Chapter6a.pdf

CHAPTER 6

Tds Equations

Entropy and Available Energy

Consider two identical bodies, each of mass m, and

heat capacity c_P, but one at temperature T₁, and the other at temperature T₂. When placed in contact,

heat flows from the hotter to the cooler, until they come to a common temperature,

2

2 1 T

T T_f  

Entropy Change

3

1 2

1 2

2 ln

2

P

T T

S mc

T T

 _ 

  _{ }

 

1 2

1 2

ln ln

f f

T T _{f f}

P P

T T

T T

dT dT

S mc mc

T T T T

 _{   }

 

  _  _  _{  }  _{ } 





2 2

1 2 _{1 2 1 2}

ln f ln f 2 ln f

P P P

T T T

mc mc mc

T T _{T T T T}

 _{ }  _{ }

  _{ }

 _{ }  _{ }  _{ }  _{ } 

Entropy Increased

4

1 2

2

T

 

T

TT

2 1 2

(

T



T

)



0

2 2

1 2

2

0

T



T



TT



2 2

1 2

2

4

“Lost” Work

5

2 2

1 2

2 1 2 1

ln Tf 0 Tf 1 T_f T T

T T T T

 

   

     

Run a Carnot engine between the two bodies.

2 1 2 1

2 1 2 1

0 _P

dQ dQ dT dT

mc

T T T T

 

   _  _

 

2 1

2 1

2 1 2 1

ln ln 0

f f

T T

f f

T T

T T

dT dT

T T T T

   

  _{ }  _{ } 

   

Amount of Work

6

1 2

1 2

2

2

P

T T

W  mc _   T T _

 

2 1

W  Q  Q

2 1

( ) ( )

P f f

W  mc _ T T  T T _

1 2 2

P f

W  mc _T  T T _

1 2 2 1 2

P

Tds Equations

7

v P

P v

P v

c c

T T

Tds c dv c dP dv dP

v P v



 

 

   

 _{ }  _{ }  

 

   

v v

v

P T

Tds c dT T dv c dT dv T

 



 

  _{ }  



 

P P

P

v

Tds c dT T dP c dT Tv dP

T 



 

  _{ }  



Internal Energy

8

v

T

u

du

c dT

dv

v









  







v T

u

u

du

dT

dv

T

v





































Thermo & Stat Mech - Spring 2006 Class 7

9

v

T

u

Tds

c dT

P dv

v



































Internal Energy

Tds



du



Pdv

v

T

u

du

c dT

dv

v









  



First Tds Equation 10 v v c s T T   _{ }  _    v T u

Tds c dT P dv

v _  _    _{ }  _      1 v T c u

ds dT P dv

T T v

_  _    _{ }  _      v T s s

ds dT dv

T v        _{ }  _{ }       1 T T s u P v T v

 

 

  _   _

First Tds Equation (cont.)

11

1

v

T

T _v

c u

P

v T T T v

 _{ } 

  

  _   _

 _{ } 

 _  _ _{  } 

  _{   }

v

v

c s

T T



 _{ }

 _ 

 

1

T T

s u

P

v T v

 

 

  _   _

 _   _     _  _

2 2

s s

v T T v

 



First Tds Equation (cont.) 12 2 1 1 v v T _T

c u u

v T v T T T v T

        _   _      _  _{   }     _{ } 2 2 2

1 1 1

v T

u u P u

P

T v T T T v T T v

      _  _{ }  _  _  _{  }  _{ }  _   _  _  _{  }  _{ } 2 2

1 1 1

T _v v T

u u P u

P P

T T v T T v T T v

 _{    } _{         }       _{    } _{         }               2 1 1 v T P u P

T T T v

 

 

  _   _

 _   _  

13

First Tds Equation (cont.)

v T

P u

T P

T v

 

 

  _   _

 _   _  

  _  _

2

1 1

v T

P u

P

T T T v

 

 

  _   _

 _   _  

  _  _

v

T

u

Tds c dT P dv

v

_  _    _{ }  _



 

 

v

v

P

Tds c dT T dv

T



 

 _{  }



The First Tds Equation (cont.)

14

v

v

P

Tds

c dT

T

dv

T









 





c_P – c_v

15

p v

T

u

c c P v

v 

_  _    _{ }  _



 

 

p v

T

u

c c P v

v 

_  _    _{ }  _



 

 

v

T

u

dq c dT P dv v

_  _ 

  _{ }  _



 

c_P – c_v

16

2

p v

Tv c c 



 

p v

v T P

P P v

c c T v Tv

T  v T 

  

        _{ }  _{   }

  

     

p v

T

u

c c P v

v 

_  _    _{ }  _

  

 

1 ,

T P

P v

v

v v T 

 

  _{ }   _  _   _ 

Enthalpy

17

P

T

h

Tds dh vdP c dT v dP

P

_{  } 

    _{ }  _



 

 

P

P T T

h h h

dh dT dP c dT dP

T P P

  

     

 _{ }  _{ }   _{ }

  

     

, so

h

 

u

Pv

2nd _{Tds Equation} 18 P P c s T T   _{ }  _    p T h

Tds c dT v dP

P _  _    _{ }  _      1 p T c _h

ds dT v dP

T T P

_  _    _{ }  _      P T s s

ds dT dP

T P        _{ }  _{ }       1 T T s h v P T P

 

 

2nd _{Tds Equation (cont.)}

19

1

P

T

T _P

c h

v

P T T T P

 _{ } 

  

  _   _

 _{ } 

 _  _ _{  } 

  _{   }

P

P

c s

T T



 _{ }

 _ 

 

1

T T

s h

v

P T P

 

 

  _   _

 _   _     _  _

2 2

s s

P T T P

 



2nd _{Tds Equation (cont.)} 20 P T v h T v T P       _{ }   _  _   _     _  _ 2 1 1 P P T _T

c h h

P T P T T T P T

        _   _      _  _{   }     _{ } 2 2

1 1 1

T _P P _T

h h v h

v v

T T P T T P T T p

   _{    } _{        }       _{    } _{     } _{  }      _{ }         2 1 1 P T v h v

T T T P

 

 

  _{ }   _

 _   _  

2nd _{Tds Equation (cont.)}

21

p

P

v

Tds

c dT

T

dP

T









 









P

T

h

Tds c dT v dP P

_{  } 

  _{ }  _



 

Third Tds Equation

v

v v

c

s T

P T P

 

  _    _   _     

22

Consider s  s P v( , )

v P

s s

ds dP dv

P v

 

     _{ }  _{ }

 

   

1

v v v v v

s s T T s T

P T P T T P

    

Third Tds Equation

23

P

P P

c

s T

v T v

 

  _  

 _   _ 

   

v P

s s

ds dP dv

P v

 

     _{ }  _{ }

 

   

1

P P P P p

s s T T s T

v T v T T v

    

Third Tds Equation

24

v P

v P

T T

Tds c dP c dv

P v

 

   

 _{ }  _{ }

 

   

v P

s s

ds dP dv

P v

 

   

 _{ }  _{ }

 

   

v P

v P

c T c T

ds dP dv

T P T v

 

   

 _{ }  _{ }

 

THE 3 Tds EQUATION

25

P v

P v

T T

Tds c dv c dP

v P

 

   

 _{ }  _{ }

 

   

v

v

P

Tds c dT T dv

T 

 

 _{  }



 

P

P

v

Tds c dT T dP

T



   _{  }



Joule Coefficient 

26

v

T

u c

v

   

 _{  }



 

v T

P u

T P

T v

 

 

  _   _

 _   _  

  _  _

v

v

P

c

T

P

T





















Joule Thomson Coefficient 

27

P

T

h c

P

   

 _{  }



 

P T

v h

T v

T P

 

 

  _{ }   _

 _   _  

  _  _

P

P

v

c

v T

T











  





Various thermodynamic relation from the TdS Eq

28

v

v

P

Tds c dT T dv

T 

 

 _{  }



 

,

v v

c s

T T



 _{ }

 _ 

 

T v

s P

v T

 

  _  

 _   _ 

29

Various thermodynamic relation from the TdS Eq cont…

P

P

v

Tds c dT T dP

T



   _{  }



 

,

P P

c s

T T



 _{ }

 _ 

 

T P

s V

P T

 

  _{ } 

 _   _ 

30

Various thermodynamic relation from the TdS Eq cont…

P v

P v

T T

Tds c dv c dP

v P

 

   

 _{ }  _{ }

 

   

P

P P

c

s T

v T v

 

  _  

 _   _ 

   

v

v v

c

s V

P T P

 

  _{ }  

 _   _ 