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Thermo%207%202009

# Thermo%207%202009 - Mat E 510 Thermodynamics of Solids Mat...

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Mat E 510 Thermodynamics of Solids Lecture #7 Yury Gogotsi A.J. Drexel Nanotechnology Institute and Department of Materials Science & Engineering, Drexel University, Philadelphia, Pennsylvania, USA Mat E 510 Thermodynamics of Solids

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Non regular Solutions S mix - R ( x A ln x A + x B ln x B ) occurs for many reasons. E.g. Change of Vibrational Entropy with x B . So non ideal entropy if ν A ν B ν S s stands for “solution” Use of the Einstein model of Solid: Partition function. Z = exp(- h ν /2 kT ) / {1-exp(- h ν / kT )} At high T , T >> h ν / k . Z -> {1 - (- h ν /2 kT ) } / {1 - (-h ν / kT )} -> kT / h ν + 0.5 But kT/ h ν Z -> kT/ h ν . S = 3 R ln Z + (3 RT /Z ) ( dZ/dT ) dZ/dT = k/h ν T/Z = h ν / k S = 3 R {ln( kT/h ν ) + 1} = 3 R {ln( kT / h ) + ln(1/ ν ) +1}
Applications Phase change pure component. H αβ > 0 S αβ = H αβ /T β > 0 S α b = S β - S α S = 3R{ln(kT/h) + ln(1/ ν ) +1} S αβ = S β - S α = 3R{ln(1/ ν β ) - ln(1/ ν a )} S αβ = S β - S α = 3R ln( ν a / ν β ) > 0 n α > n β So h ν α > h ν β H αβ = H β - H α > 0, H β > H α Strong bonding Weaker bonding Widely spaced levels Closely spaced levels Sn diamond cubic Sn Metallic fcc Na bcc Na

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Gibbs Phase Rule Imagine P' phases, and C' components. H 2 O + NaCl C'=2 P'=4 - Ice, liquid, vapor, salt Each phase: a, b, g… will have atomic fraction of each element: Fe + C C' = 2, P' = 4 - bcc, fcc, Fe 3 C, graphite x A (a), x B (a),x C (a)… C' variables x A (b), x B (b), x C (b)… P' phases + P a , T a for each phase Fe + C + Mn C' = 3 # of variables = P' (C' +2) = P'C' + 2P' For each phase. P
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Thermo%207%202009 - Mat E 510 Thermodynamics of Solids Mat...

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