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

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Physics 202

Professor Lee Carkner

Ed by CJV

Lecture -last




  • 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:

    DS = Sf –Si = ∫ (dQ/T)

Isothermal entropy

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):

    DS = (1/T) ∫ dQ

    DS = Q/T

  • This is also approximately true for situations where temperature changes are very small

    • Like heating something up by 1 degree

State function

State Function

  • Entropy is a property of system

    • Like pressure, temperature and volume

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

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

  • 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

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:

    DSbox = (-Q/Tbox) DSres = (+Q/Tres)

Second law of thermodynamics entropy

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:


  • 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



  • 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



  • 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

  • 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

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

Configurations and Microstates

Configuration I

1 microstate

Probability = (1/16)

Configuration II

4 microstates

Probability = (4/16)



  • 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



  • The multiplicity of a configuration is the number of microstates it has and is represented by:

    W = N! /(nL! nR!)

    • Where N is the total number of molecules and nL and nR 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

Microstate Probabilities

Entropy and multiplicity

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 10-23 J/K)

  • Sometimes it helps to use the Stirling approximation:

    ln N! = N (ln N) - N



  • 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

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

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 = QL/W

  • Energy is conserved (first law of thermodynamics), so the heat in (QL) plus the work in (W) must equal the heat out (|QH|):

    |QH| = QL + W

    W = |QH| - QL

  • This is the work needed to move QL out of the cold area

Refrigerators and entropy

Refrigerators and Entropy

  • We can rewrite K as:

    K = QL/(QH-QL)

  • From the 2nd law (for a reversible, isothermal process):

    QH/TH = QL/TL

  • So K becomes:

    KC = TL/(TH-TL)

    • 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

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