12. For a given macrostate of a system, let P, be the probability that the system is in the microstate >. The corresponding Gibbs entropy is given by 5-k En PinPn. We can obtain the familiar Boltzmann entropy S = k In from the given Gibbs entropy if the probability distribution is, ( is the number of accessible microstates), A) P₁₂ = PR E = 1/₂hw NEF ==WEE B) P₁ = e-a C) P₁ = e D) P₁ = E) P₁ = 2² 13. For a system with linear dispersion E(k)= hvk, in three dimensions, the density of states at energy E depends on energy as -* (en (k) E(K) = hvk + 1 h nk + 1 huk A) E B) E2 C) E D) E E) Independent of energy E of an ideal بدارية بمرايت ~/M + + S = -K EnPnlupn. S = klnn rim is confined to half of a boy s 12 lukk moved and Joule (free)

a box of mass 10.0 kg and intaial velocity 3.0 m/s slides along a floor and comes to a rest due to friction. estimate the total cahnge in entropy of the universe. assume everthing occures at room tempeture of 293 k

Calculate the efficiency of the reversible cycle in figure for an ideal monoatomic gas where the transformation from 1->2 is described by the equation P2 V = A= constant (Assume V2 = 4 V1). P 3 1 V1 2 V2 V Hint: You might need to use the following integral: = √ + GA little bit of a problem can be found for Q1-2 = AU + W where the integral above may result useful to know.

Calculate the number of microstates that are available in a single atom of carbon in graphite.The standard molar entropy (S°) of carbon is 5.7 J/(mol · K) and the Boltzmann constant (kB) is1.381 × 10 ―23?/?.   HELP PLEASE