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Deriving S(T,V) for an ideal gas

  1. Apr 17, 2015 #1
    1. The problem statement, all variables and given/known data
    Given: Ideal gas equations:
    Find S(T,V) for an ideal gas

    2. Relevant equations
    Ideal gas equations:
    [tex]\begin{array}{l}
    {\rm{}}\\
    U = \frac{3}{2}N{k_B}{\left( {\frac{N}{V}} \right)^{2/3}}\exp \left[ {\frac{S}{{\left( {3/2} \right)N{k_B}}} - {s_0}} \right]{\rm{ }}\\
    T = {\left( {\frac{{\partial U}}{{\partial S}}} \right)_{V,N}} = \frac{U}{{\left( {3/2} \right)N{k_B}}}\\
    \\
    {\rm{Find: }}\\
    {\rm{S = S}}\left( {T,V} \right){\rm{ }}\\
    \\\end{array}[/tex] for an ideal gas

    The answer, according to the book (David Goodstein's new book "Thermal Physics: Energy and Entropy")
    [tex]S = \frac{2}{3}N{k_B}\log T{\left( {\frac{V}{N}} \right)^{2/3}} + {s_0} = S\left( {T,V} \right)[/tex]

    3. The attempt at a solution
    I'm not sure if the answer given in the book is correct and I'm missing something, or if it is an error.
    [tex]\begin{array}{l}
    \\
    U = \frac{3}{2}N{k_B}{\left( {\frac{N}{V}} \right)^{2/3}}\exp \left[ {\frac{S}{{\left( {3/2} \right)N{k_B}}} - {s_0}} \right]{\rm{ }}\\
    T = {\left( {\frac{{\partial U}}{{\partial S}}} \right)_{V,N}} = \frac{U}{{\left( {3/2} \right)N{k_B}}} = \frac{{\frac{3}{2}N{k_B}{{\left( {\frac{N}{V}} \right)}^{2/3}}\exp \left[ {\frac{S}{{\left( {3/2} \right)N{k_B}}} - {s_0}} \right]}}{{\frac{3}{2}N{k_B}}}\\
    = {\left( {\frac{N}{V}} \right)^{2/3}}\exp \left[ {\frac{S}{{\left( {3/2} \right)N{k_B}}} - {s_0}} \right]\\
    \exp \left[ {\frac{S}{{\left( {3/2} \right)N{k_B}}} - {s_0}} \right] = \frac{T}{{{{\left( {\frac{N}{V}} \right)}^{2/3}}}} = T{\left( {\frac{V}{N}} \right)^{2/3}}\\
    \frac{S}{{\left( {3/2} \right)N{k_B}}} - {s_0} = \log \left[ {T{{\left( {\frac{V}{N}} \right)}^{2/3}}} \right]\\
    \frac{S}{{\left( {3/2} \right)N{k_B}}} = \log \left[ {T{{\left( {\frac{V}{N}} \right)}^{2/3}}} \right] + {s_0}\\
    S = \frac{3}{2}N{k_B}\log \left[ {T{{\left( {\frac{V}{N}} \right)}^{2/3}}} \right] + \frac{3}{2}N{k_B}{s_0}\\
    \\
    \\

    \end{array}[/tex]
     
    Last edited: Apr 17, 2015
  2. jcsd
  3. Apr 18, 2015 #2

    robphy

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  4. Apr 19, 2015 #3

    BvU

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    Book is a typo.

    I like David Stroud's treatment here (more here, etc.)
     
  5. Apr 20, 2015 #4
    Thanks BvU and robphy for your opinions and the links you both postsed. I agree that it is a typo. It is kind of funny that this is actually the first problem in the book and there is a typo. I hope that the rest of the book isn't plagued by errors..... fortunately if that turns out to be the case, my class is using a different book, Classical Statistical Thermodynamics by Ashley Carter.
     
    Last edited: Apr 20, 2015
  6. Apr 20, 2015 #5

    BvU

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    Goodstein has a mail address; I'm sure he'll appreciate if you point out stuff he can improve for the next edition !
     
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