(d) Suppose we define: SE = E – < E > Show that: < (8E)² > kT² (e) Show that: < E² > - < E >²= -d < E > dB
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- The probability of a system being in the ith mi- crostate is P₁ = e/Z, (14.56) where E is the energy of the ith microstate and ẞ and Z are constants. Show that the entropy is given by S/kB = In Z+BU, where U=PE is the internal energy. (14.57)wouldn't it be -7.5J, if you consider ground as 0 and anythng higher is postive, the jar losses energy right?The energy generation rate for the CNO thermonuclear cycle is given by the following propor- tionality relation 1/37 exp[-152 (109)]. T ECNO X T-2/3 exp-152 If the energy generation rate is approximated by a power law as e x Tº at a given temperature T, then a is given by a = din. Hence show that ECNO x T¹8 at T = 2.0 × 107 K. In T
- For a dilute gas of N monatomic particles with mass m and total energy E, use the Sackur- Tetrode equation for the entropy S V = log + NkB to derive expressions for the pressure and internal energy in terms of the temperature T and volume V. [You may use that X₁ = 3πh² N/(mE).] thAccording to the Debye equation the heatthe capacity of a solid is proportional to the temperaturecubed at low temperatures, Cp(T) = σT3, where σis a constant. Show that the entropy at a temperatureT0 is given by S(T0) = Cp(T0)/3 when the Debyeequation holds.Consider a system containing N distinct, stationary and non-interacting particles. Each atom has only two energy levels 0 and ξ>0. If the average energy per atom E/N for N is close to 0 determine (a). Sum of microstates (b) entropy per atom