ChE 122 Problem 3.1

 

Expres Expr esss the the volum olume e exp xpan ansi sivi vity ty an and d th the e isot isothe herm rmal al comp comprres essi sibi bili lity ty as fu func ncti tion onss of de dens nsit ity y  ρ and its partial derivatives. For water at 50⁰C and 1 bar,   ƙ  = 44.18x1 x10 0-6 bar-1 . To what pressu ssure must water be comp mprres esse sed d at 50⁰C to change its densit sity by 1%? Assume that   ƙ  is independ inde pendent ent of   P.

Volume olume exp expans ansivi ivity ty consta constant: nt:     β =  ( ) Therma The rmall compre compressib ssibili ility ty constan constant: t:  

ƙ=   ()

 

ρ=

 

since m = constant, ρ =

 

Substitute Subst itute in into to volum volume e expansivi expansivity ty equati equation: on:  

β= (

 

)

  

= (    ) (

  ) 

= (    )

β =

− 

  ( )





 

ρ=

 

since m = constant, ρ =

 

Substitute into thermal compressibility equation:  

ƙ=   ()  = -(

= -( ƙ=



  



  )

 



 

( )

  

  )

 

For water at 50⁰C and 1 bar, ƙ  = 44.18x10-6 bar -1 From this equation,  

ƙ= (

) @ constant T: ƙdP =

 

  

so, 

  

  ƙ =  



ƙ(    ) = ln ln   ln 

     = (

   

ƙ

 ) change of density is 1%,   = 1.01

  1 = (

 

 . . 

44.18x10-6

  = 226.22 bar 

 )