Quantitative relationship between entropy and number of microstates The connecti

Question

Quantitative relationship between entropy and number of microstates The connection between the number of microstates,W, and the entropy, S, of a system is expressed by the Boltzmann equation S = klnW where k is the Boltzmann constant per molecule (particle), 1.38 times 10^- 23 J/(K particle). From this equation the entropy change, deltaS, for a system can be related to the change in the number of microstates as deltas = kln W_final/W_initial where W_final in the final number of microstates and initial W_initial is the initial number of microstates. Part B A gaseous system undergoes a change in temperature and volume. What is the entropy change for a particle in this system if the final number of microstates is 0.563 times that of the initial number of microstates? Express your answer numerically in joules per kelvin per particle. For the decomposition of calcium carbonate, consider the following thermodynamic data (Due to variations in thermodynamic values for different sources, be sure to use the given values in calculating your answer ): Calculate the temperature in kelvins above which this reaction is spontaneous. Express your answer to four significant figures and include the appropriate units. Relationship between free energy and the equilibrium constant The standard free energy change, deltaG degree, and the equilibrium constant K for a reaction can be related by the following equation: delta G degree = -RT ln K where T is the Kelvin temperature and R is equal to 8.314 J/(mol middot K)

Explanation / Answer

Part-A)

Reaction is spontaneous when Standard Gibbs free energy change for the given transformation is –ve.

i.e. for spontaneous reaction, 0G = -ve or 0G <0

G0, H0, S0 and T related by equation,

G0 =H0 - TS0

For spontaneous processes,

0G <0

i.e. H0 - TS0 < 0

H0 < TS0

(H0 / S0) < T

Let us put, H0 = 178.5 kJ/mol = 178500 J/mol

S0 = 161.0 J/molK

178500 / 161 < T

1108.7 < T

i.e. T > 1108.7 K

T > (1108.7 – 273.15) 0C

T > 835.6 0C

Try entering answer as >835.6 0C (use “greater than” sign if possible)

Or T = 835.7 0C

Any one of this should certainly work.

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Part-B)

Given data,

Final number of microstates (Wfinal) = 0.563 x Initila number of microstates (Winitial)

i.e. Wfinal = 0.563 x Winitial

i.e. Wfinal / Winitial = 0.563………….(1)

and Boltzmann constant k = 1.38 x 10-23 J / (K.particle)

S = kln(Wfinal / Winitial)

S = (1.38 x 10-23) ln(0.563)

S = - 7.928 x 10-24 J/K.particle)

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