How to Learn How to Derive a Mathematical Expression for the Entropy of an Ideal Gas of Photons


This article shows the details of the derivation of the entropy of an ideal gas of photons based on the equation of forces that govern the motion of photons.

Steps

  1. Image titled Learn How to Derive a Mathematical Expression for the Entropy of an Ideal Gas of Photons Step 1
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    Examine the mathematical derivation of the entropy of an ideal gas of photons is described based on the recently developed equation of forces that govern the motion of light in the space. This equation has the following form: F=F1*(c/v)
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    Learn about this equation which was derived in a previous article from the relativistic energy expression of a photon. The mathematical details of this expansion is also described in detail in that article.
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    Check into one of the immediate consequences of this equation which is the invalidity of the assumption of the theory of relativity about the constancy of the speed of light. Also in a different article the kinetic energy of a given photon was developed using this equation of forces also. The value of the kinetic energy of a photon was shown to follow this equation: (1/2)*m*(v**2)=F1*L
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    Calculate the kinetic energy of a given photon as given by the expression: (1/2)*m*(v**2) . The physical meaning of this equation is describing a form of kinetic energy associated with the force F type. Also it says that we can associate a hypothetical mass m with the given photon. This form of kinetic energy is shown to be equivalent to the work that is done by the force F1 along the distance L.
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    Follow this as it is shown how to develop a mathematical expression for the entropy of an ideal gas of photons based on this equation of forces.
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    Equate this expression of work with the work of expansion of an ideal chemical gas.
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    Take an infinitesimal value of the work F1*dl and equate to an infinitesimal value of the volume expansion work PdV done by the ideal chemical gas. This then gives the following expression: F1*dl=PdV
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    Use the ideal gas equation PV=nRT and isolate P in terms of the volume one gets: P=nRT/V
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    Substituting this expression of the pressure in the equation above gives: F1*dl=(nRT/V)dV
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    Integration of both sides gives the following expression: F1*L=nRT*ln(V)
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    Use this equation to relate the work that is done by the force F1 that acts on the photon to the volume expansion work that is done by an ideal gas of photons. This expression was developed because it is necessary for the derivation of the entropy of the ideal gas of photons.
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    Compare this to entropy of an ideal chemical gas as it is usually derived from the second law of thermodynamics that has the following infinitesimal value: dE=TdS-PdV
    • At constant temperature or for an isothermal process this value is equal zero. Therefore we have at T=Constant
      Image titled Learn How to Derive a Mathematical Expression for the Entropy of an Ideal Gas of Photons Step 12Bullet1
    • CvdT=0
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    Use this to obtain the following expression about the second low of thermodynamics:
    • TdS=PdV
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    Divide both sides of the equation by the temperature T one gets the following expression about the entropy:
    • dS=(P/T)*dV
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    Use the ideal gas equation PV=nRT and isolate the value of P/T one gets the following expression:
    • (P/T)=nR/V
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    Substitute this value of P/T into the above equation gives the following differential equation:
    • dS=(nR/V)*dv
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    Integrate both sides one gets the following expression for the entropy:
    • S=nR*ln(V)
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    Remember that
    • F1*L=nRT*ln(V)
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    Obtain a value of the entropy in terms of the work of the force F1.
    • F1*L=T*S
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    Isolate the entropy from this expression to give the following equation for the entropy:
    • S=F1*L/T
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    Notice that this equation shows the dependence of the entropy of the photons on the temperature.
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    Recall that Q=TdS, then one can immediately get the relation:
    • Q=F1*L
      Image titled Learn How to Derive a Mathematical Expression for the Entropy of an Ideal Gas of Photons Step 22Bullet1
  23. Image titled Learn How to Derive a Mathematical Expression for the Entropy of an Ideal Gas of Photons Step 23
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    Interpret the equation to say that the work of the force F1 is equal to the heat content of the ideal gas.

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