Find an expression for S(T,x) from tension and specific heat

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acorazarxf · Accepted Answer

The route of finding             ∂      S              ∂      x      and             ∂      S              ∂      T       is the more practical route.             ∂      S              ∂      x       can be found using a simple Maxwell relation for the Helmholtz free energy:  d  A  =  d  U  −  d  (  T  S  )  =  J  d  x  −  S  d  T                                    ∂          A                          ∂          x                    |        T    =  J          a    n    d                                          ∂          A                          ∂          T                    |        x    =  −  S  .So then   −            ∂      J              ∂      T        =            ∂      S              ∂      x        =  b  −  c  x. The other relation is obtained from the equation for specific heat:  C  =            δ      Q              ∂      T        =      C    x    =  A  T  T  d  S  =  δ  Q  =  A  T  d  T            ∂      S              ∂      T        =  A  .Integrating both expressions gives   S  (  T  ,  x  )  =  b  x  −            c              x        2              2    +  f  (  T  )  =  A  T  +  g  (  x  )  =  A  T  +  b  x  −            c              x        2              2    +  C. Lastly, you simply put in   (  T  ,  x  )  =  (  0  ,  0  ) and get that   C  =      S    0

Find an expression for S(T,x) from tension and specific heat

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