[試題] 100-2 陳義裕 統計物理導論

[harveyhs (Hango)]

課程名稱︰統計物理導論
課程性質︰系必修
課程教師︰陳義裕
開課學院：理學院
開課系所︰物理系
考試日期（年月日）︰April 11, 2012.
考試時限（分鐘）： 180 mins
是否需發放獎勵金：是
（如未明確表示，則不予發放）

試題 :

1. (This is from Carter's textbook.) A container is initially separated by

   a diathermal wall into two compartments of equal volume. The left com-

   -parment is filled with 1 kilomole of neon gas at a pressure of 4 atm

   and the right with argon gas at 1 atm. The gases may be considered ideal.

   The whole system is initially at temperature T=300K, and is thermally

   insulated from the outside world. Suppose that the diathermal wall is

   removed.

   (a) (4 points) What is the new temperature of the system? The new pressure?

   (b) (4 points) What is the change in the Gibbs function of the system?

   (c) (8 points) What is the change in the entropy of the system?


2. (a) (5 points) Please describe in detail the third law of thermodynamics.

   (b) (5 points) Please argue why the third law of thermodynamics implies

                  that it is impossible to reach absolute zero.

   (c) (5 points) Please argue why the third law of thermodynamics implies

                  that the specific heat of a material tends to zero as T

                  approaches 0 K.

   (d) (5 points) Please argue why the third law of thermodynamics implies

                  that the thermal expansivity of a material tends to zero

                  as T approaches 0 K.


3. Suppose the equation of state of a gas satisfies

                               (P+f(v))g(v)=RT,

   where f(v) and g(v) are some differentiable functions of the molar volume

   v of the gas.

   (a) (6 points) Use a suitable Maxwell's relation to find (ðS/ðv)|
                                                                      T
   (b) (6 points) Find the most general form of the entropy S in terms of v

                  and T.

   (c) (6 points) Setting T=0, argue that the value of S still depends on v

                  so that the third law of thermodynamics does not hold for

                  this system.

   (d) (6 points) Does van der Waals gas obey the third law of thermodynamics?

                  Please explain.


4. The coefficient of viscosity η, the thermal conductivity κ, and the

   diffusion constant D for a gas are defined by


                          shear stress = η(ðu_x/ðz)

                             heat flux = κ(ðT/ðz)

                         particle flux =  D (ðρ/ðz),

   where we have assumed that all macroscopic quantities involved, such as the

   x-component u_x of the bulk velocity of the gas, the temperature T, and the

   number density ρ only vary in the direction of z.

   (a) (15 points) Please use the kinetic theory of gases to argue why, in

                   terms of order of magnitude,
                                          _
                                    η ~ mv/σ,                   (1)
                                               _
                                    κ ~ (c_v)*v/σ,              (2)
                                         _
                                    D  ~ v/ρσ.                  (3)
                                           _
                   In the above , m, σand v denotes the mass, the cross sec-

                   -tion, and the mean speed of a gas particle, respectively,

                   whereas c_v denotes the isochoric specific heat per particle.

   (b) (5 points) In the expressions above we see that the tranport coefficient

                  all vary with 1/σ, which means that a smaller particle

                  actually has a larger transport property, everything else

                  assumed equal! Please explain in physical terms why this

                  is to be expected.

   (c) (5 points) A vacuum flask (or commonly called a thermos bottle) can

                  hold hot or cold food in the bottle for an extended period

                  of time. The flask has an inner bottle to fold the food,

                  and it is separated from the outer bottle by a thin space

                  in partial vacuum. The pressure inside this space may be

                  of order 10^(-6) Pa. It is usually said that the flask

                  insulates heat so well because of the presence of this

                  vacuum space. However Eqn. (2) above says that the thermal

                  conductivity actually is independent of the gas density!

                  So why does the flask work so well?


5. This problem concerns the phenomenon of effusion.

   (a) (5 points) (This problem can take up quite some time! So be careful

                  with your time management.) Let b be a positive contant

                  (=m/2kT) and n≧0. Define


                                          ∫v^(n+2) exp(-bv^2)dv
                                    g_n=－－－－－－－－－－－－－－.
                                           ∫ v^n exp(-bv^2)dv

                  (v from 0 to infinity) Please show that g_3>g_2.

   (b) (5 points) (You could still answer this question even if you did not

                  successfully finish part (a).) Use the result of (a) to

                  argue why, in effusion, the particles coming out of the tiny

                  hole of the chamber containing a gas at the temperature T

                  have an average kinetic energy that is greater than that

                  for the particles inside the chamber.

   (c) (5 points) Please provide a physical explanation of why the effusion

                  can be used to separate isotopes.





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