, ∆S = 3/2 Nk ln(T2/T1)

• If ∆S = 0, TV2/3 = constant


Documents

Thermodynamics Lecture 26