适用于 2011 级本科物理学基地班 （2013-2014 学年度第一学期）
1. (20 points) Show that
C p ? CV ?
1 ?V )T and the thermal v ?p
where the isothermal compressibility ? is given by ? ? ? (
? is defined as ? ?
1 ?V ( )p V ?T
2. (20 points) The partition function of a hyponthemal system is given by
ln Z ? aT 4V
where a is a constant. Evaluate the mean energy E , the pressure P and the entropy S .
3. (15 points) A gas of melocules,each having mass m ,is at rest in thermal equilinrium at absolute temperature T .Denote the velocity of a melocule by v ,its three Cartesian components of velocity by vx , v y , v z ,and its speed by v .Find the following mean values: (a) vx , (b) v , (c) v vx , (d) vx v y , (e) (vx ? bv y ) , where b is a constant.
2 2 2
4. (15 points) The density of metal Li is 0.534 g/cm
and the standard atomic weight of Li is 6.94.
Calculate the Fermi energy ? F , the Fermi temperature TF and the Fermi pressure pF .
5. (15 points) Estimate of the Bose condensation temperature. (a) What is the approximate value of Tc for an ideal Bose gas at a density of ? ? 125 kg/m ,
the density of liquid
He ?Take m ? 6.65 ?10?27 kg .
(b) The value of Tc for a collection of
Rb (rubidium) atoms is about 280 nK (2.8 ?10?7 K) .
What is the mean separation between the atoms? 6. (15 points) The heat capacity Cn of a normal metal at a very low absolute temperature is of the form Cn ? ? T where ? is a constant characteristic of the metal. If such a metal is superconducting below a critical temperature Tc , then its heat capacity Cs in the
superconducting state in the temperature range 0 ? T ? Tc is approximately given by the relation CS ? ?T
? is some constant. No heat is absorbed or given off when a
metal is transformed from its normal to its superconducting state at the critical temperature Tc . Hence it follows that at this temperature Sn ? S s , where S n and S s denote the entropies of the metal in its normal and superconducting states, respectively. (a) What statements can you make about the entropies S n and S s in the limit as T ? 0 ? (b) Use the answer to part(a), and the connection between heat capacity and entropy, to find a relation between Cs and Cn at the critical temperature Tc .
Mathematical Formulas and Physical Constants
Electron mass me ? 9.11?10
Electron Charge e ? 1.6 ?10
Planck’s constant h ? 6.63 ?10
Boltzmann’s constant k ? 1.38 ?10
dx x n e? ax ?
n! a n ?1
1 ?( ) ? ? 2
lim ln N ! ? N ln N ? N
Fundamental thermodynamics relation
dU ? TdS ? pdV ? ?dN
Thermal de Broglie wavelength
? h2 ? 2 ? ?? ? ? 2? mkT ?