Statistical vs. Thermodynamic Definition of Entropy - Edubirdie

University:
The Pennsylvania State University
Course:
CHEM 450 | Thermodynamics
Academic year:

2024
Views:

237

Pages:

2
Author:

Alexis H.

Statistical vs. Thermodynamic definition of entropy S = kB ln W dS = Statistical Ludwig Boltzmann 1844-1906 dqrev T Thermodynamic Sadi Carnot 1796-1832 Entropy is an extensive property and a state function Use statistical to determine probability: Y let Probability that 1 molecule is on the left: P rob = 1 2 Probability that 2 molecules are on the left: V2 P rob = 1 1 1 £ = 2 2 4 Entropy as an extensive property Each molecule is either on the left or right: w=2 for each N For N molecules: S1 = kB ln 2 = kBN ln 2 For 2N molecules: S2 = kB ln 22N = 2kBN ln 2 Twice the molecules increase entropy by a factor of 2: extensive property Entropy of expansion (isothermal) Si = kB ln (V1) Sf = kB ln (V2) N Assume V(1) and V(2) are proportional to the N N ¢S = Sf ¡ Si = kB ln (V2) ¡ kB ln (V1) µ ¶N V2 = kB ln V1 = kBN ln = nR ln ¢Sm = R ln V2 V1 N kBN = nR V2 V1 V2 V1 Isothermal, reversible expansion of a perfect gas to get delta S from thermodynamic definition ¢S = Z isothermal: f i dqrev 1 = T T Z f dqrev i ¢U = 0 = q + W W = ¡q qrev = T Wrev = ¡ =¡ Z Z f pexdV i f i = ¡nRT nRT dV V Z f dV = ¡nRT ln i Vf Vi Delta S for surroundings qrev = nRT ln nRT ln ¢S = Vf Vi T ¢Sm = R ln Vf Vi Vf = nR ln Vi Vf Vi If the surroundings is an infinite heat bath: dSsurr = dqsurr;rev Tsurr dSsurr = dqsurr Tsurr ¢Ssurr = qsurr Tsurr For an adiabatic process: ¢Ssurr = 0 Tsurr =0 ¢U = q + W ¢Usurr = qsurr + Wsurr For constant volume conditions: ¢Usurr = qsurr + Wsurrno

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