Total: 150%
1. Please write down the three laws of thermodynamics with brief interpretation. (15%)
2. Please prove the relation, pV ^γ= a constant, during an adiabatic process of an ideal gas, where γ= Cp / Cv（p, v 下標）, the ratio of the molar specific heats for the gas. (5%)
3. A solid cylinder is attached to a horizontal massless spring so that it can roll without slipping along a horizontal surface (Fig.1) The spring constant k is 3.0 N/m. If the system is released from rest at a position in which the spring is stretched by 0.25m, find (a) the translational kinetic energy and (b) the rotational kinetic energy of the cylinder as it passes through the equilibrium position. (c) Show that under these conditions the center of mass of the cylinder executes simple harmonic
motion. What is its period? (15%)
Fig.1
| k M ←╮（逆時鐘方向）
|~~~~⊙
4. (a) Please derive the entropy change: ΔS = S_f – S_i
= n R ln( V_f / V_i ) + n Cv ln( T_f / T_i ) for all reversible process that take the gas from state i to state f. (10%)
(b) Please use this relation to calculate the change in the entropy for a free expansion process from V to 2V. Please also give the reason that you may do in this way. (10%)
(c) Derive this increase of entropy with statistical mechanics (using the Boltzmann's entropy equation S = k ln W, where k is the Boltzmann's constant, W the multiplicity of the configuration). (10%)
(You can use the Stirling's Approximation: ln N! = N ln N - N .)
5. Please express (in string tension τ and mass length density μ) and derive the equation for the wave speed v on stretched string from Newton's second law. (10%)
6. An apparatus that liquifies helium is in a room maintained at 300K. If the helium in the apparatus is at 4.0K, what is the minimum ratio of the heat delivered to the room to heat removed from the helium? (15%)
7. Please construct the plots of P versus V, T versus S, and S versus E_internal（internal 下標）for the isothermal expansion and isobaric expansion thermodynamic process. (15%)
8. One mole of an ideal gas are expanded from V_1 to V_2 = 3 V_1. If the expansion is isothermal at temperature 300K, find (a) the work done by the expanding gas and (b) the change in its entropy. (c) If the expansion is reversibly adiabatic instead of isothermal, what is the change of entropy of the gas? (15%)
9. What is the entropy change (a) for any reversible CLOSED cycle and (b) for the irreversible process whose final T and V are the same as the initial ones. (20%)
10. (10%) Consider a damped simple harmonic motion with the total force
ΣF = - kx - bv, where k is force constant of the spring, x the displacement, v the velocity, b the damping constant. Please write down and solve the (differential) equation of motion from the Newton's second law.
11. Good luck!!! and Happy NEW YEAR!
他真的很喜歡出推導公式題
更重要的是..他喜歡出考古題...
--
※ 發信站: 批踢踢實業坊(ptt.cc)
◆ From: 140.112.7.59