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### Topic: ?S mixing for Perfect Gas Mixture  (Read 4353 times)

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#### Donaldson Tan ##### ?S mixing for Perfect Gas Mixture
« on: May 22, 2006, 07:53:22 PM »
I am stuck at trying to derive the expression for ?Smixing

For a perfect gas mixture, the chemical potential of component i is given as:
?i,mixture = ?i,REF + RT.ln(Pi/PREF)

from G = H - TS, we arrive at dG = dH - TdS - SdT
since dH = TdS + V.dP, then dG = VdP - SdT

At constant pressure, dP = 0 => dG = -SdT => S = -(dG/dT)P

Hence, si,mixture = -(d?i/dT)P

si,mixture = -(d?/dT)P = -d(?i,REF)/dT - (d/dT)(RT.ln(Pi/PREF)) = -d(?REF)/dT - (d/dT)(RT.ln(niRT/PREFV))

si,mixture = si,REF - R.ln(niRT/PREFV) - RT/T = si,REF - R.ln(niRT/PREFV) - R

si,mixture = si,REF - R( ln(niRT/PREFV) + 1 ) =si,REF - R( ln(Pi/PREF) + 1 )

Then I got stuck here...
« Last Edit: May 22, 2006, 10:17:04 PM by geodome »
"Say you're in a [chemical] plant and there's a snake on the floor. What are you going to do? Call a consultant? Get a meeting together to talk about which color is the snake? Employees should do one thing: walk over there and you step on the friggin� snake." - Jean-Pierre Garnier, CEO of Glaxosmithkline, June 2006

#### Donaldson Tan ##### Re: ?S mixing for Perfect Gas Mixture
« Reply #1 on: May 22, 2006, 10:23:34 PM »
Think I figure it out already...

Prior to mixture, the chemical potential of each compound is given as

?i,initial = ?i,REF + RT.ln(P/PREF)

si,initial = -(d?i,initial/dT)P = si,REF - R( ln(P/PREF) + 1 )

?Smixing = sum{ nisi,mixture } - sum{nisi,initial} = sum{ni.(si,mixture - si,initial)} = sum{ -ni.R.ln(Pi/P) }

?Smixing =  sum{ -ni.R.ln(Pi/P) } = sum{ -niR.ln(xi) }

since 0 < xi < 1, then ln (xi) < 0
?Smixing > 0 for all possible values of xi

« Last Edit: May 22, 2006, 10:29:44 PM by geodome »
"Say you're in a [chemical] plant and there's a snake on the floor. What are you going to do? Call a consultant? Get a meeting together to talk about which color is the snake? Employees should do one thing: walk over there and you step on the friggin� snake." - Jean-Pierre Garnier, CEO of Glaxosmithkline, June 2006