Statistical Physics: Solving Exam Questions and Using External Resources

[murat386]

Almost no one has answered.To take some external help, i write all questions here, maybe someone have an answer.

1)An aluminium cube cooled down to 90 Kelvin.At this point there is too many microstates available.how much heat we need, to increase these microstates in factor of 10$$^{10}$$ ? Answer in Joule unit.

2)In a simple October's forecast, days are tagged as rainy or shiny. Probability of a shiny day followed a shiny day is 0.8 .Probability of a rainy day followed by shiny air is 0.4 .Probability of 1st October to be shiny is 0.75
a)Find probability of 2nd and 3rd October be shiny.
b) If 1st October is shiny, what's probability is 3rd October to be rainy?

3)Two heat tanks, with constant volume and heat capacity, are used to power up a heat machine.Their startup temperature is T$$_{a}$$ and T$$_{b}$$.Assume that this heat machine gives as most possible product.(Net entropy change is zero).What is highest work that can be obtained from this machine?

4) A solid, has N count, 1/2 spinned atoms.In a enough temperature, these spins rotated randomly.But enough low temperatures, they are acted ferromagnetically.Due this reason, all spins are rotated same in T->0 K .In a rough given formula spin oriented addition to heat capacity is C(T)

C(T) = C$$_{1}$$ = ($$\frac{2T}{T_{1}}$$ - 1 ) if T$$_{1/2}$$ < T < T$$_{1}$$
( 1 and 1/2 are subscripts. Sown wrongly)
0 else

So take maximum probable of C$$_{1}$$ using entropy. (C subscript 1)

5) In a fermi gase,related to $$\epsilon$$$$_{f}$$ , state density D($$\epsilon$$$$_{f}$$) , show that heat capacity is written as

C$$_{v}$$ = $$\frac{\pi^{2}}{3}$$ D($$\epsilon$$$$_{f}$$) k$$^{2}$$ T

 

[LightHero]

6) An ideal gas has C_{v} = 3/2 k.If it is heated up to T, it's entropy will be S(T) .Calculate entropy in a lower temperature T_{1}, using this equation.

 

[piercas]

1) To increase the microstates by a factor of 10^10, we need to add an amount of heat equal to 10^10 times the initial heat content of the system. This can be calculated using the formula for the heat capacity of a solid, C = γNk, where γ is the specific heat capacity, N is the number of atoms, and k is the Boltzmann constant. Thus, the heat needed is 10^10γNk.

2) a) The probability of 2nd October being shiny is 0.8, as it follows a shiny day. The probability of 3rd October being shiny is 0.8*0.8 = 0.64, as it follows two consecutive shiny days.
b) If 1st October is shiny, the probability of 3rd October being rainy is 0.4, as it follows a shiny day and then a rainy day.

3) The highest work that can be obtained from the heat machine is given by the Carnot efficiency, which is equal to 1-Tb/Ta. This can be calculated using the formula for the efficiency of a heat engine, η = (T1-T2)/T1, where T1 is the hot reservoir temperature and T2 is the cold reservoir temperature.

4) The maximum probable value of C1 can be determined by maximizing the entropy of the system, which is given by S = k ln(Ω), where Ω is the number of microstates available to the system. In this case, Ω is equal to 2^N, as each spin can have two possible orientations. Thus, the maximum probable value of C1 is when all spins are aligned, which occurs at T = 0 K. Therefore, C1 = Nk.

5) The heat capacity of a fermi gas can be calculated using the formula C = γNk, where γ is the specific heat capacity, N is the number of particles, and k is the Boltzmann constant. In this case, the number of particles is given by the state density, N = D(εf)V, where V is the volume and D(εf) is the state density at the fermi energy. Thus, the heat capacity can be written as C = γD(εf)Vk. Using the relation between the fermi energy and temperature, εf = π^2kT