[試題] 102下 陳振中 物理化學三

[oray914 (oray)]

課程名稱︰物理化學三
課程性質︰必修
課程教師︰陳振中
開課學院：理學院
開課系所︰化學系
考試日期（年月日）︰103/4/18
考試時限（分鐘）：130min
是否需發放獎勵金：是
（如未明確表示，則不予發放）

試題 :
Section A (90%)
------------------------------------------------------------------------------
1. (20%) Given that the translational partition function is

   q =Σg exp(-ε/kT) and U=ΣN ε ,derive the equayion U=NkT^2(∂lnq /∂T).
    t  i i      i            i i i                                   t
                                                          _
2. The energy of a proton in amagnetic field is equal to -hνB/2 when the

   proton's magnetic dipole is parallel to the magnetic field, and hνB/2

   when antiparallel.

   (a) Derive the energy  E of N independent protons in a magnetic field B as

   a function of T.(20%)

   (b) Calculate the results in (a) when the temperature is absolute zero and

   infinity high.(10%)

   (c) Explain the results.(5%)

3. The ground state of Cl    is fourfold degenerate. The first excited state
                         (g)
   is 875.4 cm^-1 higher in energy and is twofold degenerate. What is the
                                                   o
   value of the electronic partition function at 25 C? At 1000 K?(15%)

4. The thermal wavelength is defined as Λ=(V/q )^(1/3), which is different
                                               t
   from the de Brogile wavelength λ=h/p.

   (a) What is the de Brogile wavelength for hydrogen atoms at 3000 K?(10%)

   (b) How does the result in (a) compare with the thermal wavelength at the

   same temperature (3.175×10^-11 m)?(15%)

   (c) How does the thermal wavelength compare with the mean distance between

   hydrogen atoms in a gas of hydrogen atoms at 3000 Kand 1 bar.(5%)


Section B (10%)
------------------------------------------------------------------------------
1. On the basis of what you have learned in statistical mechanics, discuss

   whether or not the following quantities depend on temperature. Give a brief

   explanation (one or two sentences) in your answer.

   (a) partion function

   (b) energy levels

   (c) average energy

   (d) molar heat capacity of helium gas

2. When a form of energy depends on a degree of freedom x, the average energy

   of that component is 〈ε〉=Σ    ε(x)P(x)
                                all x
   where P(x) is the probability that the system is found in x.

   (a) Rewrite 〈ε〉 in term of the explicit expression of P(x).

   (b) Suppose x can be positive or negative and the summation can be

   approximated by integration, rewite 〈ε〉 interms of integrals.

   (c) Square-law relations holdfor many types of degrees of freedom. For

   translations and rotations, the energy depends on the square of the

   appropriate quantum number:
    translation            rotation
   E           α n^2 and E        α L(L+1)
    n                      n
   That is, the energy is a square-law function: ε(x)=cx^2, where c>0.

   Calculate the corresponding result of 〈ε〉 and discuss its dependence

   on c.

   (d) When equipartition theorem applies, you need only count the number of

   degrees of freedom to compute the total average energy. Therefore you have

   energies of 3×(1/2)kT fortranslation in three dimensions and (1/2)kT for

   every rotational degree of freedom. Under what conditins does the

   equipartition theorem become invalid?

   (e) Calculate the fluctuation of x.


Constants & integration table
------------------------------------------------------------------------------
                                    2   -2 -1
Boltzmann constant k: 1.38×10^-23 m kgs  K
                                  2   -1
Planck constant h: 6.626×10^-34 m kgs
          2   -2
Joule J: m kgs

Speed of light c: 3×10^8 ms^-1

bar: 1×10^5 Pa

mass of a mole of hydrogen atoms: 1.0078 g
 ∞  2 -ax^2               3
∫  x e   dx = (1/2)√(π/a ) (a>0)
 -∞
 ∞  -ax^2
∫  e     dx = √(π/a) (a>0)
 -∞

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