Entropy Physics 202 Professor Lee Carkner Lecture 17

PAL #16 Internal Energy  3 moles of gas, temperature raised from 300 to 400 K  He gas, isochorically  Q = nC V  T, C V = (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 = nC P  T, CP = C V + R = (5/2) R  Q = (3)(5/2)R(100) = 6333 J  # 2 for heat, energy in translational and work  H2 gas, isochorically  Q = nC V  T, 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 = nC P  T, CP = C V + 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 = S f –S i = ∫ (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:  S box = (-Q/T box )  S res = (+Q/T res )

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(V f /V i ) + nC V ln(T f /T i )   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 I 1 microstate Probability = (1/16) Configuration II 4 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! /(n L ! n R !)  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:  Homework: Ch 20, P: 6, 7, 21, 22