Entropy Physics 202 Professor Lee Carkner Lecture 15

PAL #14 Internal Energy 3 moles of gas, temperature raised from 300 to 400 K He gas, isochorically Q = nCVDT, CV = (f/2)R = (3/2) R Q = (3)(3/2)R(100) = 3740 J He gas, isobarically Q = nCPDT, CP = CV + R = (5/2) R Q = (3)(5/2)R(100) = 6333 J H2 gas, isochorically Q = nCVDT, CV = (5/2) R, f = 5 for diatomic H2 gas, isobarically Q = nCPDT, CP = CV + R = (7/2) R Q = (3)(7/2)R(100) = 8725 J

PAL #14 Internal Energy 4 moles of N2 gas isobaric expansion from 0.45 m3 to 0.78 m3 and 457 K pressure = p =nRT/V = (4)(8.31)(457)/(0.78) = 19475 Pa initial temp = T = pV/nR = (19475)(0.45)/(4)(8.31) = 263.7 K W=pDV = (19475)(0.78-0.45) = 6427 J Q=nCp DT = (4)(7/2)(8.31)(457-263.7) =22489 J adiabatic process starts at the same point, ends where V= 0.78 m3. piVig = pfVfg pf = piVig /Vfg = (19475)(0.45)1.4/(0.78)1.4 = 9017 Pa

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: Perfume diffuses throughout a room 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: DS = Sf –Si = ∫ (dQ/T)

Isothermal Entropy DS = (1/T) ∫ dQ DS = Q/T Like heating something up by 1 degree

Heat Reservoir Something that is too big to change temperature A heat reservoir can gain or lose heat without changing temperature Since Q = mcDT, if m is very large, DT can be very small

Second Law of Thermodynamics (Entropy) Consider objects A and B that exchange heat Q with each other isothermally: We always find that the positive term is always a larger than the negative term, so: DS>0 Entropy always increases

Entropy Problems Using Q/T Need to find heat Sign of DS is sign of Q (positive in and negative out) T constant for phase change or heat reservoir For total entropy, must add all sources and sinks of heat

DS = nRln(Vf/Vi) + nCVln(Tf/Ti) General Entropy From the first law and the ideal gas law, we get DS = nRln(Vf/Vi) + nCVln(Tf/Ti) Note that we only need to know the initial and final conditions, not the path

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

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: Direction in which entropy increases Direction that you do not remember 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 Head towards the Big Crunch Will the others reverse as well? Expand forever

Heat Death Left with white dwarfs, neutron stars and radiation Everything in the universe trying to be same temperature Universe gets more and more disordered Left with white dwarfs, neutron stars and radiation Can live off of compact objects, but eventually will convert them all to heat

Next Time Read: 20.5-20.7

Suppose it is 0 F outside today Suppose it is 0 F outside today. What would the temperature need to be outside tomorrow (in F) to be twice as hot? -34 100 458 510

How much heat does it take to change the temperature of one mole of a monatomic ideal gas 1 degree K in a constant volume process? How much heat does it take to change the temperature of one mole of a monatomic ideal gas 1 degree K in a constant pressure process? 1 J : 1 J 1 J : 12.5 J 12.5 J : 12.5 J 12.5 J : 20.8 J 8.3 J : 16.6 J

What is the change in internal energy for an ideal monatomic gas whose temperature increases 1 degree K in a constant volume process? What is the change in internal energy for an ideal monatomic gas whose temperature increases 1 degree K in a constant pressure process? 1 J : 1 J 1 J : 12.5 J 12.5 J : 12.5 J 12.5 J : 20.8 J 8.3 J : 16.6 J