Entropy

Physics 202Professor Lee

CarknerLecture 17

PAL #16 Internal Energy 3 moles of gas, temperature raised from 300 to

400 K He gas, isochorically

Q = nCVT, CV = (f/2)R = (3/2) R Q = (3)(3/2)R(100) = 3740 J # 4 for heat, all in translational motion

He gas, isobarically Q = nCPT, CP = CV + R = (5/2) R Q = (3)(5/2)R(100) = 6333 J # 2 for heat, energy in translational and work

H2 gas, isochorically Q = nCVT, CV = (5/2) R, f = 5 for diatomic Q = (3)(5/2)R(100) = 6333 J # 2 for heat, energy into translational and rotational

motion H2 gas, isobarically

Q = nCPT, CP = CV + R = (7/2) R Q = (3)(7/2)R(100) = 8725 J # 1 for heat, energy, into translation, rotation and work

Randomness Classical thermodynamics is deterministic

Every time!

But the real world is probabilistic

It is possible that you could add heat to a system and the temperature could go down

The universe only seems deterministic because the number of molecules is so large that the chance of an improbable event happening is absurdly low

Reversible

Why? The smashing plate is an example of an

irreversible process, one that only happens in one direction

Examples: Heat transfer

Entropy

What do irreversible processes have in common?

The degree of randomness of system is called entropy

In any thermodynamic process that proceeds from an initial to a final point, the change in entropy depends on the heat and temperature, specifically:

S = Sf –Si = ∫ (dQ/T)

Isothermal Entropy In practice, the integral may be hard to compute

Let us consider the simplest case where the

process is isothermal (T is constant):S = (1/T) ∫ dQ

S = Q/T

Like heating something up by 1 degree

Entropy Change Imagine now a simple idealized system

consisting of a box of gas in contact with a heat reservoir

If the system loses heat –Q to the reservoir and the reservoir gains heat +Q from the system isothermally:Sbox = (-Q/Tbox) Sres = (+Q/Tres)

Second Law of Thermodynamics

(Entropy)

S>0 This is also the second law of thermodynamics Entropy always increases Why?

The 2nd law is based on statistics

State Function

Entropy is a property of system

Can relate S to Q and W and thus P, T and V

S = nRln(Vf/Vi) + nCVln(Tf/Ti)

Not how the system changes ln 1 = 0, so if V or T do not change, its

term drops out

Statistical Mechanics

We will use statistical mechanics to explore the reason why gas diffuses throughout a container

The box contains 4 indistinguishable molecules

Molecules in a Box There are 16 ways that the molecules can

be distributed in the box

Since the molecules are indistinguishable there are only 5 configurations

If all microstates are equally probable than the configuration with equal distribution is the most probable

Configurations and Microstates

Configuration I1 microstate

Probability = (1/16)

Configuration II4 microstates

Probability = (4/16)

Probability

There are more microstates for the configurations with roughly equal distributions

Gas diffuses throughout a room because the probability of a configuration where all of the molecules bunch up is low

Multiplicity The multiplicity of a configuration is the

number of microstates it has and is represented by:

W = N! /(nL! nR!)

n! = n(n-1)(n-2)(n-3) … (1)

For large N (N>100) the probability of the equal distribution configurations is enormous

Microstate Probabilities

Entropy and Multiplicity The more random configurations are most

probable

We can express the entropy with Boltzmann’s entropy equation as:

S = k ln W

Sometimes it helps to use the Stirling approximation:

ln N! = N (ln N) - N

Irreversibility Irreversible processes move from a low

probability state to a high probability one

All real processes are irreversible, so entropy will always increases

The universe is stochastic

Arrows of Time

Three arrows of time: Thermodynamic

Psychological

Cosmological

Direction of increasing expansion of the universe

Entropy and Memory

Memory requires energy dissipation as heat

Psychological arrow of time is related to the thermodynamic

Synchronized Arrows Why do all the arrows go in the same direction?

Can life exist with a backwards arrow of time?

Does life only exist because we have a universe with a forward thermodynamic arrow? (anthropic principle)

Fate of the Universe

If the universe has enough mass, its expansion will reverse

Cosmological arrow will go backwards

Universe seems to be open

Heat Death

Entropy keeps increasing

Stars burn out

Can live off of compact objects, but eventually will convert them all to heat

Next Time

Read: 20.5-20.7 Homework: Ch 20, P: 6, 7, 21, 22