Entropy Physics 202 Professor Lee Carkner Ed by CJV Lecture -last

Entropy  What do irreversible processes have in common?  They all progress towards more randomness  The degree of randomness of system is called entropy  For an irreversible process, entropy always increases  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  Need to know Q as a function of T  Let us consider the simplest case where the process is isothermal (T is constant):  S = (1/T) ∫ dQ  S = Q/T  This is also approximately true for situations where temperature changes are very small  Like heating something up by 1 degree

State Function  Entropy is a property of system  Like pressure, temperature and volume  Can relate S to Q and thus to  E int & W and thus to P, T and V  S = nRln(V f /V i ) + nC V ln(T f /T i )  Change in entropy depends only on the net system change  Not how the system changes  ln 1 = 0, so if V or T do not change, its term drops out

Entropy Change  Imagine now a simple idealized system consisting of a box of gas in contact with a heat reservoir  Something that does not change temperature (like a lake)  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)  If we try to do this for real we find that the positive term is always a little larger than the negative term, so:  S>0  This is also the second law of thermodynamics  Entropy always increases  Why?  Because the more random states are more probable  The 2nd law is based on statistics

Reversible  If you see a film of shards of ceramic forming themselves into a plate you know that the film is running backwards  Why?  The smashing plate is an example of an irreversible process, one that only happens in one direction  Examples:  A drop of ink tints water  Perfume diffuses throughout a room  Heat transfer

Randomness  Classical thermodynamics is deterministic  Adding x joules of heat will produce a temperature increase of y degrees  Every time!  But the real world is probabilistic  Adding x joules of heat will make some molecules move faster but many will still be slow  It is possible that you could add heat to a system and the temperature could go down  If all the molecules collided in just the right way  The universe only seems deterministic because the number of molecules is so large that the chance of an improbable event happening is absurdly low

Statistical Mechanics  Statistical mechanics uses microscopic properties to explain macroscopic properties  We will use statistical mechanics to explore the reason why gas diffuses throughout a container  Consider a box with a right and left half of equal area  The box contains 4 indistinguishable molecules

Molecules in a Box  There are 16 ways that the molecules can be distributed in the box  Each way is a microstate  Since the molecules are indistinguishable there are only 5 configurations  Example: all the microstates with 3 in one side and 1 in the other are one configuration  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  The equal distribution configurations are thus more probable  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:  = N! /(n L ! n R !)  Where N is the total number of molecules and n L and n R are the number in the right or left half n! = n(n-1)(n-2)(n-3) … (1)  Configurations with large W are more probable  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  They also have the highest entropy  We can express the entropy with Boltzmann’s entropy equation as: S = k ln W  Where k is the Boltzmann constant (1.38 X J/K)  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  Because of probability, they will not move back on their own  All real processes are irreversible, so entropy will always increases  Entropy (and much of modern physics) is based on statistics  The universe is stochastic

Engines and Refrigerators  An engine consists of a hot reservoir, a cold reservoir, and a device to do work  Heat from the hot reservoir is transformed into work (+ heat to cold reservoir)  A refrigerator also consists of a hot reservoir, a cold reservoir, and a device to do work  By an application of work, heat is moved from the cold to the hot reservoir

Refrigerator as a Thermodynamic System  We provide the work (by plugging the compressor in) and we want heat removed from the cold area, so the coefficient of performance is: K = Q L /W  Energy is conserved (first law of thermodynamics), so the heat in (Q L ) plus the work in (W) must equal the heat out (|Q H |): |Q H | = Q L + W W = |Q H | - Q L  This is the work needed to move Q L out of the cold area

Refrigerators and Entropy  We can rewrite K as: K = Q L /(Q H -Q L )  From the 2nd law (for a reversible, isothermal process): Q H /T H = Q L /T L  So K becomes: K C = T L /(T H -T L )  This the the coefficient for an ideal or Carnot refrigerator  Refrigerators are most efficient if they are not kept very cold and if the difference in temperature between the room and the refrigerator is small