Chemical Forums
Chemistry Forums for Students => Physical Chemistry Forum => Topic started by: Donaldson Tan on May 22, 2006, 07:53:22 PM
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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...
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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